The Structure of
Comp osition Algebras
Die Struktur von Komp ositionsalgebren
Dissertation
zur Erlangung des akademischen Grades
eines Doktors der Technischen Wissenschaften
vorgelegt von
Dipl Ing Erhard Aichinger
Angefertigt am Institut f ur Algebra Sto chastik und
wissensbasierte mathematische Systeme
der technisch naturwissenschaftlichen Fakult at
der Johannes Kepler Universit at Linz
b ei
O Univ Prof Dr G unter Pilz
Gef ordert durch ein Doktorandenstip endium der
osterreichischen Akademie der Wissenschaften
Linz im Mai
Begutachter
Prof Dr G unter Pilz
Institut f ur Algebra Sto chastik und wissenbasierte mathematische Systeme
Johannes Kepler Universit at Linz Osterreich
Prof Dr Kalle Kaarli
Institut f ur Mathematik
Universit at Tartu Estland
Vorwort
Als ich im Mai mit der Arb eit an meiner Dissertation b egann wollte ich
Komp ositionsringe untersuchen Komp ositionsringe entstehen wenn man die
Menge aller Funktionen auf einem Ring mit den Op erationen der punktweisen Ad
dition und Multiplikation sowie mit der Op eration der Hintereinanderausf uhrung
von Funktionen versieht
Ein Anliegen der Algebraiker ist es interessante und erstaunliche Eigenschaften
einer Klasse von Algebren herzuleiten Verfolgen wir diese Arb eit am Beispiel der
Fastk orper Fastk orper sind zun achst durch recht steril klingende Bedingungen
de niert Sie sind jene algebraische Strukturen die b ei der Ko ordinatisierung b e
stimmter pro jektiver Eb enen auftreten Es ist nun H Zassenhaus gelungen alle
endlichen Fastk orper zu b eschreiben man erh alt diese sch onen Strukturen durch
Verdrehen der K orpermultiplikation eines endlichen K orpers bis auf sieb en
Ausnahmefastk orper die Zassenhaus explizit angeb en konnte
Komp ositionsringe hab en mit Fastringen gemeinsam da sie als Algebren von
Funktionen auf einer anderen Algebra entstehen Daher ho te ich den Komp osi
tionsringen mit den Werkzeugen der Fastringtheorie b esonders mit den Dichte
s atzen erfolgreich zu Leib e r ucken Nat urlich war schon einiges uber Komp osi
tionsringe b ekannt Man wu te da jeder Komp ositionsring als Komp ositions
ring von Funktionen auf einem Ring aufgefa t werden kann was Cayleys Satz in
der Grupp entheorie entspricht I Adler hatte im Jahr alle Komp ositions
ringe von Funktionen auf einem K orper b estimmt die zumindest jede konstante
Funktion enthalten Die Dichtes atze in der Hand glaubte ich einige weitere Fra
gen kl aren zu k onnen Darunter war zum Beispiel die Bestimmung aller einfachen
Komp ositionsringe Was mir dazu eingefallen ist erf ahrt der Leser im f unften
Kapitel dieser Dissertation
Gleichzeitig interessierte ich mich f ur ein anderes Problem Wie lassen sich Poly
nomfunktionen auf Grupp en und auf universellen Algebren b eschreiben Welche
Funktionen lassen sich von Polynomfunktionen an einer gewissen Anzahl von
Stellen interpolieren Eine Bedingung die eine Polynomfunktion erf ullen mu
ist da sie Kongruenzen erh alt Also ist es nat urlich zu fragen wann jede
kongruenzerhaltende Funktion eine Polynomfunktion ist Algebren b ei denen
das der Fall ist nennen wir a n vol lst andig Herrn K Kaarli verdanke ich den
VORWORT
Hinweis da eines der b edeutendsten Resultate zu diesem Thema in einer Ar
b eit von J Hagemann und Chr Herrmann enthalten ist Da deren Arb eit ab er
Konzepte verwendet die erst sp ater gr undlich studiert wurden so zum Beispiel
die Kommutatortheorie f ur universelle Algebren lassen sich die Resultate heute
eindringlicher als zur Zeit ihrer Entstehung formulieren Zum anderen sind mir
die Beweise in dieser Arb eit bis heute ein R atsel Da also kein Weg an Hagemanns
und Herrmanns Resultaten vorbeif uhrte sah ich mich gezwungen sie erneut zu
b eweisen Dazu hab e ich die Beweise da b estimmte Ringe und Grupp en a n
vollst andig sind in die Sprache der universellen Algebra ubersetzt Dort wo diese
Ub ertragung den Inhalt verschleiert hat hab e ich auch die Beweise angegeb en
von denen ich ausgegangen bin Ich hab e meine Einsicht in diese Resultate durch
die universelle Sichtweise ab er do ch wesentlich verbessern k onnen Im letzten
Kapitel dieser Arb eit hab e ich die Resultate von Hagemann und Herrmann uber
a n vollst andige Algebren no ch einmal kurz zusammengestellt
Eigentlich bin ich zum Kommutator ab er ganz anders als durch die universelle Al
gebra gekommen Was man b ei der Interpolation einer Funktion auf einem K orper
mithilfe der Multiplikation scha t geht in der universellen Algebra mithilfe des
Kommutators Multiplikationen auf Algebren die eine K orpermultiplikation im
itieren wurden schon von H K Kaiser verwendet Den Dichtesatz f ur Fastringe
kann man recht anschaulich b eweisen wenn man Funktionen in passender Weise
multipliziert Auch S D Scott multiplizierte allerdings nicht Elemente o der
Funktionen sondern Ideale in Grupp en In dieser Arb eit erkl are ich wie sich
diese Multiplikationen auf den Kommutator wie er in der universellen Algebra
studiert wird zur uckf uhren lassen
G Betsch S V Polin D Ramakotaiah und H Wielandt hab en eine sehr system
atischen Strukturtheorie der Fastringe gescha en die sich auf Interpolationsre
sultaten aufbauen l a t Welche Auswirkungen k onnen also die Interpolationsaus
sagen der universellen Algebra auf die Strukturtheorie von Komp ositionsringen
o der uberhaupt auf andere Algebren die den Fastringen ahnlich sind hab en
Da ich Verallgemeinerungen dieser Interpolationsresultate zur Verf ugung hatte
konnte ich viele Resultate von Fastringen auf rechtsdistributive universellen Al
gebren die dann Komp ositionsalgebren hei en ubersetzen Die wesentliche
Botschaft dieser Arb eit ist da sich viel aus der Fastringtheorie auf die rechts
distributiven Algebren hin uberretten l a t deren Addition sich zumindest in einer
Hinsicht wie eine Grupp enop eration verh alt die Kongruenzen der additiven
Struktur dieser Algebra m ussen b ez uglich des Relationspro dukts vertauschbar
sein Das ist schon b ei Lo ops das sind nichtassoziative Grupp en immer der
Fall Diese Verallgemeinerungen sind ab er nicht nur f ur jene Algebren frucht
bar deren additive Struktur armer als die der Fastringe ist Was sie bringen
kommt auch dann heraus wenn man die universell algebraischen Resultate auf
Komp ositionsringe anwendet
VORWORT
W ahrend der Entwicklungen der theoretischen Resultate hab e ich immer wieder
einzelne Fragestellungen am Computer untersucht Die brauchbaren Programme
stehen im Paket SONATA f ur das Grupp entheoriesystem GAP und sind daher
nicht in dieser Arb eit enthalten
Ich danke G unter Pilz f ur die Betreuung und der osterreichischen Akademie der
Wissenschaften f ur die Finanzierung dieser Arb eit eb enso danke ich Kalle Kaarli
und Pawe l Idziak f ur hilfreiche Anleitungen und meinen Kollegen Franz Binder
Tim Boykett J urgen Ecker und Christof N obauer f ur zahlreiche Diskussionen
und Kommentare Den Dank f ur private Unterst utzung m ochte ich p ers onlich
abstatten lediglich meinen Eltern Gertraud und Bruno Aichinger sei die Freude
nicht verwehrt hier namentlich erw ahnt zu werden
Linz im Mai
VORWORT
Preface
When I started to work on this thesis four years ago my idea was to investi
gate comp osition rings Those arise if one provides the set of all functions on a
ring with the op erations of p ointwise addition and multiplication and with the
op eration of functional comp osition
An algebraist s goal is to detect interesting and surprising prop erties of a class of
algebras Near elds provide a go o d example for the algebraists work reading
the way they are de ned they might seem as yet another structure of the quasi
infra semi hemi para skew near type On a closer view one nds out that
near elds naturally arise in the co ordinatization of certain pro jective planes It
was H Zassenhaus who succeeded in describing all nite near elds these b eau
tiful structures can b e constructed from nite elds by doing some harm to the
multiplication except for seven sp oradic near elds that Zassenhaus managed
to describ e explicitly
Comp osition rings and near rings have in common that they are algebras of func
tions on another algebra This suggested to me that the to ols of near ring theory
esp ecially the density theorems might prove useful also for comp osition rings
What was known ab out comp osition rings It was known that every comp osi
tion ring can b e embedded into a comp osition ring of functions on a ring This
corresp onds to Cayley s result in group theory In I Adler determined all
comp osition rings of functions on a eld that contain all constant functions Us
ing the density theorems I hop ed to settle some other questions Among these
is the determination of all simple comp osition rings What I have b een able to
do in this direction can b e found in the fth chapter of this thesis
At the same time I was interested in the following type of questions How can one
describ e and distinguish p olynomial functions on groups and universal algebras
Which functions can b e interpolated by a p olynomial function at a xed number
of places We know that every p olynomial function preserves the congruences
of an algebra So we may ask on which algebras every congruence preserving
function is p olynomial Algebras where this is the case have b een called a ne
complete K Kaarli p ointed out to me that J Hagemann and Chr Herrmann have
proved a central result on this topic Since they have used concepts that were
to b e studied more thouroghly only later such as the theory of commutators
PREFACE
in universal algebra their results can b e formulated more easily to day On the
other hand Hagemann s and Herrmann s pro ofs are di cult to follow Given the
imp ortance of these results I have provided new pro ofs for them which I have
obtained by translating the pro ofs that certain classes of groups are a ne com
plete into the language of universal algebra What we thereby gain in generality
is sometimes outweighed by added notational complications so at some places
I have included the original pro ofs However to me the universal algebraic
viewp oint is often the clearest one
In this thesis the commutator plays an imp ortant role The rst place where
I met commutator theory however was not universal algebra if one tries to
interpolate functions then what one can do in elds using the eld multiplica
tion can often b e done in universal algebras using commutators Multiplications
that emulate eld multiplications have already b een studied by H K Kaiser The
density theorem for near rings can b e proved in an intuitive way by introducing a
suitable way of multiplying functions And also S D Scott liked to multiply but
not elements in an algebras or functions but rather ideals in groups In the
present thesis I explain what these multiplications have to do with commutators
in universal algebra
G Betsch S V Polin D Ramakotaiah and H Wielandt have developed a very sys
tematic structure theory for near rings that can b e based on interpolation results
So it seemed likely that the interpolation results in universal algebra might have
impacts on the structure of comp osition rings or other similar universal algebras
And really many results for near rings can b e translated to right distributive
universal algebras which are called comp osition algebras One of the central
messages of the present thesis is that many results for near rings still hold for
structures whose addition b ehaves like a group op eration at least in one re
sp ect the congruences of the additive structure must commute with resp ect to
the relation pro duct This is already the case for lo ops these are non asso ciative
groups These generalizations are not only fertile for those algebras that have
less structure than near rings What they are go o d for also comes out if one
applies the universal algebraic results to comp osition rings
During the theoretical part of this research I have computed lots of examples on
the computer The usable functions arising from this enterprise can b e found in
the package SONATA for the group theory system GAP and are therefore not
contained in this thesis
I want to thank G unter Pilz for sup ervising and the Austrian Academy of Science
for nancing the work on this thesis Kalle Kaarli and Pawe l Idziak have provided
helpful suggestions and Franz Binder Tim Boykett J urgen Ecker and Christof
N obauer have sp ent a lot of time discussing this material with me Also p ersonally
I owe a great deal to several p ersons but the reader will allow me to thank them
PREFACE
p ersonally at this place I just want to thank my parents Gertraud and Bruno
Aichinger for their supp ort
Linz May
PREFACE
Contents
Vorwort
Preface
Chapter Prerequisites
Algebras and Varieties
Congruences and Mal cev conditions
Commutators
Ab elian algebras
Neutral algebras
groups
Various concepts of commutator op erations for groups
Ab elian groups
Chapter An interpolation result for function algebras
Function algebras
The interpolation result
Chapter Comp osition algebras
Adding comp osition
Lo cal interpolation algebras
Mo dules of comp osition algebras
The structure of comp osition algebras with constants
The structure of some comp osition algebras with a left identity
Chapter Tame comp osition algebras on groups
CONTENTS
Tame near rings
Chapter Comp osition Rings
The de nition of comp osition rings
Simple Zero Symmetric Comp osition Rings
Comp osition rings with constants
Chapter On Hagemann s and Herrmann s characterization of strictly
a ne complete algebras
Varieties generated by algebras that have only neutral subalgebras
Strictly a ne complete algebras
A ne complete algebras
Some consequences for p olynomial interpolation
Bibliography
Index
Tabellarischer Leb enslauf
CHAPTER
Prerequisites
In the present chapter we give a brief rep etition of the algebraic and notational
prerequisites that we shall need in the sequel We write N for the set of natural
numbers f g N for N f g and Z for the set of integers A set whose
cardinality is nite or countably in nite will b e called countable
Algebras and Varieties
We use the notation of Ihr which is very similar to the notation used in stan
dard references on universal algebra such as BS and MMT A short and
concise introduction to the notions of universal algebra is also given in HM
n
pp An n ary operation on a set A is a function from A to A A type of
algebras is a pair F where F is a set and a function from F to N The
elements of F are called operation symbols for f F the number f denotes
the arity of f The element f is then called a f ary operation symbol The
set F ff F j f ng is called the set of al l n ary operation symbols
n
An algebra of type F is a pair A A F where A is a non empty set
and F f j f F is a family of op erations on A where an op eration f
A A
f
A A is assigned to each f F The set A is called the universe of A
and the elements of F are called fundamental operations of A For the algebra
A A F with F f f we will also write A f f if it is clear
how the op erations f f are assigned to the op eration symbols in F A
subset B of A is called subuniverse of A i it is closed under all fundamental
op erations of A The algebra B is called a subalgebra of A i A and B are of
the same type the universe B of B is a subuniverse of A and each fundamental
op eration of B is the restriction of the corresp onding fundamental op eration of
A Sometimes we want to forget ab out some op erations of the algebra A If A
is an algebra of type F and B is an algebra of type F then B is a
reduct of A i B A F F and and for all op eration symbols in F we
have If B is a reduct of A then A is called an expansion of B
B A
We will now de ne three notions that will b e imp ortant throughout this the
sis These are terms term functions and polynomial functions The de nitions
of these notions can also b e found in MMT De nitions and
PREREQUISITES
and Ihr De nitions and We adopt BS De
nitions and Let F b e a type and let X b e a set In this
context we call the elements of X variables Then the set T X of terms of type
F over X is the smallest set such that
X ff F j f g T X This means that every variable and every
nullary function symbol is a term
If p p p T X and f is an n ary function symbol of F then the
n
string f p p p is an element of T X
n
Let X fx x x g and let A b e an algebra of type F Then every
k
k
term t of the same type induces a function from A to A This function is denoted
by t and is de ned as follows
A
If t x then
i
t a a a a for all a a a A
A k i k
If t f with f F f then
t a a a f for all a a a A
A k A k
n i
If t f t t t with f F f n and all t T X then
t a a a
A k
n
f t a a a t a a a t a a a
A k k k
A A A
for all a a a A
k
Every such function is called a k ary term function The set of all k ary term
functions on A will b e abbreviated by T A
k
Polynomial functions arise from term functions by plugging in some constants
k
A function p A A is called a k ary polynomial function on A i there exist
n b b b A and a k n ary term function f on A such that
n
p a a a f a a a b b b for all a a a A
k k n k
The set of all k ary p olynomial functions on A will b e abbreviated by P A
k
Small b oldface letters stand for vectors hence a stands for a a a
k
A class of algebras of the same type F is called a variety i it is closed under
the formation of direct pro ducts subalgebras and homomorphic images This is
the case i the class can b e describ ed using identites
Definition Let A b e an algebra and let t t b e terms of type F
over the variables x x x Then t t is an identity of A i for all
k
k
a A we have t a t a
A A
CONGRUENCES AND MAL CEV CONDITIONS
For example the group G G is ab elian i x x x x is an identity
of G
Congruences and Mal cev conditions
The set of all congruences of an algebra A is denoted by Con A For two congru
ences and of A we denote the largest equivalence relation that is contained
in b oth and by It is given by
It is easy to see that is again a congruence on A We denote the smallest
equivalence relation that contains b oth and by It is given by
Here denotes the relation pro duct f a b j z A a z and z b g
The relation is again a congruence on A Now Con A is a lattice
We abbreviate this lattice by Con A In this thesis we are mainly considered
with algebras whose congruences satisfy which is the case for all
groups If we have
For Con A and a A the set fa j a a g is denoted by a The congru
ence relation f a a ja Ag will usually b e denoted by the congruence relation
A
A A by The congruence generated by a b with a b A will b e denoted by
A
a b Let b e a congruence on the algebra A If a b A are congruent mo d
A
k
ulo we shall write a b a b or a b mo d For a b A we say
a b mo d i a b mo d for i k Furthermore by a b
i i A
we denote the congruence on A generated by a b a b a b
k k
Let A b e a family of algebras of the same type Then a subalgebra D of
i i I
Q
A is a subdirect product of the family A i for each i I the pro jection
i i
i I
from D to the ith comp onent is surjective
In this section we lo ok at functions that go from some set D into an algebra
A Of course we cannot comp ose two functions of that kind unless we imitate
the construction of sandwich near rings but we will nevertheless get a useful
interpolation result Let A b e a universal algebra and let D b e a set We
D
form the algebra A that consists of all functions from D to A with op erations
de ned p ointwisely This algebra is called the direct product of A indexed by D
D
A subalgebra B of A forms a subdirect product i for all d D the mapping
1
We will sometimes see groups as algebras with one binary op eration at other times
groups will b e regarded as algebras with a binary op eration a unary op eration of forming
the inverse and a nullary op eration giving the identity of the group
PREREQUISITES
D
B A b b d is surjective We also call every subalgebra of A a
d
function algebra from D to A
D
Let D and A b e sets and let F b e a subset of A For each cardinal number n
we form the set L F of all functions with domain D that can b e interpolated at
n
n places by a function in F Formally this reads as follows
D
Definition Let D and A b e sets and let F b e a subset of A Then for
every cardinal number n we de ne the set L F by
n
L F fl D A j S D jS j n f F f j l j g
n S S
D
Clearly for all cardinals s b with s b we have F L F L F A
b s
We shall make use of the following two ternary partial functions on an algebra A
The Mal cev operation m of A is de ned on f x y y j x y Ag f y y x j x y
Ag by
m y y x m x y y x
The Pixley operation p of A is de ned on f x y y j x y Ag f x y x j x y
Ag f y y x j x y Ag by
p x y y p x y x p y y x x
We abbreviate the Pixley op eration on A by Pix A and the Mal cev op era
tion on A by Mal A On some algebras the Mal cev and the Pixley op era
tions are restrictions of term functions For example on every group G we have
Mal G x y z x y z On the two element eld GF we can write the
Pixley op eration as Pix GF x y z x z xy xz y z
For any partial function f we denote its domain by dom f
The imp ortance of Mal cev and Pixley op erations lies in the following fact If
an algebra has a term function which is a Mal cev op eration or a Pixley op era
tion then its congruences b ehave in a certain way We are going to make this
more precise in the following two prop ositions We call an algebra congruence
permutable if for each pair of congruences of A the relation pro duct
is equal to the pro duct If and ful l then their join in
the lattice Con A is given by Ihr Bemerkung
Proposition cf Ihr Satz If Mal A can be interpolated at
each subset of its domain with at most two elements by a term function in T A
then A is congruence permutable
Proof We x Con A Let a b Let c A b e such that a c
and c b Now let t b e a term function on A that interpolates the Mal cev
op eration Mal A at a b b and b b c Then we have
a t a b b t a b c t b b c c
CONGRUENCES AND MAL CEV CONDITIONS
This implies a b
We call an algebra arithmetical if it is congruence p ermutable and if its lattice of
congruences is distributive This is for example the case in the ring of integers
Other algebras with the same prop erty can b e found with the help of the following
prop osition
Proposition cf Ihr Satz If Pix A can be interpolated at ev
ery subset of its domain with at most four elements by a term function in T A
then A is arithmetical
Proof Since the Mal cev op eration on an algebra is a restriction of the Pixley
op eration Prop osition yields that A is congruence p ermutable For proving
that Con A is a distributive lattice lattice theory in particular MMT
Theorem tells that that it is su cient to prove
for all Con A
We x Con A and a b A such that a b Since A is
congruence p ermutable is equal to and therefore we have a c A
such that a c and c b Let p b e term function on A that interpolates
the Pixley op eration at f a a b a b b a c a a a a g Then we have the
following relations
a p a a a p a p a c a a p a p a c b b This holds b ecause of
a b
a p a b b p a p a a b b p a p a c b b This holds b ecause of
a c
p a p a c b b p b p b c b b b
p a p a c b b p a p a b b b p a a b b This holds b ecause of
b c
Putting together the rst two congruences we obtain
a p a p a c b b mo d
Putting together the third and the fourth congruence we obtain
p a p a c b b b mo d
Altogether we get a b which we had to prove
For A let SP A denote the class of all subalgebras of direct pro ducts of nitely
f
many copies of A The following term conditions describ e whether the class
SP A is congruence p ermutable or arithmetical A class of algebras is congru
f
ence p ermutable arithmetical i each of its members has this prop erty
Proposition For a universal algebra A the fol lowing are equivalent
PREREQUISITES
SP A is congruence permutable
f
Mal A can be interpolated at every nite subset of its domain by a term
function on A
Proof Let D b e a nite subset of dom Mal A and let T b e
the function algebra from D to A with universe ftj j t T A g Since T
D
SP A it is congruence p ermutable Let x y z b e the elements of T de ned by
f
x d d d d y d d d d z d d d d We de ne two congruences
on T by
f t t j x y D t x y y t x y y g
f t t j x y D t y y x t y y x g
Then we have x y mo d and y z mo d Altogether this gives
x z mo d
and hence by congruence p ermutablility
x z mo d
This yields an element m T with x m mo d and m z mo d
The last two conditions imply that m interpolates Mal A on D in fact for all
x x y y m x y y and hence x m x y y in the same x y D we have
way we get m y y x z y y x x
k
Let B b e a subalgebra of A for some natural number k N We want
to show that B is congruence p ermutable For this it is su cient to show that
Mal B can b e interpolated at every nite subset D of its domain by a function
in T B Actually by Prop osition we could even restrict ourselves to the
case that D has two elements We de ne D to b e the subset of A given by
D f x y z j x y z D i k g
i i i
The set D is nite and a subset of the domain of the Mal cev op eration on A
hence there is a term t such that the induced term function t interpolates the
A
Mal cev op eration Mal A on D Then one can easily convince oneself that the
function t which is the function that t induces on B interpolates Mal B on
B
D
Proposition For a universal algebra A the fol lowing are equivalent
SP A is arithmetical
f
Pix A can be interpolated at every nite subset of its domain by a term
function on A
Proof Let D b e a nite subset of dom Pix A and let T b e the
function algebra from D to A with universe ftj j t T A g Since T SP A
D f
CONGRUENCES AND MAL CEV CONDITIONS
x y z b e as in the pro of of Prop osition and let it is arithmetical Let
b e the congruence on T de ned by
f t t j x y D t x y x t x y x g
x z mo d It is obvious that we also have x As ab ove we have
z mo d Altogether we get x z mo d and therefore by
arithmeticity x z mo d Hence there exists an element
p in T such that x p mo d and p z mo d The rst
condition gives p x y y x and p x y x x the second gives p y y x x
and again p x y x x Hence p interpolates the Pixley op eration at D
We replace Mal cev op eration with Pixley op eration and rep eat
the pro of of of Prop osition
Given an algebra A and a subset D of A we say that a ternary function f is
a Mal cev function on D i f x Mal A x for all x D dom Mal A
We say that f is a Pixley function on D i f x Pix A x for all x D
dom Pix A Actually Prop osition tells the following SP A is congruence
f
p ermutable if and only if there is a Mal cev function f T A on D for every
nite subset D of A
If the class SP A is congruence p ermutable the following helpful lemma makes
f
it easy to check for relations whether they are congruences
Lemma Let A be an algebra for which SP A is congruence permutable Let
f
be a relation on A such that
is a re exive relation
For al l a b A with a b mo d and p P A we have p a
p b mo d
Then is a congruence relation on A
This lemma is of course applicable to all algebras that lie in a congruence p er
mutable variety
Proof By assumption is re exive
For symmetry we assume that a b lies in Let t T A b e the term
function that interpolates the Mal cev function on f a a b a b b g and let p
b e the p olynomial function in P A de ned by
p x y t a x b
Then condition gives that p a a p b b lies in But we have p a p b
t a a b t a b b b a and hence b a lies in
2
One can weaken the rst condition from is re exive to is not the empty set
PREREQUISITES
For showing that is transitive we assume that a b and b c lie in Let
t T A b e the term function that interpolates the Mal cev function on
f a c c b b c g and let p b e the p olynomial function in P A de ned by
p x y t x y c
Since is symmetric we know that c b lies in Hence condition gives that
p a c p b b lies in But we have p c p b t a c c t b b c a c
and hence a c lies in This shows that is also transitive Hence is an
equivalence relation
For showing that is a congruence let a b and let f b e an n ary funda
mental op eration of A We x x x x x x A and show
i i n
f x x x a x x f x x x b x x mo d
i i n i i n
We consider the binary p olynomial function de ned by
p u v f x x x u x x
i i n
Of course the p olynomial function p do es not dep end on v By the fact that is
preserved under binary p olynomial functions we have p a a p b b mo d
which implies
Commutators
In the textb o ok FM on the theory of commutators in universal algebra sev
eral commutator op erations are de ned The commutator de nition that we are
going to use in the present thesis is the one of HM p Exercises
The same de nition can also b e found in MMT p For congruence mo d
ular varieties this commutator op eration is the same as the op eration denoted
by in FM p
Definition MMT De nition Let b e two congruence re
lations on the universal algebra A Then the commutator of and is
the smallest congruence on A for which the following condition holds
k
For all k N t T A a b A c d A we have
k
a b mo d
c d mo d
t b c t b d mo d
t a c t a d mo d
At rst glance it is not obvious that a smallest congruence with these prop erties
really exists The reason for the existence of a smallest congruence with these
prop erties is given in MMT Lemma
COMMUTATORS
Commutators are the universal algebraic formulation of concepts that had b een
widely used in classical algebra b efore The concept sp ecializes to the concept of
commutator subgroups of two normal subgroups in the theory of groups and to
the ideal pro duct in the case of rings MMT p Exercises and A
description of the commutator in certain algebras with group reduct is given in
Prop osition
We want to recall brie y the main prop erties of the commutator op eration
Con A Con A Con A for congruence mo dular varities cf FM Prop osi
tion We assume that A is an algebra living in a congruence mo dular variety
and that are congruences of A
cf FM Prop osition If f is an epimorphism from A onto B
then we have
ker f ker f ker f
Here is the mapping from the congruences of A ab ove ker f to the con
gruences of B de ned as follows
f f a f a j a a g
By the diamond lemma is a bijection and therefore it is meaningful to
sp eak also of its inverse We observe that the rst commutator in
is taken in Con B and the second one in Con A
We want to give one application of this commutator calculus in which group
theorists will discover a known lemma due to H Wielandt
Lemma Wielandt Let A be an algebra in a congruence modular variety
and let Con A Then we have
In group theory this lemma reads as follows If A B C are normal subgroups of
the group G then the group de ned by
A C B C A B C
is ab elian H Wielandt proved a stronger version of this result Pil Prop osi
tion
PREREQUISITES
Proof
Definition An algebra A is called abelian i
A A A
Let us single out ab elian algebras in some interesting varieties All statements
will b e consequences of Prop osition
A group is ab elian i it satis es the equation x y y x
A ring is ab elian i it satis es the equation x y
A near ring Pil is ab elian i it satis es the equations x y y x
and x y x All those near rings can b e obtained as follows
Start with an ab elian group G and and an endomorphism e on
G that satis es e e e Then de ne x y e x
What we call an ab elian near ring is not what is called an ab elian near
ring in classical near ring theory We stay consistent with universal algebra
terminology another motivation to distinguish near rings that satisfy the
identites x y y x and x y x is given by IK
Let R b e a ring Then every R mo dule is ab elian
Furthermore from the de nition of commutators one sees that every algebra that
has only unary fundamental op erations is ab elian
Ab elian algebras
Definition Let A A b e a group with identity element Then a
k
function f A A is called a ne with resp ect to
k
x y A f x y f y f x f
Here we have used a b as an abbreviation of a b and a b as abbreviation
of a b a b a b
k k
Actually a ne functions are the sum of a homomorphism and a constant func
tion
k
Proposition Let A A be a group Then a function f A A
is a ne with respect to i the function g de ned by g x f x f is a
k
homomorphism from A to A
ABELIAN ALGEBRAS
Proof If f is a ne we can compute
g x y f x y f
f x y f y f y f
f x f f y f
g x g y
On the other hand if x f x f is a homomorphism we have
f x y f y f x y f f f y
f x f f y f f f y
f x f
This proves the result
Now we can state the main result ab out ab elian algebras in congruence p er
mutable varieties Compare also MMT Theorem
Proposition Gum Let A be an abelian algebra in a congruence per
mutable variety with Mal cev term d Let be any element in A and de ne an
addition on A by
a b d a b
A
Then the fol lowing conditions hold
A is an abelian group
The inverse of a in the group A is given by a d a
We have d a b c a b c for al l a b c A
A
Every polynomial function p P A is a ne with respect to
k
Proof Ihr Satz
For the converse we have the following well known result
Proposition Let A be an algebra of type F with a function
P A such that A is an abelian group and al l fundamental operations of A
are a ne with respect to Then A is abelian
Proof We observe that under these assumptions not only the fundamental op
erations but actually all p olynomial functions on A are a ne with resp ect to
We use the de nition of commutators given in De nition for proving
k
Let k N a b A c d A and let t T A such that
A A A k
t a c t a d The function t is a ne and we get
t a c t a t t c
In the same way we get
t a d t a t t d
PREREQUISITES
Hence the equality t a c t a d implies t c t d Therefore we have
t b c t b t t c
t b t t d
t b d
This implies
A A A
Just for getting familiar with these notations let us do a little exercise A
more general version of this result is given in MMT Lemma As
in MMT p we use M to denote the diamond lattice with ve elements
Lemma MMT Let A be an algebra in a congruence modular variety
that has congruences with the property that f g is the
A A
universe of a sublattice of Con A isomorphic to the diamond lattice M Then
A is abelian
Proof We have
A A
A
Hence we have
A A
For the same reason we have and therefore
A A A A
This proves that A is ab elian
A
Neutral algebras
Definition An algebra A in which all congruences satisfy
is called neutral
Let us give some examples of neutral algebras
Let R b e a nite ring which is isomorphic to a direct pro duct of elds
Then R is neutral
A nite group is neutral if and only if it has a principal series in which all
the factors are non ab elian
Every simple non ab elian algebra is neutral
Every Bo olean algebra is neutral MMT p Exercise
Every semilattice is neutral MMT p Exercise
NEUTRAL ALGEBRAS
Furthermore it is proved in FM Exercise that a variety V of congruence
mo dular algebras is congruence distributive i every algebra of V is neutral A
single neutral algebra in a congruence mo dular variety has distributive congru
ences FM Exercise
For checking whether an algebra is neutral we do not have to check all pairs of
congruences
Lemma Let A be a universal algebra Then A is neutral i for
al l Con A
Proof The only if part is obvious For the if part let and b e two
congruences of A By its de nition the commutator is monotonous in b oth
arguments i e and implies Hence we have
Let us quickly rep eat two prop erties of neutral algebras that we need in the
sequel
Proposition FM Exercise Let A be a family of nitely
i i I
many neutral algebras in a congruence modular variety and let D be a subdi
rect product of A Then D is neutral
i i I
Definition BS De nition Let D b e a sub direct pro duct of
A A A Then D is skew free i for every congruence Con D there
n
are congruences with Con A such that
n i i
D
n
In this de nition we use the notation
n
n
for the relation on A de ned by
a a a b b b a b for i n
n n i i i
Proposition Let A be a family of nitely many neutral algebras
i i n
in a congruence modular variety and let D be a subdirect product of A
i i n
Then D is skew free
Proof By BS Lemma it is su cient to prove that for every congruence
Con D we have
n
i
i
where is the kernel of the pro jection of D to A But by Prop osition and
i i
the fact that neutral algebras in congruence mo dular varieties have distributive
PREREQUISITES
congruences can b e rewritten as
n
i
i
T
n
which is true b ecause
i D
i
groups
groups are groups with further op erations which ful l one restriction For each
k N and for every k ary fundamental op eration of an group we claim
groups were introduced by Hig and are also studied in Kur Many
algebraic structures can b e seen as groups such as e g groups rings near
rings and ring mo dules One of the pleasant facts ab out groups is that a
congruence on an group is completely describ ed by the congruence class of
Actually the congruence class of is always a subalgebra of the group
Those subuniverses of an group that arise as congruence classes of for some
congruence are called ideals Ideals can easily b e describ ed by intrinsic prop erties
these topics are discussed e g in Kur or Pil Before continuing our
discussion of groups let us rst give a clean de nition of groups
Definition Let F b e a type Then an algebra V of type is
called an group i
The set F of op eration symbols contains the elements and
The op eration symbol is binary is unary and is nullary
The algebra V is a group
V V V
The set f g is a subuniverse of V i e for all k N and for all k ary
fundamental op erations of V we have
Using vector notation can b e written as So one can say that an
group is an algebra with a group reduct and a one element subuniverse f g
For reason of simplicity we will write for and we will use
V V V
also as a binary op eration symbol in the usual way As in De nition we will
k
also use for the comp onentwise addition in V
Definition A non empty subset I of V is called an ideal of V i for all
k
k N for all k ary fundamental op erations of V for all i I and for all
k
we have v V
v i v I
GROUPS
For a subset S of V the smallest ideal of V that contains S is denoted by I S
V
We will also write I v for I fv g
V V
Note that an ideal I is automatically a subuniverse of V take v and
furthermore a normal subgroup of V This holds b ecause if we take
v
i
v V i I and x x x and if we choose w and j we
have
w j w v i v
v i v
But this is precisely the closure prop erty that a normal subgroup has to ful ll
Let Id V b e the set of all ideals of V It is not to o hard to see that for ideals
I J Id V the sets I J and I J fi j j i I j J g are again ideals of V
Furthermore the structure Id V is a lattice and the mapping de ned
by
Id V Con V
I I
with
v v mo d I v v I
is a lattice isomorphism from Id V to Con V
Of course the commutator notation for universal algebras is immediately avail
able for the congruences of groups however we want to have a commutator
op eration for ideals and not only for congruences Then the commutator op era
tion on the set Id V should b ehave in a way that the mapping de ned in
is also a homomorphism from Id V to Con V Hence we
de ne the commutator of two ideals A B of V by
A B A B
A nicer description of the commutator using p olynomial functions is given in
Prop osition There is another thing that we should pay attention to Let V
b e an group and let W b e a subalgebra of V Supp ose that V has an ideal I
with I W It is easy to see that I is then also an ideal of W However if we
write I I it is not clear whether the commutator has to b e taken in the algebra
V or in its subalgebra W We will therefore write I I if the commutator is
V
taken in V and I I if the commutator is taken in W
W
Note that a similar distinction is not vital for congruences b ecause a congruence
relation on V can never b e a congruence on a prop er subalgebra W of V This
holds simply b ecause for every v V W the pair v v is in and therefore
is not even a subset of W W
PREREQUISITES
Various concepts of commutator op erations for groups
The imp ortance of the concept of commutators has already b een observed in
Hig and Kur However their de nition gives a commutator concept
which is di erent from the commutator arising from the universal algebra com
mutator due to Smi and FM which is the concept that we use in the
present thesis Still it must b e said that Higgins s commutators coincide with the
universal algebra commutators for the imp ortant cases of groups or rings Many
years later Stuart D Scott felt the necessity of a go o d concept of commutators
for groups Sco and he introduced an ideal multiplication that di ered
from the commutator of Hig His de nition proved sucessful in his work
which is not to o surprising b ecause we will show that the Scott commutator
is actually a re invention of the universal algebra commutator A very similar
characterization of the commutator op eration using p olynomials can b e found in
GU
Nevertheless we think it is instructive to give a short comparision of the various
commutator concepts in the following section b efore forsaking Higgins s concept
of commutators for the rest of this work and staying exclusively with the universal
algebra concept that has proved most successful For this discussion let V always
b e an group and let A and B b e two ideals of V
For introducing Scott s commutator concept and comparing it to the one used in
universal algebra we need the following set of zero symmetric p olynomials on
an group V
Definition Let k N with k and let V b e an group Then
ZP V is de ned by
k
k
ZP V fp P V j p v p w for all v V w V g
k k
A p olynomial function is in ZP V if it is zero whenever either the rst argument
k
or all the other arguments are zero These p olynomial functions can b e used as
multiplications note that in every ring R the function x y x y lies in
ZP R In groups x y x y x y is an example of such a zero symmetric
p olynomial function For a near ring N the function x y x y x lies in
ZP N These p olynomial functions can b e used to describ e the commutator
Proposition Let V be an group let k be a natural number with k
and let A B be ideals of V We de ne a set S as fol lows
k
k
S fp a b j p ZP V a A b B g
k k
Then the ideal of V that is generated by S is equal to A B
k
V
For k this prop osition yields that the commutator of A and B is generated
by all elements p a b where a is taken in A b in B and p is a binary p olynomial
VARIOUS CONCEPTS OF COMMUTATOR OPERATIONS FOR GROUPS
function of V with the prop erty p x p x for all x V This is the
actual de nition given by Stuart Scott Sco
Proof Let k b e a natural number with k We will prove the following subset
relations
A B I S I S I S A B
V j V V k
V V
j N j
Let us start with the rst relation namely with
S A B I
j V
V
j N j
Let
I
V
j N j
b e the congruence corresp onding to the ideal
I S
V j
j N j
and let and b e the congruences A and B which corresp ond to A and
B What we are going to show is
I S
V j
j N j
By the de nition of the commutator op eration we know that is the smallest
congruence satisfying the implication given in De nition Therefore we
are done if we show that this implication also holds for
I S
V j
j N j
To this end we take n N a term function t T A two elements a b of
n
n
V with a b mo d and two vectors c d of V with c d mo d We
assume that
t a c t a d mo d
Now we de ne the following p olynomial function q P V
n
q x y t a x c t a x c y t a c y t a c
n
Since q y q x for all x V y V the de nition of S tells us
n
that q a b c d lies in S and hence we have
n
q a b c d mo d
Replacing q with its de nition this is equivalent to
t b c t b d t a d t a c mo d
PREREQUISITES
This implies t b c t b d mo d Therefore lies ab ove the commutator
which proves
Now we prove the next inclusion of namely
I S I S
V j V
j N j
We will prove the following fact by induction on n
n S I S
n V
Base case n Obvious
n
Induction step n Let a A b B and let q P V with q y
n
n
q x for all x V y V We have to prove
q a b I S
V
R
We now write the vector b as b b From the equation
R R R
q a b q a b b q a b q a b
we see that it is su cient for to prove
R R
q a b b q a b I S
V
and
R
q a b I S
V
R R
Considering the p olynomial r x y q x y b q x b we see that r lies in
ZP V and therefore r a b is in I S But this is just the condition given
V
in For proving we observe that the p olynomial function s P V
n
n
de ned by s x y q x y x V y V is zero whenever either x
R
or y By induction hypothesis it follows that s a b is in S Hence we
also have the condition given in This nishes the pro of of
Altogether we have proved the second inclusion in Let us now attack the
third inclusion of namely
I S I S
V V k
In order to prove this inclusion it is su cient to prove S S But this can
k
b e seen immediately Let p b e in ZP V a A and b B We will show that
p a b lies in S To this end we de ne a p olynomial q P V by
k k
q x y p x y
Then p a b q a b b b and q lies in ZP V Therefore p a b also lies in
k
S which nishes the pro of of k
VARIOUS CONCEPTS OF COMMUTATOR OPERATIONS FOR GROUPS
The fourth inclusion of is
I S A B
V k
V
We show that the generators of I S lie in A B For that purp ose let
V k
V
k
p ZP V a A b B We can nd a natural number n k a vector
k
n k
v in V and a term function t in T V such that the p olynomial function p
n
can b e written as
p x y t x y v
Note that v is the vector of elements of V that app ear in a term representation
of the p olyonomial function p Since p ZP V we have
t a v t b v
n
Setting a b a and de ning the vectors c and d in V by c b v
and d v we get
t a c t b v p b
Similarly we get
t a d t v p
By the de nition of the commutator we get
t b c t b d mo d A B
V
This means
t a b v t a v mo d A B
V
which gives p a b A B This nishes the pro of
We are now going to give some consequences of Prop osition
Corollary Let l and m be natural numbers Each of the fol lowing sets
generates the commutator A B of the ideals A and B of V
V
l m l m
G fp a b j a A b B p P V x A y B
l m
p y p x g
l m
G fp a b j a A b B p P V p b p a
l m
p g
G f p a p a b p b p j a A b B p P V g
Proof Since G dep ends on l and m let us write G l m for G in this pro of
We rst prove
I G l m A B
V
V
Now we pro of by induction on l If l then the set G is equal to set S
l
R
de ned in Prop osition For l we write a as a a and then we have
R R R
p a b p a a b p a b p a b
PREREQUISITES
R R
It is easy to see that that p a a b p a b lies in G m We also see
R
that p a b lies in G l m Now induction hypothesis gives that b oth
summands lie in A B
V
For let s b e in S where S is the set de ned in Prop osition Then s
can b e written as p a b with a A b B and p ZP V Obviously the
p olynomial function p P V de ned by
l m
y y y p x y p x x x
z z
m times
l times
is if x or y is zero Hence s p a b lies in G This nishes the pro of of
For proving that G generates A B we observe that G G We will now
V
whow that G is actually equal to G To this end let g b e an element of G
l m
Then g can b e written as g p a b with a A b B and p P V
l m
where p a p b Now we de ne the p olynomial q P V by
l m
q x y p x p x y p y p
By the fact that q x y is zero whenever x or y is zero we see that q a b lies
in G But q a b p a b and hence g lies in G
For proving that G generates A B we prove that G is equal to the set S
V
de ned in Prop osition It is obvious that S is a subset of G For the other
side let p a p a b p b p b e a typical element of G Then the
p olynomial function q P V de ned by
q x y p x p x y p y p
lies in ZP V Therefore q a b lies in S
Now we want to compare this notion of commutators to the commutators studied
in Kurosh s b o ok Kur There the following commutator op eration is de ned
Definition Kur Let V b e an group and let A and B b e ideals
k
of V Then for each k ary fundamental op eration of V and for all x y V
we de ne
x y a b a b
Then we de ne K A B to b e the ideal of V generated by the set
k k
f a b j is a k ary fundamental op eration of V a A b B g
Since every generator of K A B obviously lies in the set G de ned in Corol
lary we know that K A B A B The other inclusion do es not
V
necessarily hold
Proposition There is an group V with ideals A and B such that
K A B A B V
ABELIAN GROUPS
Proof of Proposition We take V Z where Z is the cyclic
group of order with the elements f g The binary op eration is de ned
by and x y else Now we consider A B f g The
set A is an ideal of V b ecause it is a normal subgroup of Z and is a
compatible i e congruence preserving function on Z b ecause it maps Z
into a minimal normal subgroup of Z
Since is trivial on A each of the generators of K A B given in De nition
is equal to From this it follows that K A B f g
Now let us compute A B We consider the p olynomial
V
p x y y x y x
It can easily b e seen that p lies in ZP V From p
w we see that A A
We close with one imp ortant application of the commutator
Proposition Let p P V with p a A b B Then
p a b p a p b mo d A B
V
Proof of Proposition We consider the p olynomial q x y p x y p y
p x Then we see that q ZP V Hence q a b A B which implies
V
the congruence stated in
Ab elian groups
In this section we describ e ab elian groups
Proposition Let V be an algebra of type F such that of arity
resp lie in F Furthermore we suppose that V is a group Then
the fol lowing are equivalent
V is an abelian algebra in the sense of De nition
V is an abelian group and for al l k N al l k ary fundamental
operations of V are a ne with respect to
Proof follows from Prop osition and the fact that the Mal cev
term for the variety generated by V is given by d x y z x y z
follows from Prop osition
If V is an group then we know that each fundamental op eration of V xes
and hence we can restate Prop osition as follows
PREREQUISITES
Proposition An group V is abelian in the sense of De nition
i the group V is an abelian group and for al l k N al l k ary funda
k
mental operations of V are homomorphisms from V to V
From now on ab elian will always mean ab elian in the sense of De nition
CHAPTER
An interpolation result for function algebras
In this chapter we are going to prove and explain several imp ortant interpolation
results due to HH Prop osition will b e crucial for the entire interpolation
and structure theory that we shall develop in this thesis
Function algebras
Recalling the de nition of function algebras from a set D to the algebra A we
notice that a function algebra from D to A is just a subalgebra of the cartesian
D
pro duct A we prefer to view its elements as functions This is mainly b ecause
of the following result
The interpolation result
In this section we give an interpolation result of the type
Interpolation at places Interpolation at n places
Due to the imp ortance of this result we shall rst state and prove it for groups
Then we will give the universal version of this result
The interpolation result for groups
Proposition Let D be a set and let V be an group Let F be a function
D
algebra from D to V i e a subalgebra of V Assume that al l subalgebras of
V are neutral Let l be a function from D to V that can be interpolated at each
subset S of D with jS j by a function in F Then l can be interpolated at
every nite subset of D by a function in F
D
Proof We show that for n N n each function in V that can b e interpo
lated at n p oints by a function in F can also b e interpolated at n p oints by
a function in F In other words we show L F L F
n n
For this purp ose we x n l L F and x x x D We want to
n n
construct a function f F that interpolates l at x x x
n
AN INTERPOLATION RESULT FOR FUNCTION ALGEBRAS
To start with we choose a function s F that interpolates l at x x x
n
Then we are left with a function l L F de ned by l l s which is zero on
n
x x x For nding the function in F that interpolates l it is su cient
n
to prove B S where B und S are the two subsets of V that are given by
B fl x j l L F and l x for i ng
n n i
S ff x j f F and f x for i ng
n i
Since F L F it is easy to see that S B
n
We shall now show
B S
First of all we prove that B is an ideal of W which is de ned as the subalgebra
of V with universe
W ff x j f F g
n
We check the ideal prop erty given in De nition To this end let b e a
k k
k ary op eration symbol of V let v W and let b B We want to show that
v b v lies in B By the de nition of B we can nd a function l in
V V j
L F for each j k that satis es the following condition
n
l x b and l x for i n
j n j j i
k
Since v W there are functions f f f F for j k that
j
satisfy
f x v
j n j
We write l for l l l and we write f for f f f Furthermore
k k
l x abbreviates the vector l x l x l x and f x ab
n n n k n n
breviates the vector f x f x f x
n n k n
Now we can write v b v as
V V
l x f x l x
V n n V n
But taking
g D l f D f
V V
we can write the expression in Equation as g x Furthermore it is
n
D
easy to see that L F is a subuniverse V A similar result will b e proved in
n
Prop osition Hence g L F We also see that g x for i n
n i
Altogether we see that g x lies in B Therefore B is really an ideal of W
n
In the same way we see that S is an ideal of W Now supp ose that S is not equal
to B By the assumptions W is neutral and hence the commutator B B is
W
equal to B By the characterization of the commutator via binary p olynomial
functions in Prop osition we know that B B is generated by the elements
W
p b b where p is a p olynomial function in ZP W and b b lie in B Since
THE INTERPOLATION RESULT
we have supp osed S B and since by assumption we have B B B we
W
know that S is a prop er subset of B B This inclusion can only b e prop er if
W
at least one of these generators of B B of the form p b b do es not lie in
W
S In other words there must b e a binary p olynomial function p ZP W and
two elements b b B with p b b S Since b lies in B the de nition of B
allows to construct a function f F such that
f x b and f x for i n
n i
In the same way we construct a function g F such that
g x b and g x
n n
r
We take r N t T V and v W such that for all x y W we have
r
t v x y p x y
r
Since v W there is a function e in F for each j r such that
j
e x v
j n j
We will write e for e e e Furthermore for a k ary term function t of V
r
and mappings f f f from F to F we write t f f f for the map
k k
ping on V that maps v to t f v f v f v Using these simpli cations
k
of notation we de ne a function h F by
h t e f t e f g t e g t e
It is easy to calculate that we have h x for i n For h x we
i n
obtain
h x t v b t v b b t v b t v
n
p b p b b p b p
p b b
Hence h x is not an element of S But since h lies in F and is zero on
n
x x x this contradicts the de nition of S Hence the assumption S B
n
leads to a contradiction and therefore the inclusion is ful lled But S B
immediately allows us to interpolate l at x x x
n
Note that the restriction that all fundamental op erations on an group preserve
is not essential in this pro of Instead of groups we might in fact consider
any algebra with group reduct In all those algebras the congruences are deter
mined by the congruence classes of although these classes then need not b e
subalgebras For generalizing the pro of we would also have to check that the
characterization of the commutator in Prop osition carries over to any alge
bra with group reduct Instead of doing this we immediately switch to a more
general version of the same result stated in the language of universal algebra
AN INTERPOLATION RESULT FOR FUNCTION ALGEBRAS
The interpolation result for universal algebras
Proposition Let D be a set and let F be a function algebra from D to
A Assume that al l subalgebras of A are neutral and that the class SP A is
f
congruence permutable Let l be a function from D to A that can be interpolated
at each subset S of D with jS j by a function in F Then l can be interpolated
at every nite subset of D by a function in F
This Prop osition follows immediately from the literature if A lies in a congruence
mo dular variety In this case we can give the following pro of
Proof I Let x x x D and let l b e a function that can b e interpolated
n
at every subset of D with not more than two elements by a function in F
We consider the following algebra G that arises from restricting the functions in
F to X fx x x g Formally G is de ned by
n
G ff j j f F g
X
X
G is a subuniverse of the direct pro duct A and the op erations of G are de ned
X
such that G b ecomes a subalgebra of A
We have to nd a function g G that satis es
g x l x
g x l x
g x l x
n n
Since l can b e interpolated at each element subset of D by an element in F
there are functions g g g G such that g x l x for i n
n i i i
Now let b e the congruence on G de ned by
i
g h g x h x
i i i
Of course is just the kernel of the x th pro jection
i i
Then we have to nd g G as a solution of
g l mo d
g l mo d
g l mo d
n n
By the assumptions every subsystem of the system in Equation that consists
of two congruences has a solution g in G
We shall now prove that the algebra G is arithmetical i e the congruences of G
are p ermutable and Con G is a distributive lattice then the Chinese Remainder
THE INTERPOLATION RESULT
Theorem gives that the whole system in Equation has a solution and this
solution is the function g we are lo oking for
X
Since X is nite and since G is a subalgebra of A we get that G is in SP A
f
and hence congruence p ermutable
For showing that Con G is distributive we show that it is neutral For that
purp ose we de ne by
G A A A
z
n times
g g x g x g x
n
Since is injective we see that G is a sub direct pro duct of the algebras B where
i
B is the subalgebra of A with universe B fg x j g Gg By assumption
i i i
every algebra B is neutral Now FM Exercise gives that a sub direct
i
pro duct of nitely many neutral algebras is neutral Therefore G is neutral and
we are done
Remark When we use the results of Chapter of FM we have to b ear
in mind that all the results in that chapter presupp ose that we are working in
congruence mo dular varieties As Pawe l Idziak remarked in the non congruence
mo dular case Pro of I breaks at the p oint where we say that G is neutral and
hence congruence distributive In the general case a neutral algebra do es not
necessarily have a distributive congruence lattice
We now give an elementary pro of of Prop osition that do es obviously not make
use of congruence mo dularity
D
Proof II We show that for n N n each function in A that can b e
interpolated at n p oints by a function in F can also b e interpolated at n
p oints by a function in F In other words we show L F L F
n n
For this purp ose we x n l L F and x x x D
n n
We de ne a binary relation B and a binary relation S on A by
B f l x l x j l l L F and l x l x for i ng
n n n i i
and
S f f x f x j f f F and f x f x for i ng
n n i i
Since F L F it is easy to see that S B
n
We shall now show B S We notice that B is the universe of a subalgebra B of
A A Let B denote the pro jection of B to the rst comp onent Then B
is the universe of a subalgebra of A and we call this subalgebra B Actually
B is a congruence relation on B For proving this we have to prove
B B B
AN INTERPOLATION RESULT FOR FUNCTION ALGEBRAS
B is a re exive relation on B
B is a symmetric relation on B
B is a transitive relation on B
It is easy to see that b b B implies that b b b b and b b lie in
B From this and follow For let b b B and b b
B By the previous observations we know that also b b and b b lie in
B By Prop osition we can pro duce a ternary term m such that m is a
A
Mal cev function on f b b b b b b g Since B is a subuniverse of A A
m b b b b b b b b lies in B This implies the transitivity
A A
of B
With the same reasoning we obtain that S is the universe of a subalgebra S of
A A As ab ove it turns out that S is a congruence on the algebra S where
S is the pro jection of S to the rst comp onent Furthermore we know that
S ff x j f F g and B fl x j l L F g Since n we
n n n
have S B
We know that b oth B and S are congruence relations on B Supp ose that
S B Then the commutator B B taken in the algebra B is equal to B
by assumption Hence by De nition there exist k N a k ary term function
k
t on B a b B and c d B such that
a b mo d B
c d mo d B
t a c t a d mo d S and
t b c t b d mo d S
Since a b mo d B there are functions f f F with
f x f x for i n
i i
and
f x a f x b
n n
In the same way for each i f k g there are functions g g F
i i
with
g x g x
i n i n
and
g x c g x d
i n i i n i
We abbreviate the vector
g g g
k
by g and the vector
g x g x g x
k
THE INTERPOLATION RESULT
by g x Similarly we form g and g x
We use Prop osition to pro duce a ternary term m such that m is a Mal cev
A
function on D D D where D is given by
D ft f g x j k n i j g
i j k
and a ternary term m such that m is a Mal cev function on D D D
A
where D is given by D D m D D D In the sequel we simply
A
and m for b oth m write m for b oth m and m and m
D D
A A
A A
We form two functions h h F as
h m t f g m t f g t f g t f g t f g
h t f g
We will rst show that h x h x for i n For this purp ose we
i i
rst take i n Then we have
h x m t f g m t f g t f g t f g t f g x
i i
m t f g m t f g t f g t f g t f g x
i
m t f g t f g t f g x
i
t f g x
i
Now we see that the last expression is equal to h x For proving h x
i n
h x we do the following calculations
n
h x m t f g m t f g t f g t f g t f g x
n n
m t f g m t f g t f g t f g t f g x
n
m t f g t f g t f g x
n
t f g x
n
Again the last expression is equal to h x
n
Therefore h x h x lies in S Now we have
n n
h x m t f g m t f g t f g t f g t f g x
n n
m t b c m t a d t a c t b c t b d
The value of h x can b e computed by
n
h x t f x g x
n n n
t b c
So h x h x S can b e rewritten as
n n
m t b c m t a d t a c t b c t b d t b c mo d S
Since we have t a c t a d mo d S we get
m t b c m t a d t a d t b c t b d t b c mo d S
AN INTERPOLATION RESULT FOR FUNCTION ALGEBRAS
which can b e calculated as
m t b c t b c t b d t b c mo d S
Hence we get t b d t b c mo d S which is a contradiction
We now have proved S B In the remainder of this pro of we want to show how
this equality allows us to construct the function that interpolates the function
l L F We want to construct a function f F such that f x l x for
n i i
i n Since l is in L F there is a function p F such that
n
p x l x for i n
i i
By de nition we have
p x l x mo d B
n n
and hence also
p x l x mo d S
n n
This means that there exist functions p p F such that
p x p x for i n
i i
and
p x p x p x l x
n n n n
Let m b e a term such that m is a Mal cev function on D D D where D
A
is given by fp x j j i n g
j i
We de ne p F by
p m p p p
A
and obtain for i n the equality
p x m p p p x m p p p x p x l x
i A i A i i i
For x we obtain
n
p x m p p p x m p p p x p x l x
n A n A n n n
Hence p is the required interpolating function
Interpolation results are often in the centre of algebraic structure theory For
this reason it is not surprising that we may fruit Prop osition in many ways
Let us give one application of this result A more general version is given in
Chapter
Proposition Let V be an group such that I I I for every ideal I
V
k
of V let k N let D be a nite subset of of V and let f D V be a function
such that for al l v w D the di erence
f v v v f w w w
k k
THE INTERPOLATION RESULT
lies in the ideal of V generated by fv w v w v w g
k k
Then f is the restriction of a polynomial function of V
Proof The conditions that we have put on f mean that f is a congruence preserv
ing compatible function It is well known that therefore f can b e interpolated
at every two element subset of its domain by a p olynomial function cf Pil
Prop osition
We consider the algebra V This is the expansion of V that we obtain by adding
all elements of V as constant op erations The congruences of V are precisely
those of V the commutators stay the same For this pro of it is already su cient
to see that the commutators de nitely cannot b ecome smaller by adding new
op erations Therefore V is neutral The algebra V do es not have any prop er
subalgebra Now we apply Prop osition to the function algebra F given by
F fpj j p P V g
D k
Prop osition yields that f lies in F
This Prop osition implies that on a neutral algebra every congruence preserving
function can b e interpolated at every nite subset of its domain by a p olynomial
function Later we will formulate this by saying that the p olynomial functions
are dense in the compatible functions
AN INTERPOLATION RESULT FOR FUNCTION ALGEBRAS
CHAPTER
Comp osition algebras
Adding comp osition
In Pil the following metho d of constructing new algebraic structures
is describ ed
Take a universal algebra A A form the set M A of all
self maps of A and de ne the op erations of p ointwise on M A
Adding the binary op eration of comp osition yields a new algebra
M A M A f g
Starting with A b eing a group we obtain the near ring of all functions on A start
ing with A b eing a ring we obtain a comp osition ring cf Adl and starting
with A b eing a set with no op erations we obtain a semigroup It is therefore re
warding to study this pro cess in the general context of universal algebra hoping
that some of the knowledge ab out the structure of sp ecial cases might carry over
to the general case A supp ort for this hop e is the pap er Mli in which R
Mlitz gives a universal algebra pattern of Jacobson s Density Theorem for rings
Jac p and the Density Theorem for near rings Ram Pol Bet
In this chapter we shall develop some structural results for the algebras obtained
by the pro cess outlined ab ove and their subalgebras and we will see that these
results explain and generalize many results obtained for near rings and comp osi
tion rings We will investigate what the ab ove pro cess yields for those algebras
A that lie in congruence p ermutable varieties every algebra with gro op or lo op
reduct is such an algebra What happ ens if we lack congruence p ermutability
can b e found in Che
Dear reader I really do not want to lose you b ecause of the notational compli
cations that cannot b e avoided in universal algebra Therefore in these sections
written in smaller font I will keep telling you what the developed theory means
for my sup ervisor s favourite algebraic structures namely near rings
In this section we will formalize how to add this binary op eration to an algebra
Let us rst de ne the type of such algebras
Definition Let F b e a type of algebras such that do es not lie in
the set F of op eration symbols of Then we de ne a new type as the type that
COMPOSITION ALGEBRAS
contains all op eration symbols of F plus the binary op eration symbol This
new type is abbreviated by C We will call C the composition type over
Formally this de nition could b e stated as follows
C F f g
where j and In the sequel we will always assume that a type
F
from which we construct the type C do es not contain the op eration symbol
If we want to construct near rings then we have to start with the type that
de nes groups If has the op eration symbol then the comp osition type C
has the symbols and
Given an algebra of type C we sometimes want to forget ab out the added
comp osition
Definition For an algebra F of type C let F b e the reduct of F of
type
+
So if we start with a near ring F F we end up with F F
Hence this op eration of forgetting the comp osition op eration leaves us with the
additive structure of the near ring
The construction that we give here shall mo del the way in which near rings are
obtained from groups or comp osition rings from rings Hence starting with a
type we will not admit every algebra of type C as a comp osition algebra over
but we put two further conditions on it We say that a composition algebra
over type is an algebra of type C in which the comp osition is asso ciative
and is right distributive with resp ect to all other op erations Actually the
de nition given here is a sp ecial case of the de nition of comp osition algebras
given in LN Chapter
Definition Let b e a type Then an algebra F of type C is called a
composition algebra over the type i
The op eration is asso ciative i e for all f g h F we have f g h
F F F
f g h
F F
The op eration is right distributive with resp ect to all fundamental op
F
erations of F i e for all n N for all n ary op eration symbols of
and for all f f f g F we have
n
f f f g f g f g f g
F n F F F F n F
These conditions mean that F satis es the identites x y z x y z
and x y z x z y z
ADDING COMPOSITION
A di erent way of stating these asso ciative and distributive laws that are is given
in the following prop osition
Proposition An algebra F of type C is a composition algebra i
is associative
F
For each x F the mapping r F F f f x is a homomorphism
x F
from F to F
This tells that F is a comp osition algebra if is asso ciative and for every
x F the mapping r F F f f x is a group homomorphism of F
x
This homomorphism prop erty is just the distributive law
Yet this pro cess is not re ned enough to pro duce the class of all near rings as
the class of all comp osition algebras over the group type The reason for this is
that in a near ring F F the additive structure has to satisfy the group
laws for F F whereas De nition do es not put any restrictions on
F Therefore we give the following de nition
Definition Let V b e a class of algebras of type Then an algebra F of
type C is a V composition algebra i
F is a comp osition algebra over the type
F lies in the class V
Let G b e the class of all groups Then the G comp osition algebras are just the
near rings
The class V will often by a variety of algebras We note that in order to b e
able to consider groups as a variety we have to see them as algebras with the
binary op eration the unary op eration of nding the inverse and a constant
op eration that gives the group identity It is easy to see that for the variety of
groups G the class of all G comp osition algebras is the class of all near rings seen
as algebras with the op erations For the class of all rings R the R
comp osition algebras are precisely the comp osition rings which were studied in
Adl and which are going to b e investigated closer in Chapter Comp osition
near rings PV fall into the same pattern to o These last examples suggests
the following way of giving a name to V comp osition algebras If the algebras in
V are the xxxs the V comp osition algebras should b e called composition xxxs
Questions ab out these comp osition algebras that arise naturally are for example
Are there meaningful examples of comp osition algebras
For a given class V is it p ossible to describ e all simple or sub directly
irreducible nite V comp osition algebras
We will now give two constructions that pro duce a V comp osition algebra out of
an algebra in V
COMPOSITION ALGEBRAS
Definition The comp osition algebra of functions on A Let V b e a class
of algebras of type and let A V We can then de ne the V comp osition
algebra M A with universe M A ff A Ag For the op eration symbols
of we de ne the op eration on M A as the p ointwise application of the
M A
op eration The op eration is de ned by
A M A
f g a f g a for a A
M A
For a group G G written additively but not necessarily ab elian we
G
obtain M G as the near ring G where is de ned p ointwisely and
is the op eration of functional comp osition
A
At a closer lo ok we nd out that M A is the cartesian pro duct A and
is the op eration of functional comp osition We call M A the ful l function
M A
composition algebra on A
It is known that every near ring F can b e embedded into the near ring
M G for some group G A group that always works for that purp ose is the
direct pro duct of F with the cyclic group of order The same embedding
result holds if instead of near rings we consider V comp osition algebras where
V is a variety of algebras This result is stated in Prop osition
Whenever studying groups the problem arises that groups seen as algebras with
one binary op eration do not form a variety Therefore on those o ccasions
where we want to see the class of groups as a variety we consider groups as
algebas with the additional op erations inverse and identity
Proposition LN Chapter Theorem Let V be a variety of al
gebras Then for every V composition algebra F there is an algebra A V such
that F isomorphic to a subalgebra of M A
So every comp osition algebra can b e embedded into some M A
We want to continue with other examples of comp osition algebras
Definition The constant comp osition algebra on A Let V b e a variety
and let A V We can then de ne the V comp osition algebra A with universe
c
a a A by A A and a
c A
c
On every group G we can de ne an op eration such that G b ecomes
a near ring we simply de ne g g g
1 2 1
Definition The comp osition algebra of constant functions on A We de
ne the algebra M A as the subalgebra of M A whose universe is given by
C
M A fm A A j jm A j g
C
Proposition For any algebra A the algebras A and M A are isomor
c C
phic composition algebras
LOCAL INTERPOLATION ALGEBRAS
Proof We take a A and consider the mapping
M A A
C
m m a
Then is an isomorphism b etween M A and A
C c
Lo cal interpolation algebras
Throughout this section we x a class V of algebras of type Furthermore we
let A V and F b e a subalgebra of M A whose universe as usual is denoted
by F We observe that F is then a V comp osition algebra We will now see that
the set L F of those function that can b e interpolated at a xed number of p oints
n
by a function in F which was de ned in De nition is a subuniverse of M A
Proposition For any cardinal number n the set L F is a subuniverse of
n
M A
The set of those functions that can b e interpolated at any subset of n places by a
p olynomial function has b een investigated in HN N ob Aic Instead
of the set of p olynomial functions one may also start with a near ring For
F b eing a near ring L F has b een studied in Aic
n
Proof L F is closed under the op erations from F Let k N and let b e
n
k ary op eration symbol of the type Furthermore let l l l L F We
k n
have to prove
l l l L F
k n
M A
For this purp ose let S b e a set with jS j n Since l l l L F there
k n
are functions f f f F such that l j f j Now it is easy to see that
k i S i S
f f f interpolates l l l at S
M A k M A k
For proving that L F is closed under functional comp osition we take l l L F
n n
We have to prove that c de ned by
c A A
a l l a
is in L F To this end let S b e a set with jS j n There is a function f F
n
with f j l j Since jf S j jS j n there is also a function f F with
S S
f j l j Now we see that f f interpolates c at S Therefore we
f S f S M A
1 1
have c L F
n
Definition L F For a cardinal number n the subalgebra of M A with
n
universe L F is denoted by L F
n n
The algebras constructed as L F will b e called local interpolation algebras Ac
n
tually the op erator L has the following prop erties n
COMPOSITION ALGEBRAS
Proposition Let F G be subalgebras of M A and let b m n s be car
dinal numbers Then the fol lowing properties hold
F G L F L G
n n
F L F
n
L F L F if s b
b s
L L F L F
m n min m n
Proof and follow from the de nition
For we rst prove
L F L L F
min m n m n
If m n we have L F L F L L F If m n we have L F
min m n m m n min m n
L F L L F Now we prove
n m n
L L F L F
m n min m n
Let l L L F For proving that l L F let S b e a subset of A with
m n min m n
jS j min m n Since l L L F there is a function l L F such that
m n n
l j l j Since l L F there is a function l F such that l j l j
S S n S S
Therefore we also have l j l j which proves l L F
S S
min m n
We will now de ne an algebra consisting of those functions that can b e interpo
lated at any nite subset of A by a function in F
Definition We de ne
L F L F
n
n nite
Then L F is a subuniverse of M A We write LF for the corresp onding subal
gebra
We will now list some prop erties of the op erator L
Proposition Let n be a cardinal number
LL F L LF L F if n is nite
n n n
LLF LF
LL F L LF LF if n is in nite
n n Proof
LOCAL INTERPOLATION ALGEBRAS
Let us rst prove LL F L F We have
n n
LL F L L F
n m n
m N
L F
min m n
m N
L F
n
Now we prove L L F L F The relation follows from F L F For
n n
we compute L L F L L F L F
n n n n
T T
LL F L L F L F L F
m m
m N m N
If n is an in nite cardinal then we have L F L F This implies that b oth
n
LL F and L L F are subsets of LL F which is equal to L F
n n
These op erators allow an easy description of what we mean by density
Definition Let F G b e subalgebras of M A Then we say that F is
dense in G i F G LF
Proposition Let F G be subalgebras of M A Then we have
LF is the largst subalgebra of M A in which F is dense
Let n be a natural number Then F is dense in L F i LF L F
n n
Pro of follows from the de nition For we note that F is dense in L F i
n
F L F LF This holds i LF L F
n n
In the following we shall often get the result F is dense in L F By the last
prop osition this is equivalent to L F LF This equality can b e paraphrased as
follows
If a function g on A can b e interpolated by a function in F at each
set of p oints then g can b e interpolated by a function in F at each
nite subset of A
Furthermore Prop osition shows that if F is dense in L F then L F is really
the largest of all subalgebras of M A in which F is dense
We will now give a prop erty that implies LF F
Definition Let B b e a subset of A Then B is called a base of equality of
F i for all f g F with f j g j we have f g
B B
As an example let G b e an ab elian group and let P G b e the subalgebra
of M G with the unary p olynomial functions as universe We know that the
p olynomial functions are those of the form f g z g h with z Z and h G
If G has an element e of in nite order then f eg is a base of equality of P G
COMPOSITION ALGEBRAS
We will now give the example of a countable ab elian group where the p olynomial
functions do not admit a nite base of equality Fixing a prime p we consider
1
which is the subgroup of the multiplicative group of the complex the group Z
p
numbers with universe
n
p
1
g fx C j n N x Z
p
1
have no nite base We will now show that the unary p olynomial functions P Z
p
of equality Keeping multiplicative notation the unary p olynomial functions on
z
1
z Z this group are the functions of the form f x c x with c Z
p
n
p
Supp ose D is a nite set Then let n b e large enough to ensure d for all
n
p
d D The p olynomial functions f x x and g x agree on D but are
not equal which disquali es D as a base of equality
The following prop osition is an obvious mo di cation of HN Lemma
Proposition Let B be a base of equality for F and let b be the cardinality
of B Then the fol lowing holds
If b is nite then L F F
b
If b is in nite then L F F
b
Proof Supp ose that there is a function l L F that do es not lie in F
b
Let f F b e a function with f j l j Since f lies in F and l do es not
B B
they have to di er in at least one p oint say a Now the cardinality of B fag
is b hence there is a function f F with l j f j Therefore we
B fag B fag
have f j f j and f a f a But this contradicts the fact that B is a
B B
base of equality of F
Supp ose that there is a function l L F that do es not lie in F Let f F
b
b e a function with f j l j Since f lies in F and l do es not they have to
B B
di er in at least one p oint say a Now the cardinality of B fag is b hence there
is a function f F with l j f j Therefore we have f j f j and
B fag B fag B B
f a f a But this contradicts the fact that B is a base of equality of F
We give an easy application of this prop osition
Corollary Let D be an in nite integral domain and let p be a function
on D that can be interpolated at every countable subset of D by a polynomial
function Then p is a polynomial function
Proof Let b e the cardinality of the naturals Then every countable subset of
D is a base of equality of P where P is the subalgebra of M D that consists of
all unary p olynomial functions on D Hence Prop osition gives L P P
0
and therefore p is a p olynomial function
Proposition Let f be the cardinality of F Then there is a base of equality
B for F with jB j f
LOCAL INTERPOLATION ALGEBRAS
Proof For each pair f f F with f f we take an element a such
f f
1 2
that
f a f a
f f f f
1 2 1 2
Now we take
B fa j f f F f f g
f f
1 2
We will now show that B is a base of equality for F To this end let g h b e
two elements in F with g j hj But then we have g a h a which
B B g h g h
contradicts the choice of a
g h
If F is in nite then this upp er b ound f can actually b e reached As an example
1
one may again consider the unary p olynomial functions on the group Z
p
Proposition If A is nite or F is nite then we have LF F
Proof If A is nite then clearly A is a nite base of equality for F If F
is nite then Prop osition pro duces a nite base of equality for F Let b
b e the cardinality of this base of equality Then by Prop osition we have
LF L F F
b
In the next paragraph we shall see that sometimes there is a kind of reversion of
Prop osition A slightly more general version of this result is given in Aic
The investigations of the next paragraph also follow the prop osal of N ob to
investigate the cardinalities of the sets LP A for all kinds of universal algebras
t
A We also give an elementary reason for the known fact that there are uncount
ably many lo cal p olynomial functions on the integers However we have to put
restrictions on A and F
Convention We assume that A has a reduct A A which is
a group We also assume that F is a subalgebra of M A
Lemma Let F A be as in Convention Then D is a base of equality
for F i D A and every function in F that is zero at al l elements of D is zero
everywhere on A
We recall that we call a set countable i it is nite or countably in nite
Theorem Let A and F be as in Convention If A and F are both
countable and if F L F then there exists a nite base of equality D for F
Proof The result is obvious if F or A is nite Let a a a and f f f
b e complete enumerations of F and A resp ectively Furthermore we abbreviate
the set fa j i r g by A r
i
Supp ose that there is no nite base of equality for F We shall construct a
of elements of non negative integers and a sequence g sequence n
m m N m m N
0 0
of F with the following prop erties
COMPOSITION ALGEBRAS
m N g j f j
m A n m A n
m m
m N n n
m m
m N g j g j
m m
A n A n
m m
We construct the sequences inductively Let g F such that g f Let n b e
f a minimal in N with g a
n n
0 0
If we have already constructed g and n we construct g and n as follows
m m m m
In the case g j f j there exists a function h F with g j
m m m
A n A n A n
m m m
hj and h f since otherwise A n would b e a forbidden base of
m m
A n
m
equality for F We set g h Now let n b e minimal with h a
m m n
m+1
f a
m n
m+1
If g j f j we set g g and n n
m m m m m m
A n A n
m m
Since for every a A the sequence g a is eventually constant we may
m m N
de ne a function l on A by
l a lim g a
m
m
The function l lies in L F and hence by assumption l lies in F So l is equal
to f for some m N Since l j g j and g j f j we
m m m m
A n A n A n A n
m m m m
obtain l j f j But this shows that l can not b e equal to f
A n m A n m
m m
Putting the last two prop ositions together we get
Corollary Let A and F be as in Convention If F and A are both
countable and if F L F then there exists an n N such that F L F
n
This prop erty can b e strengthened
Corollary Let A and F be as in Convention If L F and A are both
countable then we have
There is a nite base of equality D for F
L F F
Proof By the idemp otence of the op erator L we have L F LL F Since b oth
L F and A are countable we may apply Theorem and get a nite base of
equality D for L F Since F is a subset of L F the set D is also a base of equality
of F This proves the claim in now follows by Prop osition
Corollary Let R be a countably in nite integral domain Then LP R
is not countable
Proof We supp ose that LP R is countable Then there exists a nite base of
Q
equality D for P R and hence the p olynomial p x x d induces
d D
LOCAL INTERPOLATION ALGEBRAS
the zero function on R This is imp ossible b ecause R is an in nite integral
domain
For p olynomial functions on groups we obtain the following corollary
Corollary Let V be an group If LP V is countable then there exists
a nite base of equality for P V
Proof The result follows from Corollary if we observe that if LP V is
countable then V is countable as well
1
We can apply this result to the unary p olynomial functions on the group Z
p
1
is not countable Then we obtain that LP Z
p
So we have seen that LF can b e a lot bigger that F Another example of this
phenomenon is p erhaps the following Let D b e any in nite eld and let F b e
the comp osition ring of all p olynomial functions on D Then we have jF j jD j
However since every function can b e interpolated at a nite number of p oints
by a p olynomial function we have LF M D and therefore jL F j jD j It is
therefore surprising that LF and F ful l precisely the same identities
Proposition Let V be a variety of algebras let A be an algebra in V and
let F be a subalgebra of M A Then F and LF generate the same subvariety of
the variety of al l V composition algebras
If we take a group G G and F as a subnear ring of
M G then Prop osition tells that L F satis es all
the identites satis ed by F For example if F is a ring then
L F must b e a ring as well
Proof Let V b e the variety generated by F and V b e the variety generated by
LF Since F LF we obviously have V V For V V we show that LF
ful lls all equations satis ed by F But this will follow from Prop osition
For studying the equations satis ed by LF we x the variables x x x
k
and we let T b e the set of all terms of type C over x x x Let N b e an
k
algebra of type C and let t t b e terms in T We recall from De nition
that the equation t t is an identity of N i
N t t
k k k
N N
i
k i
where t denotes the function from N to N that is induced by the term t
N
For instance given C as the type with the binary symbols and the unary
symbol and the nullary symbol and given a near ring N of this type we say
that N is a ring i x x x x and x x x x x x x are
b oth identites of N
Definition Let t b e a term Then for l f k g we de ne O cc l t
as the number of o ccurrences of x in the term t For any terms t t we de ne l
COMPOSITION ALGEBRAS
the complexity C of the equation t t by C t t max fO cc l t
O cc l t j l k g
So C t t n means that no variable o ccurs more then n times in the
equation As an example C x x x x x x x
Proposition Aic Aic Let A be an algebra let F M A and
let t t be terms such that t t is an identity of F Let c C t t
Then t t is an identity of L F for al l cardinals n with n c
n
Pro of Let t t b e an equation that is not an identity in L F Hence there
n
are l l l L F such that
k n
t l l l t l l l
k k
L F L F
n n
Both sides of are elements of L F hence functions on A Thus we have
n
a A with
t l l l a t l l l a
k k
L F L F
n n
If we actually want to compute t l l l a and t l l l a
k k
L F L F
n n
then each l gets evaluated at at most c places Since l lies in L F and since
j j n
c n we can nd an f in F that is equal to l at these c places Hence
j j
i i
t l l l a t f f f a
k k
L F L F
n n
for i From this we conclude that f f f violate the equation t
k
t which is therefore not an identity of F
If F is in a variety K of comp osition algebras that can b e describ ed by equations
that have all complexity less or equal to n we may even conclude that L F is in
n
K
Taking again A to b e a group and F to b e a sub near ring of M A that is a
ring we obtain that L F is a ring as well A ring can therefore never b e
dense in a non ring Actually a subalgebra of M A can only b e dense in a
subalgebra with precisely the same identites
Mo dules of comp osition algebras
As in the structure theory of rings near rings and comp osition rings we get
information ab out the structure of a V comp osition algebra F by interpreting it
as an algebra of functions on an algebra A in V
Definition Mo dule op erations Let A b e an algebra of type and let
F b e a comp osition algebra of type C Then the op eration
F A A
MODULES OF COMPOSITION ALGEBRAS
is called a module operation of F on A i the mapping F M A de ned by
f A A
a f a
is a homomorphism from F to M A
Remarks
Both algebras F and M A are of type C Hence we claim that is a
homomorphism of algebras of the comp osition type over and therefore
in particular also a homorphism with resp ect to
The fact that is a homomorphism is equivalent to the following For all
f f F a A for all k N and for all k ary op eration symbols of
type we have
a f f a f f a
F
b f f f a f a f a f a
F k A k
For a group G G and a near ring F F the op eration F
G G is a module operation if the identities f f g f f g and
1 2 1 2
f f g f g f g hold So in this context mo dule op erations are just
1 2 1 2
N group op erations
Let us brie y outline where mo dule op erations arise in familiar contexts To this
end we compare mo dule op erations to the notion of N groups used in near ring
theory cf Pil De nition and to the notion of modules used in ring
theory
If A is a group F is a near ring and is a mo dule op eration of F on A then A
can b e seen as an F group On the other hand every F group A gives rise to a
mo dule op eration of F on A
If A is an ab elian group F is a ring and is a mo dule op eration of F on A
then A can b e seen as an F group in the sense of near ring theory However A
do es not have to b e a ring mo dule in the sense of Jac This is shown by
the following example Let A b e the group Z Z and let e b e the mapping
x
de ned by e and e on all other three places Then f eg
y
is the universe of a subalgebra F of M A The op eration f a f a is a
mo dule op eration of F on A and F is a ring However since e is not a linear
function on A A is not an F mo dule in the sense of ring theory
This natural view that sees an F mo dule X as a pair of an algebra and a mo dule
op eration is not practical when we want to examine mo dules with the to ols of
universal algebra For this reason we shall adopt a di erent p oint of view For
1
Note that Jacobson lets rings op erate from the right on their mo dules but apart from
notation there is also real di erence b etween the mo dules used in ring theory and the mo dules
that we are going to consider here as we shall explain in the sequel
COMPOSITION ALGEBRAS
each f F we shall interprete the op eration x f x as a new unary op eration
on the algebra X To this end we rst de ne a type that contains a unary
op eration symbol for each f F
Definition The type of F Mo dules Let F b e a comp osition algebra over
the type Then we de ne a new type as the type that contains all op eration
symbols of plus the unary op eration symbol S f for every f F We abbre
viate this type by M F and call it the mo dule type of F
In other words starting from a comp osition algebra F over type F the
type M F is de ned by M F F where F and are de ned by
F F F F
F F fS f j f F g
F
and by for all F and S f for f F The ab
F F
breviation S f can b e read as the operation symbol produced from f In order
to prevent notational complications we will throughout assume that the original
type do es not contain any of the symbols S f with f F Given an algebra
X of the mo dule type of F we sometimes want to forget ab out the op eration of
F
Definition Let F b e a comp osition algebra over the type and let X b e
an algebra of the mo dule type M F of F Then by X we denote the reduct
of X of type
Recall that out of a type we have constructed two new types Adding the
binary op eration symbol we have obtained the type C and using an algebra
F of this type we could de ne the mo dule type M F Given an algebra A of
either type the algebra A has b een de ned to b e the reduct of type
Hence we have completed the rst step in de ning F mo dules We have de ned
the type that these algebras must have
Definition Let F b e a comp osition algebra over the type and let X b e
an algebra of the mo dule type M F of F We say that X is an F module i the
mapping de ned b elow is a homomorphism from F to M X
F M X
f f
and f is de ned as
f X X
x S f x
X
Since this de nition is quite technical it is worth giving some comments
MODULES OF COMPOSITION ALGEBRAS
Since X is an algebra of type the algebra M X a comp osition algebra
over the type Hence the mapping is a homomorphism b etween algebras
of type C in particular it is a homomorphism also with resp ect to
comp osition
The fact that is a homomorphism could b e rewritten as follows For all
f f F x X k N and for all k ary op eration symbols of the type
we have
a
S f S f x S f f x
F
X X X
b
S f f f x S f x S f x S f x
F k X k
X X X X
An equivalent p ossibility of introducing F mo dules is given by the following
prop osition
Proposition Let F be a composition algebra over the type and let X be
an algebra of type M F Then X is an F module i
S f S f x S f f x
F
X X X
For each x X the mapping r de ned by
x
r F X
x
f S f x
X
is a homomorphism from F to X
This prop osition also settles the question what the distributive law means if we
have nul lary op eration symbols in F If is such a nullary op eration symbol
and X is an F mo dule we have for all x X
x
F X
If we are given a comp osition algebra over the type algebra A of type and
a mo dule op eration of F on A then we can construct an F mo dule X from
these ingredients in the following way X will b e algebra of type M F We take
A as universe for X furthermore the reduct X of X to the type shall b e
equal to A The unary op erations S f are de ned by S f a f a for all
X X
f F a A It is immediate to check that X is an F mo dule If is clear from
the context we abbreviate this mo dule by A
F
On the other hand every F mo dule X gives rise to a mo dule op eration of F on
X This mo dule op eration can b e de ned as f x S f x
X
Here are some examples of mo dules Let F b e a comp osition algebra over the
type Then we have the following two F mo dules
COMPOSITION ALGEBRAS
We take the reduct F with type and de ne a mo dule op eration of F
on F by
f g f g for f g F
F
We construct an F mo dule from this op eration by the pro cess outlined
ab ove We will usually abbreviate this mo dule by F
F
We take A to b e an algebra of type with precisely one element say a
Then we de ne a mo dule op eration by f a a for f F a A
If F is a subalgebra of M A for some algebra A then the op eration F A
A f a f a is a mo dule op eration The F mo dule resulting from this
op eration will b e denoted by A
F
Before we start doing real work with mo dules we collect some easy observations
Lemma For al l n ary terms of type M F and for al l h h h
n
F x X we have
+
t h h h x t h x h x h x
n X n
F
F
Sketch of the proof We induct on the depth of t and use the identities
f x S f x
X
and
S f f f x S f x S f x S f x
F k X k
X X X X
The latter identity has b een stated after De nition
The following metho d allows to construct sub F mo dules of X
Lemma Let F be a composition algebra and let X be an F module and let
x X Then F x ff x j f F g is a subuniverse of the F module X
For every element g of the F group G the set ff g j f F g is the universe of
a sub F group of G
Proof We have to prove that F x ff x j f F g is a subuniverse of X To
this end let f f f F and a k ary op eration symbol in F Then by
k F
Lemma we have
+
f x f x f x f f f x
X n k
F
F
Hence f x f x f x F x
X n
We will denote this mo dule also by F x
At rst glance it seems that F x is the universe of the sub F mo dule of X
generated by x This is however not true in general since x itself do es not have
to b e an element of F x
MODULES OF COMPOSITION ALGEBRAS
We still need to re ne De nition for a reason that we explain in the following
example Let R b e the variety of all rings with identity and let D b e the dihedral
group with elements We let I D b e the near ring of all zero symmetric
p olynomial functions on the group D cf Pil Since D is nilp otent of
class we know that I D has ab elian addition The op eration I D D
D p d p d is a mo dule op eration But the resulting I D mo dule do es
not have an ab elian addition Sometimes we want to claim that this mo dule also
has ab elian addition This will b e made p ossible by the following de nition
Definition Let V b e a variety of algebras and let F b e a V comp osition
algebra Then an F mo dule X is called an F module with reduct in V i X V
Let us examine the previous example under this new light To this end let A b e
the variety of all ab elian groups The ring I D is an A comp osition algebra
Considering again the mo dule op eration I D D D p d p d the
resulting I D mo dule is not an I D mo dule with reduct in A
Prop osition tells that for every variety V and for every V comp osition algebra
F there is an interesting F mo dule with reduct in V Interesting will mean
that we can distinguish the elements of F by their actions on this mo dule
Proposition Let F be a V composition algebra Then the fol lowing classes
form varieties of algebras of type M F
The class of al l F modules
The class of al l F modules with reduct in V
Proof By the de nition of mo dules in De nition we se that F mo dules are
de ned by equational conditions Hence the class of all F mo dules is a variety
The class of all F mo dules with reduct in V is also a variety b ecause it is the
intersection of the variety of all F mo dules with the variety of those algebras of
type M F whose reduct is in V
Starting with a near ring N the class of all N groups is a variety
The class of all N groups that have ab elian addition is a subvariety of this variety
Definition For a given variety V and a V comp osition algebra F we
abbreviate the variety of all F mo dules with reduct in V by V
F
The subalgebras of an F mo dule X are called sub F modules of X
We call a mo dule faithful if di erent elements of F b ehave di erently on X
Definition An F mo dule X is called faithful i the mapping in De
nition is injective
COMPOSITION ALGEBRAS
We see that X is faithful i for all f g F the implication
x X f x g x f g
holds Referring to the mo dule op eration of F on X that arises from the
mo dule X we say that F op erates faithfully on X by i X is faithful
Let us compare the notions faithful and base of equality If A is an algebra
and F is a subalgebra of M A then the op eration f a f a is a mo dule
op eration and gives rise to the mo dule A Now we consider a submo dule B of
F
this mo dule A Then B is a faithful F mo dule if and only if B is a base of
F
equality for F
A near ring F op erates faithfully on a group G i g G f g implies
f
If a mo dule X is faithful every equation that is satis ed in X is automatically
satis ed in F
F
Proposition If X is a faithful F module then F lies in the variety of
F
F modules generated by X
A near ring that op erates faithfully on an ab elian group will also have ab elian
addition
Proof The mapping de ned by
F M X
f f
with f x S f x is not only a homomorphism from F into M X
X
X
but also a homomorphism of F mo dules from F into X If is injective
F
X
then F is isomorphic to a subalgebra of X and hence F lies in the variety
F F
generated by X
If a sub F mo dule Y of X has a universe of the form F x then every equation
that is satis ed in F is automatically satis ed in Y
F
Proposition Let X be an F module and let F x be the submodule of X
with universe F x Then F x lies in the variety generated by F
F
Proof The mapping r F X f S f x is not only a homomorphism
x
X
from F to X but also a homomorphism of F mo dules from F into X Hence
F
F x is a homomorphic image of F and thus lies in the variety generated by
F
F
F
Starting again with the near ring I D we therefore know that all I D
mo dules generated by a single element have ab elian addition
MODULES OF COMPOSITION ALGEBRAS
Let us now give a metho d that pro duces a congruence of the comp osition algebra
F from a congruence on the F mo dule X
Proposition Let F be a composition algebra and let X be an F module
Let Con X Then the relation F F de ned by
f g x X f x g x
is a congruence on F
If we start with an F group G and the congruence corresp onds to the F
ideal I then the congruence pro duced in this prop osition corresp onds to the
No etherian Quotient I G
F
Proof We have to prove that is a congruence relation on F To this end we x
x X and consider the mo dule homomorphism from F to X de ned by
x F
F X
x
f f x
It is easy to see that is a mo dule homomorphism Let k b e the kernel of
x x x
We then see that
k
x
x X
This implies that is a congruence of the mo dule F
F
For showing that is a congruence on the comp osition algebra F we still have
to show that for all f f g F we have
f f f g f g
F F
To this end we show that for all x X we have
f g x f g x
F F
Let x g x Then what we have to prove is the following
f x f x
Since f f the de nition of implies Condition
Sometimes we can obtain a congruence of an F mo dule in the following way
Proposition Let X be an F module that is contained in a congruence
permutable variety of F modules and let x x X Furthermore we assume
that for al l c X there is an f F such that
f x c and
f x c
Then the set
f f x f x j f F g
is a congruence on X
COMPOSITION ALGEBRAS
For an F group G the assumptions are ful lled if every constant mapping on
G lies in L F The congruence that we pro duce on G then corresp onds to the
2
ideal x x
1 F 2
The condition that there is an f with f x f x c could b e understo o d as
follows Each constant function can b e interpolated at fx x g by a function in
F Let us now give the pro of of Prop osition
Proof We x x x X Let
f f x f x j f F g
Since X lies in a congruence p ermutable variety we know from Lemma that
it is su cient to show that is a re exive relation that is invariant under the
application of binary p olynomial functions on X Re exivity is immediate For
c C take f F such that f x f x c then we have
c c f x f x
For showing that is closed under the application of p olynomial functions we
let f x f x f x f x and let p P X We have to show
p f x f x p f x f x
Now note that X is an F mo dule hence an algebra of type M F Thus there
are elements d d d X and a m ary term t of type M F such that
m
for all x X we have
p f x f x t d d d f x f x
X m
Now we choose g g g F such that we have
m
g x g x d for i m
i i i
Hence if we take f t g g g f f we have
F m
f x t g g g f f x
i F m i
Using the distributive law stated in Lemma we get for i
p f x t d d d f x f x
i X m i i
t g x g x g x f x f x
X i i m i i i
t g g g f f x
F m i
f x
i
Therefore Condition can b e rewritten as
f x f x
which is true by the de nition of Hence is a congruence relation on X
A result that stands at the basis of the theory of comp osition algebras and their
mo dules is the following density result It is again a result of the type
THE STRUCTURE OF COMPOSITION ALGEBRAS WITH CONSTANTS
Interpolation at places Interpolation at n places
Theorem Let A be an algebra in a congruence permutable variety and let
F be a subalgebra of M A If al l sub F modules of A are neutral then F is
F
dense in L F
Proof We show L F L F To this end let l L F and let D b e a nite subset
of A We prove that there is an f F with f j l j Our aim is to apply
D D
D
Prop osition We take G to b e the subalgebra of A with universe
F
G ff j j f F g
D
Hence G is a function algebra from D to A By assumption all subalgebras
F
of A are neutral and A lies in a congruence p ermutable variety Since l lies
F F
in L F l can b e interpolated at each subset of D with at most two elements by
a function in G Now Prop osition tells us that l can b e interpolated at the
whole set D by a function in G This means that there is a g G with g l j
D
But by the de nition of G we therefore have an f F with f j l j
D D
Hence F is really dense in L F
The structure of comp osition algebras with constants
We will now single out those elements of a comp osition algebra F that b ehave
similar to the constant functions in a comp osition algebra of functions The
following de nition go es back to Adl p
Definition Let F b e a comp osition algebra Then an c F is called a
constantly behaving element of F i
x F c x c
F
If F is a near ring then an element c is constantly b ehaving i c c
Proof It is obvious that the condition c c is necessary Now we assume
that c c Fixing f F we get c f c f c f c c
and hence c b ehaves constantly
We will collect the set of all constantly b ehaving elements of F in the set F
C
Proposition Given a composition algebra F the set F of al l constantly
C
behaving elements of F is a subuniverse of the composition algebra F
This means that for every near ring F the set ff j f x f for all x F g
is the universe of a subnear ring of F
Proof We have to show that F is closed under the op erations of the comp osition
C
algebra F Let b e rst a k ary fundamental op eration of F that is di erent
COMPOSITION ALGEBRAS
from Let k b e a natural number and let c c c F We have to
F k C
show
c c c F
k C
For that purp ose we x x F Then we have
c c c x c x c x c x
k F F F k F
c c c
k
This proves Condition It remains to show that F is closed under For
C F
that purp ose let c c F We x x F Then c F implies c c x
C C F F
c c which proves c c F
F F C
Proposition Given a composition algebra F the set F of al l constantly
C
behaving elements of F is a subuniverse of the F module F
F
This means that for a near ring F the set of elements F fc F j c
C
cg is closed under addition and also under multplication with elements of F from
the left So if c c then f c f c for all f F
Proof From Prop osition we know that F is closed under all op erations of
C
F For proving that F is closed under the mo dule op eration of F we x
C
f F c c Let x b e an element in F We then have
f c x f c x
F F F
f c
F
f c
Hence the constantly b ehaving elements of F are closed under multiplication from
the left with arbitrary elements of F
The aim of this section is to give a characterization of some comp osition alge
bras with at least two constantly b ehaving elements We succeed in giving a
description if the additive structure F b ehaves su ciently group like This
su cently group like b ehaviour is guaranteed if F has a Mal cev term What
we are actually going to do is to describ e the nite simple comp osition algebras
F for which F or F lies in a congruence p ermutable variety
F
So let s consider comp osition algebras F where F and its constantly b ehaving
elements gathered in C are sub ject to the following restrictions which we state
here for easier reference
Convention We assume that F and C are as follows
F a comp osition algebra of type C
C the set of all constantly b ehaving elements of F
ff F j x F f x f g
F
C the sub F mo dule of F with universe C
F
Furthermore we assume that C is not empty we have to claim this b ecause
otherwise the de nition of C go es wrong
THE STRUCTURE OF COMPOSITION ALGEBRAS WITH CONSTANTS
Proposition Let F and C be as in Convention If C is faithful then
the F modules F and C generate the same variety of algebras of type M F
F
Proof By de nition C is a subalgebra of F Hence C is in the variety gener
F
ated by F On the other hand the mapping de ned by
F
F M C
f f
with
f C C
c f c
C
is an embedding of the F mo dule F into the F mo dule M C C There
F
fore F lies in the variety generated by C
F
In particular Prop osition gives that C lies in a congruence p ermutable va
riety i F do es
F
Let us now relate simplicity of the comp osition algebra F to the simplicity of the
mo dule C
Proposition Let F and C be as in Convention Suppose that jC j
and that F is a simple composition algebra Then C is a faithful and simple F
module
Proof Let us rst show that C is faithful Let b e the homomorphism from F
to M C that was de ned in De nition Since F is simple every homo
morphism is either injective or has a one element range Let us therefore exclude
the second case Let c c b e two di erent constantly b ehaving elements of F
Since c x c for all x F the mapping c is given by
c C C
x c
This shows that c c
For proving that C is simple let us supp ose that C has a congruence with
Now we de ne a relation on F by
C C
f g c C f c g c
By Prop osition we know that is a congruence on the comp osition algebra
F Since F is simple must b e either or Since there are
F F C
c c C with c c and c c Using the de nition of we nd out
c c This implies that cannot b e the relation and therefore we have
F
But hence for all c c C we have
F
c c
COMPOSITION ALGEBRAS
Let c b e any element of C Then the de nition of gives
c c c c
Using the fact that c c are constantly b ehaving elements of F we get
c c
This shows that is equal to which contradicts the choice of
C
If the mo dule F is ab elian and lies in a congruence p ermutable variety sim
F
plicity of F means that all elements of F are b ehaving constantly
Proposition Let F and C be as in Convention Suppose that jC j
and that F is a simple composition algebra If F is abelian and lies in a
F
congruence permutable variety then F C
For a near ring F we can interprete this prop osition as follows The fact that
+
F is ab elian means that F satis es the identites f f f f and f
F 1 2 2 1 1
f f f f f f f The set of a ne mappings on a vector space
2 3 1 3 1 2 1
is an example of such a near ring If such a near ring F has more than one
element satisfying f f which just means that F is not a zero symmetric
near ring then F cannot b e simple unless all of its elements b ehave constantly
which means that the near ring satis es the identity x y x
Proof We will rst show that C gives rise to a congruence of the F mo dule F
F
Let d b e a ternary Mal cev term of type M F Using the idea of H P Gumm s
characterization of ab elian algebras in congruence p ermutable varieties Gum
Ihr Satz we take an element C and de ne two op erations and
on F by
+
x y x y d
F
F
+
y d y
F
F
It is known that F is an ab elian group Let b e the binary op eration
on this group that maps f g to f g Actually is given by f g
+
f g Now for all k N p P F and x x x y y y d
k F k k
F
F
F we have
p x x x p y y y p x y x y x y
k k k k
We de ne a relation on F by
f g f g C
We shall now prove that is a congruence relation on F Since F is in a
F F
congruence p ermutable variety it is su cient to show that is a re exive relation
that is invariant under the application of binary p olynomial functions on F
F
For re exivity we observe that f f and was chosen to b e in C hence we
THE STRUCTURE OF COMPOSITION ALGEBRAS WITH CONSTANTS
have f f For showing that is closed under the application of p olynomial
functions let f g f g and let p P F We have to show
F
p f f p g g
which is equivalent to
p f f p g g C
We notice that there must b e m N and a m ary term t of type M F and
elements h h h F such that for all x x F we have
m
+
p x x t h h h x x
m
F
F
So Condition can b e rewritten as
+ +
h h h g g C h h h f f t t
m m
F F
F F
Using Condition this condition b ecomes
+
f g f g C t
F
F
We know that and all f g are in C Since C is a subalgebra of F it
i i F
is closed under the application of term functions which proves Condition
Therefore is a congruence relation on F
F
In the next step we show that F is also a congruence relation on F To this
F
end let f g F with f g and let h F We have to show
f h g h C
F F
Using the characterization of this b ecomes
+
f h g h C d
F F
F
F
Since is right distributive with resp ect to all term functions and since h
F F
b ecause C this can b e rewritten as
+
d f g h
F
F
F
This gets
f g h C
F
But we know that f g C and hence f g h f g C Altogether we
get that is a congruence relation on F Since F is simple and since all elements
in C are congruent mo dulo we have This implies in particular that
F
we have
f F f C
Since C this gets f C and so we have C F
COMPOSITION ALGEBRAS
In K Kaarli Kaa presented a characterization of nite simple near
rings with constants Theorem shows that his result carries over to all V
comp osition algebras where V is congruence p ermutable For stating the result
we need the following comp osition algebras
Definition Let A b e an algebra of type and let b e an equivalence
relation on A Then we de ne
M A fm A A j a a A a a m a m a g
Proposition M A is a subuniverse of M A
Proof Obvious
Definition We let M A b e the subalgebra of M A with universe
M A
Theorem Let F and C be as in Convention We assume jC j
If F is simple and F lies in a congruence permutable variety then there is a
F
subalgebra F of M C with the fol lowing properties
F is isomorphic to F
F is dense in M C where is the equivalence relation on C that is
de ned by c c f F f c f c
F F
+ +
g If Con C and then f
C C
Let us restate Theorem for near rings In this sp ecial case it reads as
follows Let F b e a simple near ring that is not zero symmetric and let C
fc F j c cg Then F is isomorphic to a subnear ring of the near ring of
all selfmaps on C This subnear ring is dense in
fm C C j c c C f F f c f c m c m c g
1 2 1 2 1 2
Proof By Prop osition we know that C is a faithful and simple F mo dule
Hence the mapping which is de ned as in De nition as
F M C
f f
where
f C C
c S f c
C
is an embedding of F into M C Let F F Obviously F satis es
let us now attack the pro of of We have to distinguish two cases as to whether
C is an ab elian algebra or not
Case C is not abelian Let us rst de ne an F mo dule C which will b e the
translation of C into an F mo dule We let the universe of C b e C and de ne
THE STRUCTURE OF COMPOSITION ALGEBRAS WITH CONSTANTS
the op erations for the op eration symbols in F such that
C C
Furthermore we de ne a mo dule op eration of F on C by
f c f c for f F c C
The F mo dule C that arises from this op eration is again simple and not
ab elian For an f F the mo dule op eration of f is describ ed by the following
equation
S f c f c f c
0
F
C
We will now show that
F is dense in L F
To this end we show L F L F Let l L F and let D b e a nite subset
of C We prove that there is an f F with f j l j Our aim is to apply
D D
Prop osition First of all we observe that C has no prop er subalgebras Let
S b e a subalgebra of C and let c C s S Then Condition and the fact
that c is a constantly b ehaving element of F give
c c s
F
S c s
0
C
and hence c S Therefore all subalgebras of C are neutral Furthermore by
the assumptions and Prop osition C lies in a congruence p ermutable variety
Now let us construct the following function algebra G from D to C We take
the universe of G to b e
G ff j j f F g
D
D
We have to show that G is the universe of a subalgebra of C The mapping
F G de ned by
F G
f f j
D
D
0
F to C Therefore G F is is a homomorphism from the F mo dule
F
D
a subuniverse of C Since l L F the function l j can b e interpolated at
D
every subset of D with no more than two elements by a function g G Now
we can apply Prop osition and using that D is nite obtain that l j is an
D
element of G Hence there is an element f F with f j l j This proves
D D
Condition Now we prove
L F M C
where c c f F f c f c For let l L F and let
c c C such that c c Then there is an f F such that f c l c
and f c l c Hence l c l c So we have l M C
COMPOSITION ALGEBRAS
For let m M C and let c c C If c c we know m c m c
Hence m c is an element of F that satis es m c c m c c
F
m c for all c C Therefore m c interpolates m at fc c g If c c
we de ne the relation on C by
f f c f c j f F g
We want to show that is a congruence relation on C Since C lies in a
congruence p ermutable variety we can apply Prop osition Let us check
whether the assumptions of Prop osition are ful lled for x c x c
We have to test whether for each c C there is an f F such that S f c
0
C
S f c c The following calculation shows that f c is a suitable
0
C
solution We have
c c c c c c
F F
S c c S c c
0 0
C C
f c f c
Therefore Prop osition yields Con C
Since c c there is an element f F with f c f c This means that
0 0
and since C is simple thus has to b e equal to cannot b e the relation
C C
But this shows that f f c f c j f F g is equal to C C and therefore the
mapping m can b e interpolated at fc c g by a function in F This concludes
the pro of of of Condition Hence Condition and Condition
prove claim of Theorem in the case where C is not ab elian Let us now
attack the pro of of claim in the case that C is ab elian
Case C is abelian Using Prop osition we notice that F lies in a congru
F
ence p ermutable variety Furthermore we observe that C is ab elian By Ihr
Aufgab e we know that the ab elian algebras of a congruence p ermutable
variety form a subvariety Hence the F mo dule F which lies in the variety gen
F
erated by C is ab elian Applying Prop osition we get F C The relation
on C is therefore equal to C C and so M C is the algebra of all constant
mappings on C It is obvious that then F M C with c c c for all
c c C is an isomorphism b etween F and M C This proves claim of
Theorem in the case where C is ab elian
Now we have to prove claim of Theorem Let b e a congruence on C
with We show that is a congruence on C To this end let c c
and let f F We have to show
S f c S f c
C C
But S f c S f c f c f c Since c c we have
F F
C C
f c f c and therefore in particular
F F
f c f c
F F
THE STRUCTURE OF COMPOSITION ALGEBRAS WITH CONSTANTS
So is a congruence on C By Prop osition has to b e either or
C C
which we had to prove
For the case of A b eing a group the simplicity of M A has b een investigated
by P Fuchs Fuc We give a universal version of one of his results
Proposition Let A be an algebra in a congruence permutable variety and
let be an equivalence relation on A such that
Con A and implies f g
A A
There are only nitely many equivalence classes modulo
Then one of the fol lowing alternatives holds
M A is simple
A is abelian and there is a binary polynomial function P A such
that A is the additive group of a vector space over GF and is the
equivalence modulo a subgroup of A of index
Sp ecial instances of this Prop osition are the near rings M Z and M Z
C
The rst near ring is simple The near ring of all mappings on the two element
group is not simple and really as the Prop osition promises the equivalence
relation is the equivalence mo dulo a subgroup of index of Z This subgroup
is trivial
Proof Without loss of generality we assume jAj Let F M A and
let A b e the F mo dule with universe A and F op erate on A in the usual way
F
First of all we show
A is simple
F
Let Con A The relation is clearly also a congruence
F A A
F F
on A and therefore the assumptions give Hence there are a a A with
a a and a a Let b b b e arbitrary elements of A Then there
is a mapping f M A with f a b and f a b But since is a
So A is congruence on A this implies b b and therefore
A F F
F
simple
Case A is not abelian Let T b e a transversal of A mo dulo First of all we
F
observe that the mapping de ned by
T
F A
f f j
T
T
is a homomorphism from the F mo dule F to the F mo dule A It is easy to
F F
see that is bijective Altogether we see that is an isomorphism of F mo dules
Now we supp ose that is a congruence of the comp osition algebra F Therefore
is also a congruence on F Hence f f g j f g g is a
F
T
congruence on A F
COMPOSITION ALGEBRAS
Now we notice that A is as a simple non ab elian algebra also neutral Further
F
T
more by the assumptions T is nite Hence Prop osition yields that A
F
is skew free By the fact that A is simple we obtain that there is a subset U
F
of T such that
g j f g f j
U U
T
If U T then is the equality relation on A If U then is
F
T T
the relation A A T Therefore we can assume
A
F
U T
Therefore there we can choose an element an element t T n U There are
T
functions l l A such that l t l t and l l Hence there
F
is precisely on extension of l to a mapping f A A such that f F and
f j l In the same way we construct f such that f F and f j l We
T T
know
f f
Now let g b e the constant mapping de ned by
g A A
a t
Obviously g F Since is a congruence on F we have
f g f g
F F
This implies that
f g j f g j
F U F U
Supp ose that U is not empty Then for u U we have
f g u f t
F
f t
f g u
F
This is a contradiction hence the assumption that U is not empty was wrong
Altogether F is simple
Case A is abelian By H P Gumm s characterization of ab elian algebras
F
Prop osition there are an element A and a binary op eration in
P A on A such that A is an ab elian group with neutral element and
F
every element f F satis es
x y f y x f x f y f
It is the contents of the following result Lemma that in this case either
A A or A is the additive group of a vector space over GF and
the relation mo dulo a subspace of co dimension
THE STRUCTURE OF COMPOSITION ALGEBRAS WITH CONSTANTS
In the case that A A all relations on A are contained in Hence the
assumptions give that the only congruences on A are and So A is simple
A A
Now we notice that M A is isomorphic to A and therefore M A is
simple
Lemma Let A be an abelian group and let be an equivalence relation
on A Every mapping in M A is a ne if and only if one of the fol lowing
alternatives holds
A A
A is the additive group of a vector space over GF and there is a
subspace R of A of codimension such that a a i a a R
Proof Let us rst prove the only if part If A A there is an element
x A with x If x x x we have a mapping f M A that satis es
f f x f x x x The linearity condition in Condition then
gives x x x and hence x which contradicts the fact that x is not
in So we know that x x x Hence for any z A there is a mapping
f M A with f f x x and f x z The linearity condition
in Condition now gives z z which implies that A is a group
of exp onent Therefore A is the additive group of a vector space over the
eld GF Now we de ne R We rst show that R is closed under
Supp ose that there are x y R such that x y R Then the function
f M A with f f x f y and f x y x is not a ne which
contradicts the assumptions hence R is closed under Now we prove
a a A a a a a R
Let us rst establish the following fact
a a t A a a a t a t
It is su cient to prove
a a t A a a a t a t
Supp ose that a a and a t a t Then either a a t
or a a t Without loss of generality we assume a a t Let
a A a Now f M A satisfying f a f a f a t
and f a t a is not a ne This proves Condition and hence
Condition Therefore is really the equivalence induced by the subspace
R
Let A b e the factor group of A mo dulo For any f M A the mapping
f A A
a f a
COMPOSITION ALGEBRAS
is well de ned Since f M A is a ne also f is a ne It is easy to see that
f j f M A g M A f
So we see that every mapping on A is a ne which implies jA j and
hence R has co dimension in A
For the if part we asume that alternative o ccurs Let f b e any mapping
from A to A such that A is constant on each of the cosets of R We may assume
f Then let a b b b e a basis of A with a R b R for all i We
i
de ne a linear map l by l a f a and l b for all i We see that l a
i
l a f a and l is constant on the cosets of R Hence l is equal to f But l is
clearly a ne and thus so is f
Concerning Prop osition we should like to remark that for an ab elian algebra
A and a relation satisfying the second alternative of Prop osition the
comp osition algebra M A is never simple
Proposition If A is abelian lies in a congruence permutable variety and
there is a binary polynomial function P A such that A is the additive
group of a vector space over GF and is equivalence modulo a subgroup of
A of index then M A is not simple
As an example we consider the subnear ring of M Z Z that consists of all
a
a
0
functions m that satisfy m m for all a b c Z Then the
2
c
b
0
2
resulting near ring it contains elements is not simple
Proof Let F M A and supp ose that F is a simple comp osition algebra
Since every mapping in F is a ne on A the algebra A is ab elian Hence
F
also F which lies in the variety generated by A is ab elian But now Prop o
F F
sition implies that every f F is a constantly b ehaving element of F But
it is easy to see that this implies that every element in M A is a constant
mapping This contradicts the fact that there are two classes mo dulo
The following result generalizes Adl Theorem It tells us that those com
p osition algebras of functions on a nite simple non ab elian algebra A in a con
gruence p ermutable variety that contain all constant functions on A are of the
form M A for some equivalence relation on A
Proposition Let A be a nite simple non abelian algebra in a congruence
permutable variety V Then the mapping de ned by
f j is equiv rel on A g fF j M A F M A g
C
M A
is a bijection
Obviously implies M A M A
THE STRUCTURE OF COMPOSITION ALGEBRAS WITH CONSTANTS
There are exactly P di erent subnear rings of M A that contain all the
constant functions on A Here P denotes the number of ways to partition
a element set and A is the alternating group on ve letters
Proof By the remarks b efore De nition we see that is really a sub
algebra of M A for every equivalence on A It is obvious that is injective
For proving that is surjective let F b e an arbitrary subalgebra of M A with
M A F We rst note that the A has only one subalgebra namely A
C F F
itself Applying Theorem we know that F is dense in L F and hence by
the niteness of A we have F L F But for function algebras that contain
all constant functions Lemma gives a characterization of the elements in
L F Since A is simple the description of L F given in Lemma b ecomes
particularly easy Actually what we obtain is
L F fl M A j a a l a l a g
A A
Note that there is no reason to study b ecause l a l a is a
A A
tautology If we set we get
A
L F M A
This shows that is also surjective
Lemma Let A be an algebra in a congruence permutable variety and let let
F be such that M A F M A Then for each congruence Con A we
C
de ne a relation by
a a f F f a f a
Then for al l Con A the relation is an equivalence relation and L F is
given by
L F fl M A j Con A a a A
a a l a l a g
Proof It is immediate that is an equivalence relation For proving the
characterization of the elements of L F we observe that of Condition is
obvious For let l b e in the right hand side of Condition For showing
that l is in L F let a a b e arbitrary elements of of A If a a the mapping
a l a is as a constant mapping an element of F and interpolates l at
fa a g fa g If a a let
f f a f a j f F g
Since each constant mapping on A lies in F Prop osition yields that is a
congruence on A and hence in particular on A By the de nition of we
F
have a a Therefore we have
l a l a
COMPOSITION ALGEBRAS
Thus we have an f F such that f a l a and f a l a This proves
l L F
Let us remark that Prop osition can b e generalized to in nite simple non
ab elian algebras A in congruence p ermutable varieties In this case each function
algebra that contains all constant functions on A is dense in a function algebra
of blo ckwise constant functions
Prop osition also shows that for a nite simple non ab elian algebra A the
interval fS j M A S M A g of the subalgebra lattice of M A is anti
C
isomorphic to the lattice of all equivalence relations on A Finite elds are
examples of simple non ab elian algebras For those we obtain precisely Adler s
result that for a nite eld D all sub comp osition rings of M D that contain all
constant functions on D are of the form M D for some equivalence on D
Furthermore Prop osition yields that every such comp osition ring is simple
For the eld GF the comp osition ring of all mappings M GF is
simple The near ring M GF however is not
The structure of some comp osition algebras with a left identity
Zero symmetric comp osition algebras In this paragraph we inves
tigate comp osition algebras that have a left identity with resp ect to comp osition
The element e F is a left identity of the comp osition algebra F if we have
e f f for all f F It is easy to see that a left identity do es by no means
F
have to b e unique We say that an F mo dule X is e unital i e x x for all
x X
We will furthermore assume that there is a nullary op eration symbol o in F First
of all we have
Proposition Let F be composition algebra over the type F and let o
be a nul lary operation symbol in F Then we have
f F o f o
F F F
For al l f F the element f o is a constantly behaving element of F
F F
For each F module X and for al l x X we have o x o
F X
For a near ring F a left identity e is an element satisfying e x x for all
x F If we consider a near ring as an algebra F then is a nullary
op eration And really As we have asserted in Prop osition x holds
in F f b ehaves constantly for every f F and if we let F op erate by as
an N group on a group G then g for all g G
Proof For we recall from Prop osition that x x f is an endomor
F
phism of F This implies that it leaves the result of nullary op erations xed
COMPOSITION ALGEBRAS WITH LEFT IDENTITY
Claim follows from by the following calculation We x f x F and get
f o x f o x
F F F F F F
f o
F F
Hence f o is a constantly b ehaving element of F
F F
Claim is proved by the remark after Prop osition
Since the case that F has at least two constantly b ehaving elements has already
b een treated in the previous section we shall here restrict ourselves to the case
that F contains precisely one constantly b ehaving element namely the element
o Note that by Prop osition the result of a nullary op eration is always
F
a constantly b ehaving element of F If we want to have only one constantly
b ehaving element we must also have f o o for all f F
F F F
This motivates that in the following section we study V comp osition algebras F
that are sub ject to the following restrictions We list these restrictions here for
easier reference
Convention
V is a congruence p ermutable variety of
algebras of type F
There is a nullary op eration symbol F
F is a V comp osition algebra
We have an element e F with f F e f f
F
For all f F we have f
F F
A comp osition algebra that satis es of Convention is called zero symme
tric
So what we are going to study here are zero symmetric near rings with a left
identity But not only those zero symmetric lo op near rings and zero symmetric
comp osition rings with identity will b e describ ed using the same techniques
In the sequel we will abbreviate b oth and for an F mo dule X
F X
simply by Let us assume that F satis es all conditions of Convention
Then in particular F is a zero symmetric comp osition algebra In this case we
have
f
for every F mo dule X So by Lemma f g F is a subuniverse of the
mo dule X This one element sub F mo dule of X is denoted by
Two density results Let us rst give a density result for simple F
mo dules This result is an immediate application of HH Prop osition
COMPOSITION ALGEBRAS
Theorem We assume that F is as in Convention and X is an F
module with reduct in V that satis es the fol lowing properties
X is faithful
X is e unital
Al l sub F modules of X with more than one element are simple and not
abelian
We de ne
Y is the collection of al l sub F modules of X
i i I
For i j I S is the set of al l isomorphisms from Y to Y
ij i j
S
M X fm X X j i I m Y Y
i i
S
ij
i j I S x Y m x m x g
ij i
S S
M X is the subalgebra of M X with universe M X
S S
ij ij
Then there is a subalgebra F of M X such that F is isomorphic to F and F
S
is dense in M X
S
ij
For near rings this result reads as follows
Theorem We assume that F F is a zero symmetric near ring
with left identity and that G is a faithful unital F group such that F is
primitive on al l sub F groups of G Furthermore we assume that for every
x G with x the structure
ff j j f F g
F x
is not a ring In other words this means that for every x there are f f f
1 2 3
in F such that
f f x f f x
1 2 2 1
or
f f f x f f f f x
1 2 3 1 2 1 3
Under these assumptions F is dense in the near ring of al l mappings on G that
preserve al l sub F groups of G and al l F isomorphisms between them
The precise meaning of preserve is to b e read in number of the statement of
Theorem
Our use of the word preserve follows the following idea A near ring N of
0
functions on an F mo dule X preserves a prop erty of X if the N mo dule X
+
that we obtain by expanding X with the elements of N as unary op erations
still satis es the same prop erty Some near ring constructions may b e seen as
closure op erations with resp ect to mo dule prop erties Starting with a near ring
F op erating on the F group X the centralizer near ring of all functions that
commute with all F endomorphisms of X is the largest near ring N such that
0
the N mo dule X has the same endomorphisms as the F mo dule X The near
ring of all compatible functions on a group X may seen as the largest near ring
such that the resulting mo dule has the same congruences as X
COMPOSITION ALGEBRAS WITH LEFT IDENTITY
One example of a near ring that satis es the assumptions of Theorem is the
subnear ring of M Z with universe fm Z Z j m f g f gg
9 9 9
Proof We know that X is faithful Hence the mapping which is de ned as in
De nition as
F M X
f f
where
f X X
x S f x
X
is an embedding of F into M X We take F F Obviously F is
isomorphic to F Let us rst de ne an F mo dule X which is the translation
of the F mo dule X into an F mo dule of the F mo dule X We let the universe
of X b e X and de ne the op erations for the op eration symbols in F such that
X X
Furthermore we de ne a mo dule op eration of F on X by
f x f x for f F x X
The F mo dule X that arises from this mo dule op eration has the same subuni
verses congruences and the same commutator op eration as the F mo dule X
We will rst show that
F is dense in L F
We see that all sub F mo dules We want to apply Theorem for A X
of X are neutral b ecause they either have one element or they are simple and
not ab elian Therefore Theorem yields Equation
Now we prove
S
L F M X
S
ij
For we observe that clearly every mapping in L F resp ects F isomorphisms
b etween sub F mo dules of X Furthermore it also preserves sub F mo dules
S
For let m M X We take x x X arbitrary but xed
S
ij
Case x x In order to nd the function that interpolates m at fx g it is
su cient to prove
S
fm x j m M X g ff x j f F g
S
ij
To this end we observe that ff x j f F g is a subuniverse of X hence equal
to a Y for an i I Since X is e unital we have e x x and therefore
i
0
x Y Now we are ready to prove the subset relation stated in Equation
i
0
S
By the de nition of M X we know that m x lies in Y Therefore there
i
S
0
ij
is an f F with f x m x
COMPOSITION ALGEBRAS
Case x x Let
Y ff x j f F g
i
Y ff x j f F g
j
Now we show that the relation
f f x g x j f F and f x g x g
is a congruence on Y We do so using Lemma First of all we note that
j
is re exive Let y b e arbitrary in Y By the de nition of Y there is an g F
j j
such that g x y Since trivially g x g x we know that g x g x
is in which means y y Furthermore is closed under the application
of binary p olynomial functions We let f f and g g b e elements in F such
that f x g x for i and let p P Y We have to show
i i j
p f x f x p g x g x
Now note that Y is an F mo dule hence an algebra of type M F F
j F F
Thus there are elements d d d Y and a m ary term t of type
m j
F such that for all y y Y we have
F F j
p y y t d d d y y
Y m
j
Now we choose h h h F such that for all i m we have
m
h x d
i i
Hence if we take u v F de ned by
u t h h h f f
F m
v t h h h g g
F m
It is now easy to see that we have u x v x Hence u x v x But
this proves Equation Hence the relation de ned in Equation is a
congruence relation on X
But since Y is simple there is not much choice for Actually in b oth cases
j
S
we will b e able to interpolate the given m M X Note that we know
S
ij
m x Y and m x Y
j j
Case Since m x Y there is a function f F such that f x
Y i
j
m x We know
f f x f x j f F and f x f x g Y Y
j j
Now let f f F b e such that
f x f x
f x m x
f x f x
COMPOSITION ALGEBRAS WITH LEFT IDENTITY
Let d b e a ternary term of the type M F of all F mo dules such that d is a
X
Mal cev function on X Then we de ne f F by
0
f f f f d
F
We have f x d f x f x f x f x m x and f x
X
d f x f x f x f x m x Hence f is a function that interpo
X
lates m at fx x g
Case In this case we know that f x f x implies f x f x
Y
j
This means that the function
Y Y
i j
f x f x
is well de ned It is easy to see that is a homomorphism of F mo dules
obviously from Y to Y By the de nition of Y is surjective Since Y is
i j j i
simple either has a one element range or is injective
Case has a one element range In this case we have Y f g Since is
i
surjective this means Y f g We also know that e x x and hence
j
x Y Thus we have x Now let f F b e any mapping such that
j
f x m x We know that f x f Furthermore m
b ecause m preserves all sub F mo dules of X and f g is a subuniverse of the
F mo dule X Therefore f interpolates m at fx x g
Case is injective In this case we have S Now let f F b e such that
ij
f x m x Then we have
f x f x
m x
m x
m e x
m e x
m x
This shows that f interpolates m at fx x g
Altogether we have proved Theorem
We will now see what this theorem yields if we have no prop er sub F mo dules
with more than one element Before stating this theorem we need a universal
version of centralizer near rings
Definition Let A b e an algebra of type F and let S b e a set of
endomorphisms on A Then
Let
M A fm A A j S a A m a m a g S
COMPOSITION ALGEBRAS
Let M A b e the subalgebra of M A with universe M A
S S
Theorem If F is as in Convention and X is an F module with reduct
in V that satis es the fol lowing properties
X is faithful
X is e unital
X has only two sub F modules namely X and
X is simple and not abelian
and S is the set of al l endomorphisms of X Then we have
There is a subalgebra F of M X such that F is isomorphic to F and F
is dense in M X
S
S S f g where
a X X x
b S is a group of automorphisms on X
c For al l S
d For al l S and for al l x X x x x id
This is the generalized version of Wielandt Betsch s Density Theorem for
primitive near rings Bet So this result having proved already fruitful in
near ring theory can b e used to lo op near rings comp osition rings comp osition
near rings For comp osition rings we milk this result in Chapter
Proof Number is an immediate consequence of Theorem Let us now
prove the claims of For proving that S is a group of automorphisms on
X we observe that if we have S X is the universe of a sub F mo dule
of X This holds b ecause the homomorphic image of a subalgebra is always
again a subalgebra Hence is surjective By the simplicity of X we get that
is injective This proves that S is a group of automorphisms on X For
proving that we observe that f g is the only prop er subuniverse of X
But since f g is again a subuniverse of X we have For the xed
p oint freeness of S we observe that the set fx j x xg is the universe of a
sub F mo dule of X Hence either is the identity function or only gets xed
by
We call an algebra A regular i every congruence Con A is uniquely
determined by One can see from GU Corollary that one condition
that garantees that every algebra in a V is regular is that there are terms
of type F such that for every algebra A V the algebra A is a
A A A
group Hence varieties of groups are always regular If X is regular and has
only the two subuniverses f g and X then X is simple Supp ose that Con X
Since is a subuniverse of X we know that f g or X In the
rst case regularity gives in the second case we see that all elements
X
of X are congruent to mo dule and therefore we have X
COMPOSITION ALGEBRAS WITH LEFT IDENTITY
The result in Theorem implies that S acts as a regular p ermutation
group on X n f g An obvious consequence of this fact is that if X is nite then
jX j is divisible by jS j If X is a group and jS j then the fact that it
admits a nontrivial regular group of automorphisms has imp ortant consequences
on the structure of the group in that case the group X is nilp otent by Thomp
son s Theorem which was originally proved in Tho it can e g also b e found
in Rob Theorem
Definition Let A b e an algebra in a variety V with the nullary op eration
Then an automorphism of A is called xed point free i for all a A
A
we have a a a A group of automorphisms on A is called a
A
regular group of automorphisms i all its elements are either xed p oint free or
the identity automorphism
Simple comp osition algebras are dense in centralizer comp osi
tion algebras Theorem allows us to determine all nite simple V comp o
sition algebras among those satisfying Convention
Theorem Let F be as in Convention Furthermore we suppose that
F is nite and simple F is not abelian and jF j Then there is an algebra
F
A V and a regular group of automorphisms with universe S on A such that F
is isomorphic to M A where S S f g
S
Proof We consider the class M of all e unital F mo dules Since F M
F
there is a nite element in M and therefore there is an element X M which
is minimal among all element M M with jM j We set
A X
S f j is an automorphism of Xg
S S f g
First of all we observe that by the minimality of jX j X is a simple F mo dule
and the only sub F mo dules of X have universe f g or X By the fact that X
is e unital and F is simple it follows that X is faithful Hence Prop osition
yields that F lies in the variety generated by X Since the ab elian algebras
F
in a congruence p ermutable variety form a subvariety this means that X cannot
b e ab elian If it were ab elian then so would b e F which is excluded by the
F
assumptions Hence all the assumptions of Theorem are ful lled This
theorem now yields that F is isomorphic to a dense subalgebra of M X
S
M A But since A is nite density implies equality which proves the
S
result
Finite centralizer comp osition algebras from regular automor
phisms groups are simple Centralizer comp osition algebras that arise from
COMPOSITION ALGEBRAS
regular groups of automorphisms with nitely many orbits are simple But b efore
we need the following easy observation ab out centralizer comp osition algebras
Proposition Let A be an algebra in a variety V with the nul lary op
eration Let S be a regular group of automorphisms on A as de ned in
A
De nition and let S S f g Let T be a transversal through the orbits
of S on A Then for every mapping l T A there is precisely one mapping
f M A with f j l
S T
Proof We de ne f t l t for all t T S The fact that S
is regular shows that f is well de ned The uniqueness of f follows from the
prop erty f M A
S
Note that this result also yields M A x A for all x
S
Definition Let V b e a variety as in Convention i e congruence
p ermutable and with constant op eration let A b e a nite algebra in V and
let S b e a regular group of automorphisms on A Then a congruence Con A
is S orbit contained i every congruence class of is contained in a single S
orbit of A
Note that it is very hard for a congruence to b e S orbit contained If is
S orbit contained then we have automatically f g In the case that A is
regular this implies
A
Proposition Let V be a variety as in Convention i e congruence
permutable and with constant operation let A be a nite algebra in V and let
S be a regular group of automorphisms on A such that
S has only nitely many orbits on A
The only S orbit contained congruence relation on A is
A
Let S S f g Then M A is a simple V composition algebra
S
For every regular group S of automorphisms on a nite group G the near ring
M G is simple MS S is given by S f g
S
Proof Let F M A and let A A n f g
S
We distinguish two cases
Case A is not abelian
F
First of all we show that the F mo dule A is simple Let Con A
A F F
F
Since Con A it is not S orbit contained Therefore there is a pair a a
such that a and a lie in di erent orbits of S on A Now Prop osition gives
that a and a can b e sent two any two elements b and b of A via a function in
F This shows that
A F
COMPOSITION ALGEBRAS WITH LEFT IDENTITY
Now we attack the pro of of the simplicity of F We supp ose that there is a
congruence Con F with f g Let T b e a transversal through the
F F
orbits of S on A Then the mapping de ned by
F T
f f
with
f T A
t f t
T
is a homomorphism of F mo dules from F to A By Theorem and
F F
Prop osition this mapping is surjective Since every mapping in F is
uniquely determined by its values on T again by Prop osition the mapping
is also injective Altogether we see that is an isomorphism of F mo dules
We know that is a congruence on F and therefore it is also a congruence on
T
F Hence f f g j f g g is a congruence on A
F F
Now we notice that A is as a simple non ab elian algebra also neutral Hence
F
T
Prop osition and the fact that T is nite yield that A is skew free By
F
the fact that A is simle we obtain that there is a subset U of T such that
F
g j f g f j
U U
T
If U T then is the equality relation on A If U then is
F
T T
the relation A A T Therefore we can assume
A
F
U T
Therefore there we can choose an element u U and an element t T n U
T
There are functions l l A such that l t l t and l l By
F
Prop osition there is precisely on extension of l to a mapping f A A
such that f M A and f j l In the same way we construct f such that
S T
f M A and f j l We know
S T
f f
Now let g b e a mapping in M A such that g u t Since is a congruence
S
on F we have
f g f g
F F
This implies that
f g j f g j
F U F U
But we have
f g u f t
F
f t
f g u
F
COMPOSITION ALGEBRAS
This is a contradiction hence the assumption that there is f g was
F F
wrong Altogether F is simple
Case A is abelian First of all let d b e a Mal cev term for the variety V Since
F
A is ab elian the algebra A where is de ned by
F
a a d a a
A
is an ab elian group Since A is ab elian and f we have
F
f a a f a f a
Using this last equality and the fact that A is faithful we get that F is
F F
a ring Now the following near ring theoretic result Prop osition yields that
F is a division ring and hence simple But since and are congruence
F F
preserving functions on the comp osition algebra F also F is simple
Proposition Let G be a group with jGj let S be a regular group of
group automorphisms on G and let S S f g Then the near ring M G
S
M G is a ring i it is a division ring
S
Proof The if part is immediate For the only if part let F M G We
S
assume that F is a ring Then let D b e the set of all endomorphisms of the
F mo dule G Then we have F M G On the other hand since S D we
F D
have M G F Hence we have
D
M G F
D
Since F is a ring and F g G for all g G n f g the group G is ab elian cf
Prop osition and every mapping f F ful lls
f g g f g f g for all g g g
This also follows from Prop osition
From these facts we can infer that the sum of two F mo dule endomorphisms is
again an F mo dule endomorphism i e for all s s D not only s s but
also s s lies in D
By the fact that the F mo dule G is simple and has no subuniverses apart from
F
G and f g all mappings in D n f g are invertible hence D D is a
division ring We also note that G b ecomes the universe of a vector space over
D if we de ne d g d g for all d D and for all g G We call this vector
space G
D
Since M G is a ring Theorem of MvdW yields that the vector space
D
G is of dimension at most one
D
Now from jGj we get dim G hence M G is precisely the set of all
D D
vector space endomorphisms of G From this we see that M G is a division
D D ring
COMPOSITION ALGEBRAS WITH LEFT IDENTITY
The ab elian case of Prop osition also follows from classical near ring theory
cf Pil Theorem c MS Actually the whole Prop osition is
well known for near rings as is by the way most of the material of this chapter
If G is nite then Prop osition can b e proved using Pil Theorem
b and Pil Theorem which is due to MPS
Another condition that makes a comp osition algebra simple
We will now give an application of these Prop ositions Supp ose that F is a V
comp osition algebra that satis es all conditions of Convention Furthermore
we supp ose that the variety V is regular Let us rst prove the following easy
lemma
Lemma Let F be a V composition algebra such that al l conditions in Con
vention are ful l led Furthermore we suppose that V is regular Let X
be an e unital F module whose only sub F modules are and X Then X is a
simple F module
The lemma retells the old story that a zero symmetric near ring with identity
that is primitive on G is automatically primitive on G
Proof We choose x The case X f g is obvious Since X F x the
mapping r F X f f x is surjective Furthermore r is a homomorphism
x x
from F to X Since F is a V comp osition algebra we know that F lies in V
and therefore its homomorphic image X lies in V Therefore each congruence
on X is uniquely determined by its class Using this fact we want to show
that X is simple Let b e a congruence on X It is easy to observe that is
a sub F mo dule of X If f g the fact that is also a congruence on X
+
If x and x and the regularity of X yield that
X
X
then the fact F x X gives that every element in X is congruent to mo dulo
This implies that Therefore X is simple
X
Now we get the following result
Proposition Let F and V be as in Convention and let X be a nite
faithful e unital F module that satis es F x X for al l x We assume that
every algebra in V is regular Then F is simple
Every nite primitive zero symmetric near ring with identity is simple In
near ring theory one gives the same pro of One shows that it is a centralizer
near ring constructed from a regular group of group automorphisms and then
one shows that such centralizer near rings are simple at least if they are nite
Proof From Lemma we obtain that X is simple If X is not ab elian The
orem forces F to b e a centralizer comp osition algebra and Prop osition
+
f f is makes it simple If X is ab elian then F with f f d
F F
COMPOSITION ALGEBRAS
a primitive ring with left identity and therefore simple by the blessings of ring theory
CHAPTER
Tame comp osition algebras on groups
In Sco Stuart Scott p oses six problems The rst and the fourth problem
b elong to the theory of tame N groups In this section we give a solution to
problem four
Since the word tame seems to app ear in many branches of algebra take e g
tame congruence theory HM the tame case in the decomp osition of p oly
nomials we will start with a short description of tame near rings Again
we will study subalgebras of the comp osition algebra M A of all functions on
an algebra A However in order to make our theory work we need that A has
at least a group structure and we will therefore only deal with subalgebras of
M V where V is an group
Tame near rings
In the following discussion we need several sets of functions on the group V
We start from a sub comp osition algebra F of M V From F we can form the
comp osition algebra of all functions that can b e interpolated at any n element
subset of V by a function in F and we obtain the comp ostion algebra L F By
n
L F M V we abbreviate the set of those selfmaps s on V that can b e written
n C
in the form s l c where l L F and c is a constant mapping on V What
n
we also need are the unary p olynomial functions of the F mo dule V which we
F
denote by P V In the following we will assume that F ful ls some particular
F
restrictions
Convention We assume that V is an group F is a sub comp osition
algebra of M V F is zero symmetric i e f for all f F and that the
identity function on V is in F
Definition Let V and F b e as in Convention and let n N Then F
is n tame i
P V L F M V
F n C
What do es it mean for a comp osition algebra F to b e n tame Usually starting
with a function f F and an element v V the function
x f v x f v
TAME COMPOSITION ALGEBRAS ON GROUPS
do es not have to lie in F As an example of such a comp osition algebra we
consider a subnear ring S of M A where A A is the alternating
group of order S consists of all multiples of the identity function so S is
given by S fz id j z Zg Supp ose that S were tame Then for any v and
x A there would b e a z Z such that id v x id v z x This would
make every subgroup of A normal which is not true
If F is n tame then the function x f v x f v can at least by interpolated
at every subset of V with at most n elements by a function in F
The usual de nition of tameness is the one suggested by the following lemma
Lemma Let G be a group and let F be a zero symmetric sub near ring of
M G that contains the identity function on G Let n be in N Then F is n tame
i for al l g G and f F the function de ned by
G G
x f g x f g
lies in L F
n
Proof For the only if part we assume that F is n tame Now we check whether
the function G G x f g x f g lies in L F It is obvious that
n
lies in P G By the assumption that F is n tame De niton yields that
F
can b e written as l c where l is in L F and c is a constant function on G
n
Now by the way we have de ned we know This yields
l c
Since l lies in L F and n the fact that F is zero symmetric yields l
n
Hence we get c and therefore we have l This implies that lies in
L F which we had to prove
n
For the if part we have to show that every p olynomial function p P V
F
can b e written in the form l c with l L F and c M G We pro ceed by
n C
structural induction on a term representation of p
If p is a constant function than p can b e written as p
If p is the identity function we get p id and have again the required
G
representation
If p p p we know by induction hypothesis that we can write p as l c l c
and l l L F c c M G We will now show that for each c C there is
n C
c
L F such that a function l
n
c
x G c x l x c
TAME NEAR RINGS
For proving Equation observe that the function de ned by
G G
x c x c
can b e written in the form x id c x id c Hence the assumptions
yield L F Now the de nition of gives c x c x and hence
n
c
c x x c Thus can b e taken as the required l L F This proves
n
Equation
Applying Equation to the term l c l c we see that this is equal to
l l l c c which implies that p p lies in L F M G as well
n C
p x for all x which means p f p we rst use the If p x S f
V
F
induction hypothesis to write p as f l c with l L F and c M G By
n C
the assumptions we know that there is a function l L F such that
n
x G c x l x c
For proving this Equation we observe that c x c id c x id c
c
and hence the assumption gives l in the same way as we have pro duced l ab ove
Now we pro duce a function l L F such that
n
x G f x c l x f c
To this end we write
f x c f c c x c
f c l x
Since by assumption there is a function l L F such that
n
x G f c x l x f c
we have
f c l x l l x f c
Hence the function l l l satis es Equation So we can write f l c
as l l f c which gives the required representation of f l c as the sum of
as function in L F and a constant function
n
We will now work towards a solution of the fourth problem p osed by S D Scott in
Sco Let us rep eat the problem here using his words Note that S D Scott
writes mappings right to their arguments
If V is a group and S a non empty collection of normal subgroups
of V then let D S b e the subnear ring of M V consisting of all
maps of V into V such that v U v U for all v in V and
U in S If V is a faithful N group then de ne D V N to b e D S
where S is taken as the set of all submo dules of V We have the
TAME COMPOSITION ALGEBRAS ON GROUPS
following theorem to app ear Supp ose V is a faithful tame N
group If N is ring free and N J N has DCCR then N regarded
as a subnear ring of M V coincides with D V N
This raises the following question Can it b e shown that with
out DCCR on N J N N is in fact dense in D V N I b elieve
this is probably true but how to pro ceed is something of a mystery
S D Scott
We give a simple answer to Scott s question Yes Let us rst prove the fol
lowing theorem It has b een obtained by Scott at least for the case that V is a
nite group
Theorem Let V and F be as in Convention and let n Further
more we assume that every submodule of V is neutral Then the fol lowing are
F
equivalent
F is tame
F is dense in the set of al l functions f on V with f that preserve
al l congruences of the module V
F
F is n tame
Proof By Theorem we obtain that F is dense in L F We will
now show that the functions L F are precisely the zero preserving compatible
functions of the mo dule V In this pro of we use the following notation For a
F
subset M of M V we de ne M by
M fm M j m g
The set of all congruence preserving functions on the mo dule V will b e denoted
F
by C V formally this reads as
F
C V f V V j Con V a b a b g
F F
What we have to show is
L F C V
F
Since the right hand side of Equation is just the set of all zero symmetric
compatible functions on the mo dule V it is equal to L P V So what we
F F
have to show is
L F L P V
F
Since is immediate we will just prove To this end we x l L P V
F
such that l Since V is neutral we know by Prop osition and it will
F
also follow from Prop osition that P V is dense in L P V Hence
F F
L P V L P V and therefore l lies in L P V So l can b e interpo
F F F
lated at every three element subset of V b e a p olynomial function in P V
F
TAME NEAR RINGS
For proving Equation we have to show that l lies in L F To this end let
x x V with x x First of all we use the fact that l lies in L P V
F
to nd a function p P V with p x l x p x l x and p l
F
Hence p l Since F is tame we can write p in the form l c with
l L F and c M V Now we take an f F that coincides with l on x
C
and x What we get is
p x f x c for i
i i
By the fact that p and f we get c and therefore f interpolates
p at x and x Thus f also interpolates l at x and x This concludes the pro of
of Equation
We have to show that P V is a subset of L F M V To this
F n C
end let p b e a function in P V The function x p x p is zero
F
reserving and compatible and hence by an element of L F Since F is dense
in L F we have L F L F and therefore aso lies in L F Thus p p
n n
is the required representation of p as a sum of a function in L F and a constant
n
function
By de nition
As a consequence we obtain the solution to Stuart Scott s fourth problem in in
Sco Let G b e an F group Then we de ne C G as the set of all unary
F F
compatible functions c on G that satisfy c
F
Corollary Let G be a group and let F be a zero symmetric sub near ring
of M G We let G be the F group with universe G and the operation of F on
F
G as expected We suppose that no homomorphic image of F with more than one
element is a ring Then F is tame i it is dense in the subalgebra of M G
with universe C G
F
Stuart D Scott has already proved this result for the case that F is nite and
more generally for the case that the lattice of congruences of the F mo dule F
F
satis es the descending chain condition
Proof The result follows directly from Theorem if we can prove that every
submo dule of G is neutral To this end we supp ose that the submo dule H of
F
G is not neutral Hence it contains an ideal I such that
F
I I I
H
We see that I is the universe of a submo dule of G We will abbreviate this
F
mo dule by I The commutator I I is an ideal of H and therefore a subuniverse
H
of G Since F is tame every subuniverse of G is an ideal of G In particular
F F F
I I is an ideal of G Therefore it is also an ideal of I For all i i I and
F
H
f F we know that b oth
i i i i
TAME COMPOSITION ALGEBRAS ON GROUPS
and
f i i f i f i
lie in I I and therefore also in I I We let I b e the F mo dule de ned by
I H
I I I I
H
We see that I is an ab elian group and that
f i i f i f i
holds for all f F i i I
Now we consider the mapping de ned by
F M I
f f
where f is de ned by
f I I
i I I f i I I
H H
It is easy to see that is a near ring homomorphism and that F is a ring
However it would b e truly interesting to know the solution of Problem of
Sco For example the author would like to see a near ring that is tame
but not compatible We recall that a near ring of functions on V is compatible if
its carrier set is the set of unary p olynomial functions for some expansion W of
V
CHAPTER
Comp osition Rings
In this chapter we rst characterize nite simple zero symmetric comp osition
rings K with an identity with resp ect to and K K f g This result
has already b een published in Aic Then we characterize nite simple non
zero symmetric comp osition rings
The de nition of comp osition rings
A composition ring is an algebra K where K is a ring K is
a near ring and f g h f h g h for all f g h K We shall refer to the
op eration as multiplication and to the op eration as composition Comp osition
rings arise from studying functions on rings Let R b e a ring Then the
set M R of all selfmaps on R b ecomes a comp osition ring if we de ne and
p ointwise on R and as comp osition of functions
Comp osition rings are also a sp ecial instance of the comp osition algebras studied
in Chapter If we start with the variety of rings R then the R comp osition
algebras are precisely the comp osition rings This allows us to test the results
obtained in Chapter for this interesting situation
Studying simple comp osition rings the following construction plays an imp ortant
role Let S b e a set of ring endomorphisms on the ring R Then the set
M R fm M R j s S r R m s r s m r g
S
gives rise to a sub comp osition ring of M R Analogous to the situation
of near rings we call these comp osition rings centralizer composition rings
A comp osition ring K K is called zero symmetric i k for all
k K
Similar to the near ring case centralizer comp osition rings help to determine
simple comp osition rings Before stating the main result we should like to rep eat
the de nition of regular automorphism groups given in Gor p
Definition Let G b e a group and let b e a group of group automor
phisms on G Then is regular i for all n fid g and for all g G n f g
G
we have g g
COMPOSITION RINGS
We shall use the concept of regularity also for groups of ring automorphisms
Definition Let R b e a ring and let b e a group of ring automor
phisms on R Then is regular i for all n fid g and for all
R
r R n f g we have r r
Simple Zero Symmetric Comp osition Rings
Let K K b e a comp osition ring and R b e a ring Then the
function K R R is a module operation i it satis es the identites k k
r k k r k k r k r k r and k k r k r k r
If we expand R by all those unary op erations we arrive at a K mo dule as
de ned in De nition By a K ring we denote a K mo dule M whose reduct
M is a ring In the language of Chapter K rings are the K mo dules with
reduct in R where R is the variety of all rings
For rings and K rings the notion of b eing ab elian De nition has an unex
p ected interpretation A ring R R is ab elian in the sense of De nition
i it has zero multiplication If R is ab elian then Prop osition yields that
for all x x y y R we have
x y x y x x y y
Setting y we obtain
y x
which implies that R has zero multiplication On the other hand every ring with
zero multiplication is obviously ab elian A K ring X is ab elian i X is a
zero ring and the action of each element of K on X can b e written as the sum
of a constant map and an endomorphism of X
In this section we discuss zero symmetric comp osition rings K that have a left
identity with resp ect to i e an element e such that for all x K we have
e x x If the comp osition ring K has nonzero multiplication and it is faithful
on the K ring X then the ring X X has nonzero multiplication as well
by Prop osition Hence for comp osition rings Theorem can b e stated
as follows
Theorem Let K be a composition ring with nonzero multiplication that sat
is es k for al l k K Suppose that K has a left identity element e K
that satis es e k k for al l k K and that X is a faithful e unital K ring
whose only sub K rings are X and
Let S be the set of al l endomorphisms of the K ring X and let R be the ring
X X Then we have
SIMPLE ZERO SYMMETRIC COMPOSITION RINGS
S S f g where S is a regular group of automorphisms on the
ring R
K is isomorphic to a sub composition ring K of M R that is dense in
M R
S
In the rst pro of we show how this result follows from the result for comp osition
algebras developed in the previous chapters The second pro of is a stand alone
version of this pro of We give it here b ecause we think that it is easier to under
stand
Proof I We observe that the K ring X is not ab elian b ecause it has nonzero
multiplication It is simple b ecause it contains no honest sub K rings Now we
can apply Theorem and obtain the result
Proof II We recall that R X hence R and X denote the same set For
sake of simplicity we shall assume that K is a comp osition ring of functions on
X i e K M R Furthermore e is the identity function on X The pro of
of part is a precise copy of the corresp onding pro of in near ring theory Let
s S S n f g Then s is injective b ecause fx X j s x g is the universe
of a sub K ring of X It is surjective b ecause s X is a subuniverse of X as well
Since e S and the inverse mapping s of s is again an K ring endomorphism
on R S is really a group Furthermore S is regular b ecause for any s S
fx X j s x xg is a sub K ring of X For part we give a pro of similar
to the one of Aic Theorem First of all we notice that K is a subset
of M X We have to show that we can interpolate any m M X at each
S S
nite subset T ft t t g of X by a function k K Therefore we x a
n
c M X Let Y b e subset of T n f g that is maximal with the prop erty that it
S
do es not contain two elements of the same orbit of the group op eration of S on
X It is su cient to interpolate c on Y b ecause two mappings in M X that
S
agree on a p oint x X necessarily agree on fs x j s S g
We will now show that we can interpolate any mapping m Y X by a function
in K We do so by induction on jY j
Y fy g Since K y X any k K with k y m y satis es the
required interpolation prop erty
Y fy y g First of all we nd a k K with k y m y It is now
su cient to nd an k K with k y and k y k y m y
b ecause then k k is the required interpolating function on T Since
V fk y j k K k y g
is a subuniverse of of X we have either V X or V f g In the
case V X we immediately get the required mapping k In the case
V f g the mapping h X X k y k y is a well de ned K ring
COMPOSITION RINGS
automorphism on X that maps y e y to y e y which contradicts
the fact that Y contains at most one element of each orbit of S on X
Y fy y y g Let n Without loss of generality we assume
n
m y m y m y It is now su cient to show
n
fk y j k y k y k y g f g
n n
Let x y b e two elements of X with x y By induction hypothesis
there exists a mapping k F with k y k y k y
n
and k y x In the same way there exists an k K with k y
n n
and k y y Then k k y lies in the left hand side of but is
n n
not zero
Note that Theorem is particularly interesting if X X has a unit
with x x x for all x X
Corollary Let R be a ring with unit and let K be a sub composition ring
of
M R fm R R j m g
such that the K ring R has no subuniverses except f g and R We assume that
K
K contains the identity function id on R Then K is dense in M R
R
Under these assumptions the condition that the K ring R has no subuniverses
K
except for f g and R can b e replaced with the condition
K r R for all r R n f g
Proof of Corollary Since any ring automorphism on R xes the unit of R
fid g is the only universe of a regular group of ring automorphisms on R Now
R K
the result follows directly from Theorem
Theorem also allows us to determine all nite simple zero symmetric comp o
sition rings with a left identity with resp ect to comp osition
Theorem Let K K be a zero symmetric composition ring with
a left identity with respect to composition Furthermore we assume that the
multiplication is not identically zero i e K K f g Then the fol lowing two
conditions are equivalent
K is nite and simple
There exists a nite ring R and a regular group S of ring automorphisms
on R such that K is isomorphic to M R where S S f g
S
Proof I Since K K is a ring with nonzero multiplication
K is not ab elian in the sense of De nition and hence K is not ab elian
K
Now the result follows from Theorem
SIMPLE ZERO SYMMETRIC COMPOSITION RINGS
We want to apply Prop osition Since every ring is obviously
regular i e each congruence is determined by its class only the congrunce
is S orbit contained Now we can apply Prop osition and obtain that K
R
is simple
Proof II Let L b e a minimal sub K ring of K with jLj Then
K
we take R L Since K is simple and has a left identity K op erates faithfully
on L Now Theorem yields the result
By a result in near ring theory namely Pil Theorem b
we know that even the near ring M L is simple
S
There are several surprising facts ab out nite simple zero symmetric comp osition
rings K K with left identity with resp ect to comp osition One is that
the near ring K is automatically simple to o We state this result in the
following Prop osition
Proposition Let V be a variety of groups and let K be a V composition
algebra If K K is nite simple and zerosymmetric
and has a left identity with respect to composition then the near ring K
is simple as wel l
Isn t that surprising Given a nite not simple zerosymmetric near ring
K
with left identity you can never de ne op erations such that
K
is a simple comp osition algebra In other words the ugly that you want to choose
you will never b e ab el to ruin all congruences of K
i
But instead of enthusiasm for this result the reader might appreciate a pro of
Proof I We let L b e a minimal sub K mo dule L of K with jLj By
K
the simplicity of K and the fact that the left identity and the zero element of
K op erate di erently on L L is a faithful K mo dule Furthermore K satis es
K l L for all l Now we lo ok at the mo dule op eration of the near
ring K on the group L Since the variety G of all groups is clearly
regular Prop osition yields that the near ring K is simple
Proof II As in Pro of I we construct L with K l L for all l Now near
ring theory gives that every nite primitve near ring with identity is simple By
Pil Theorem K is isomorphic to a centralizer near ring which
is simple due to Pil Theorem d
1
The density theorems for primitive near rings is due to Wielandt and Betsch Bet or
Polin Pol for the interpolation part of the pro of see also Aic The interpolation part
has also b een stated in Ram
COMPOSITION RINGS
We will now go deep er into a p eculiar question Given a nite simple zero
symmetric comp osition ring K with left identity and nonzero multiplica
tion can it ever happ en that K is a ring and has more than one element
We shall obtain the following result
Proposition Let K be nite simple zero symmetric composition ring with
left identity with respect to composition and nonzero multiplication and suppose
jK j and K is a ring
Then K is isomorphic to Z where is the multiplication of
the two element eld Z
Proof By Theorem K is isomorphic to a centralizer comp osition ring
M R
S
for some ring R
Since we have assumed that M R is a ring Prop osition makes
S
it even b e a division ring and since it is nite a eld So we are left with the
following problem Which multiplications can b e de ned on a nite eld F
such that F b ecomes a comp osition ring We shall see in the following
results namely in Prop osition and Prop osition that only in the case that
F is the two element eld there is a nonzero multiplication that turns
F into a comp osition ring and that this multiplication is precisely the
eld multiplication on Z This completes the pro of of Prop osition
Proposition Let F be a eld If there exists a nonzero multiplication
on F such that F becomes a composition ring then char F
Proof We shall denote the identity with resp ect to by by and
by Now let x y b e elements in F Then we have x y x y
x x y y x y From this it follows that for all x y F we have
x y If char F than this implies that x y for all x y
Proposition Let F be a perfect eld with char F If there exists
a nonzero multiplication on F that turns F into a composition ring
then F is isomorphic to Z where is the eld multiplication
on Z
Before proving Prop osition we recall that in particular every nite eld is
p erfect
Proof First of all we shall prove that for all x y k F we have
x k y x y k
SIMPLE ZERO SYMMETRIC COMPOSITION RINGS
We x x y k F and denote the identity with resp ect to by Now we
compute x k y k in two ways
x k y k x k x y k y
x k y k x k y x y k x y
x y k x k y x y k x y
On the other hand we have
x k y k x y k
x y k x y
From this we get
x k y x y k
and hence equation
Since F is a p erfect eld of characteristic every element x F has a unique
p
y F with y y x As usual we denote this y by x Now we prove the
following fact ab out the multiplication
p
x y x y
We assume x y Then we apply equation and get of course fractions
are taken with resp ect to the eld op eration
r
x x
q
y x y
x
y
y
p p p
x y x y x y Computing the fractions and ro ots this is equal to
p p
x y x y But now note that is asso ciative Let x and z b e
two elements of F n f g Then we have x z z x z z By equation
we can write this as
q
p
p
x z z x z
Since the multiplication is nonzero Hence we get
p
x z z x z
and squaring and simplifying
x z z z x x z z
This implies x z hence all nonzero elements of F are equal therefore F
is isomorphic to Z From equation we see that the only nonzero
multiplication that turns Z into a comp osition ring is
At this p oint we want to mention one of the p ossible applications of Prop osi
tion
COMPOSITION RINGS
Corollary Let K be a nite simple ring with identity with more
than two elements Then there is no nonzero multiplication such that K
is a composition ring
Proof Supp ose there were such a nonzero multplication Then K is a
comp osition ring that ful lls the assumptions of Prop osition This means
that K has two elements a contradiction
Comp osition rings with constants
As in Adl we call an element c in the comp osition ring K a constant element
of K i c x c for all x K This is the case i c c We let R b e the
set of all constant elements of K and let R b e the sub ring of K K
with universe R We call R the ring of constants of K In the structure theory of
simple comp osition rings we will make use of the following kind of comp osition
rings of blo ckwise constant functions This de nition is the comp osition ring
version of De nition
Definition Let U b e a ring and let b e an equivalence relation on U
Then we de ne
M U fm U U j u u U u u m u m u g
We let M U the sub comp osition ring of M U with universe M U
Applying the theory of comp osition algebras in congruence p ermutable varieties
we have the following result It is a generalization of K Kaarli s result on simple
near rings presented in Kaa
Theorem Let K be a simple composition ring and let R be its ring of
constants We assume that jR j Then there is a sub composition ring K of
M R with the fol lowing properties
K is isomorphic to K
K is dense in M R where is the equivalence relation on R that is
de ned by r r k K k r k r
If Con R and then f g
R R
Proof I The result is an instance of Theorem
Proof II We distinguish two cases
Case K has a zero multiplication K is an abelian group and K
is a ring where K fk K k g Let b e the following relation on
K
k k k k R
COMPOSITION RINGS WITH CONSTANTS
We show that is a congruence on K To this end we recall that every comp o
sition ring K is an group as de ned in De nition Hence it is su cient to
prove that R is an ideal of the comp osition ring K According to De nition
this amounts to showing the following facts
R is a normal subgroup of K K is an ab elian group hence
every subgroup is normal Since we know that R R is a sub ring
of K K the group R is obviously a subgroup of K
K R R Since K K f g we have K R f g
R K R Since K K f g we have R K f g
R K R We show that for every r R and k K we have r k
r k
r k r k
r
r k
r k
r k
Applying the de nition that r is a constant element of K i r x r for
all x K we get using that r is a constant element of K r k y
r y r r k which proves again that r k is constant
For al l k k K and r R we have k k r k k R We write
k as l l where l k and l k k It is easy to see that
c c
l R and l K Then using the fact that K is a ring we have
c
k k r k k l k r l k r l k l k
c c
l l k l r l k l
c c
l r
This holds b ecause we have K R R which holds b ecause of k r
k r k r
Hence R is an ideal of K But K is simple hence R is either f g or K Since by
assumption jR j we have R K Hence every element k K ful ls
l K k l k
Since as the pro jection function to the rst of its arguments is always a
congruence preserving function on K we observe that the fact that K is
simple implies that even K is simple Therefore K is the zero ring
on a group of prime order
Using Equation we see that the comp osition ring K is isomorphic to the
comp osition ring of all constant functions on the ring R If we take R R
we see that therefore K is isomorphic to M R which proves claim and
claim
COMPOSITION RINGS
For claim we observe that the ring R is simple therefore every congruence
on R is either or
R R
Case K has a nonzero multiplication or K is a nonabelian group or
K where K fk K k g is not a ring Since jR j
and since K is simple the comp osition ring K op erates faithfully on the ring R
Hence the mapping de ned by
K M R
k k
where
k R R
r k r
embeds K into M R Let K K Obviously K satis es claim let us
now attack the pro of of claim First of all we de ne a multiplication on R
as follows If K has a nonzero multiplication we de ne
r r r r
if K is a nonab elian group we de ne
r r r r r r
if K is not a ring and K is an ab elian group we take elements
k k k K with k k k k k k k and de ne
r r k r k r k r r
Using the assumptions made in this case we see that satis es the following
prop erties
For all k k K the function k k de ned by
R R k k
r k r k r
is again in K
2
The fact that a non ab elian algebra allows to construct such multiplications lays at the
b eginning of my work in near ring theory I rst saw the imp ortance of such a multiplication
when studying the pro of of Wielandt and Betsch s Density theorem in Pil I then found
similar ideas in the pap ers by H K Kaiser e g Kai a IK Later K Kaarli p ointed
out to me that some results that were obtained using this metho d Kaa had also b een
obtained using metho ds of universal algebra HH and that a similar ideal multiplication
introduced by S D Scott in Sco again lo oked similar to the commutator studied in Kur
Later I proved that S D Scott s ideal multiplication was a reinvention of the universal alge
bra commutator Prop osition and that Hagemann s and Herrmann s main interpolation
result HH could b e proved using such an idea of multiplication Unfortunately Pro of II
for Prop osition hides Kaiser s multiplication idea b ehind the notational complications of
universal algebra In the present pro of the reader will nd this multiplication again
COMPOSITION RINGS WITH CONSTANTS
For all r R we have
r r
x y R with We have
x y
Another prop erty that we shall need is the following For every subset X R
and any y R the set
I fk y j k K x X k x g
is either f g or R We notice that the set I is the class of a congruence on the
0
R This can b e proved using the fact that we have all the constant K mo dule
K
mappings in K We then show that the set of all functions in K whose image
is contained in I namely
0
fk K j k R I g I R
K
is an ideal of the comp osition ring K Since K is simple it is either f g or K
0
is f g then in particular every constant function which has its image If I R
K
0
is equal in I is the zero function and therefore we have I f g If I R
K
to K then every function in K has its image contained in I In particular all
constant functions on R have their image contained in I This implies R I
and thus I R Altogether we see that I is either f g or R
We will now prove
K is dense in M R
To this end we show that for each n N we can interpolate any function in
M R at n places by a function in K We will do so by induction on n
Case n immediate since all the constant functions on R are elements of
K
Case n Let r r R and let m M R Since all constant functions
on R are in K it is su cient to nd a function k K with
k r and k r m r m r
b ecause in that case i x k x m r interpolates m at fr r g
By the remark after the set
I fk r j k K and k r g
is either f g or R Using the fact that all constant functions are in K it is not
hard to prove that I f g i r r If r r the zero mapping
satis es If r r we know that I R which immediately gives us a
function k K that satis es
COMPOSITION RINGS
Case n Let r r r R and let m M R Since by induction
n
hypothesis every function on in M R can b e interpolated at n p oints by
a function in K then after subtracting a suitable element k K from m we
are left with a function m M R that satis es m x m x
m x If m x the function k is already the required function
n n
that interpolates m So we will assume that m x and we will try to nd
n
k K that interpolates m at x x x Since m can b e interpolated at
n
every subset with no more than n elements by a function in K we have
I fk x j k K and k x k x k x g
n n
and
I fk x j k K and k x g
n n
By the remarks after we have I R and I R
We will now start to construct the function k Let x y R b e such that
x y The fact that I R allow us to construct a function p K such
that
p x for i n and p x x
i n
The fact that I R allow us to construct a function p K such that
y p x and p x
n n
Multiplying these two functions via we obtain a function
p r p r p r
with p K that satis es
p x for i n and p x
i n
The remark after yields
fk x j k K and k x k x k x g R
n n
which yields the function k that interpolates m at fx x x g
n
This nishes the pro of of claim and claim in this case We will now prove
claim Let A b e the class of i e A Then A is an ideal of the
ring R We de ne a set I by
I fk K j r R k r Ag
If is a congruence of the ring R and then I is an ideal of the comp osition
ring K It is easy to see that I is an ideal of K Now we let k k b e
elements of K and let i I The mapping i k then has its range contained in
A For the left ideal prop erty namely that
k k i k k
COMPOSITION RINGS WITH CONSTANTS
lies in I we x r R Then k k r i r is equal to k k r a for some
a A Since k r a is equivalent to k r mo dulo This implies
k k r i r k k r
and therefore k k i k k lies in I Altogether I is an ideal of K
If I K then R hence If I f g then hence
R
R
Prop osition gives the following result for comp osition rings
Proposition Let R be a ring with nonzero multiplication and let be an
equivalence relation on R such that
Con R and implies f g
R R
There are only nitely many equivalence classes modulo
Then M R is simple
This gives the following characterization of simple comp osition rings with con
stants
Theorem Let K be a nite composition ring such that R fk K
k k g has the property jR j Then the fol lowing are equivalent
K is simple
Both of the fol lowing two conditions hold
a K is isomorphic to a composition ring M R where R is a ring
and is an equivalence relation on R such that Con R
implies f g
R R
b K is not isomorphic to the composition ring M R where R is the
zero ring on two elements
Proof a Theorem yields that K is dense in a comp osition ring of
the form M R Actually Theorem gives precise information how to get
R and from K
b By Prop osition M R where R is the zero ring on two elements
is not simple Since K is simple it cannot b e isomorphic to this M R
If R has a nonzero multiplication then Prop osition gives the
required result For dealing with the case that R has zero multiplication we
need to lo ok further back to Prop osition Let K M R Actually
distingushing cases whether R has zero multiplication or not is the wrong case
distinction A real dividing line is whether the K ring R is ab elian or not
Case R is not abelian This includes the case that R has nonzero multiplica
tion Prop osition gives that the comp osition K ring M R is simple
COMPOSITION RINGS
Case R is abelian If M R is not simple then only the second alternative
of Prop osition can o ccur Hence is the equivalence mo dulo a subgroup
of index of R But since R has zero multiplication is a congruence of
the ring Hence by the assumptions on we have By its de nition
R
M R is therefore equal to M Z But this is excluded by the
assumption that K is not isomorphic to the comp osition ring of all functions over
the two element zero ring
And another example of an application of a universal result to the theory of
comp osition rings is the following result which is an instance of Prop osition
We let M R denote the comp osition ring of all constant functions on R
C
Proposition Let R be a nite simple ring with nonzero multiplication
Then the mapping de ned by
f j is equiv rel on R g fK j M R K M R g
C
M R
is a bijection
As in Prop osition we see that reverses inclusions Under the assump
tions of Prop osition R is the full matrix ring over a eld Prop osition
describ es all those sub comp osition rings of M R that contain all constant func
tions on R It would b e interesting to classify them according to isomorphism
CHAPTER
On Hagemann s and Herrmann s characterization of
strictly a ne complete algebras
The interpolation result Prop osition also implies the characterization of strict
ly a ne complete algebras in HH In this chapter we explain how these
results can b e derived from Prop osition Therefore this chapter do es not
contain original results but a new way of deriving the results in HH
Varieties generated by algebras that have only neutral subalgebras
Proposition Let A be a universal algebra Then the fol lowing are equiva
lent
Every subalgebra of A is neutral and SP A is congruence permutable
f
The class SP A is arithmetical
f
By Prop osition b oth statements are equivalent to the fact that the Pixley
op eration Pix A can b e interpolated at every nite subset of its domain by a
term function on A
Proof We rst show that the function Pix A can b e interpolated
at every nite subset of its domain by a term function in T A We x a nite
subset D of dom Pix A Let F b e the function algebra of A that has
F ftj j t T A g
D
as its universe Prop osition now yields that p can b e interpolated by a function
in F if it can b e interpolated at every subset of D with not more than two
elements by a function in F But subsets S of D with two elements fall in one of
the following classes
S f x y y x y y g or
S f x y x x y x g or
S f x y y x y x g for some x y x y A
Then t x y z x interpolates Pix A at S
S f y y x y y x g or
S f x y x y y x g for some x y x y A
Then t x y z z interpolates Pix A at S
STRICTLY AFFINE COMPLETE ALGEBRAS
S f x y y y y x g for some x y x y A
We use Prop osition to pro duce a Mal cev function m on S with
m T A Then m interpolates Pix A on S
Hence Prop osition yields that for every nite subset D of dom Pix A there
is a term function t T A such that t is a Pixley function on D Now
Prop osition implies that SP A is arithmetical
f
Supp ose that SP A is arithmetical We want to show that each
f
subalgebra of A is neutral Let B b e a subalgebra of A and let b e a congruence
on B We show that where the commutator is taken in B Therefore
we x a b B with a b mo d Let D f a a a a a b b a a b a b g
D
and let F b e the subalgebra of B with universe
F ftj j t T A g
D
Let c b e the function from D to B de ned by c a a a c a a b c b a a a
and c b a b b We want to nd a term function that agrees with c on D
We will use the following observation let f g F and let d D Then f d
g d is equivalent to f g mo d Ann d where Ann d is the congruence on
F that is de ned by
f g mo d Ann d f d g d
So the term function t F that agrees with c on D has to ful ll the following
system of congruences We write x for the function in F that maps d d d
to d y for the function in F that maps d d d to d and z for the function
that maps d d d to d Then c agrees with x on f a a a a a b b a b g
and with y on f b a a g
I t x mo d Ann a a a
II t x mo d Ann a a b
III t y mo d Ann b a a
IV t x mo d Ann b a b
D
Since F is a subalgebra of A it has distributive congruences The Chinese
Remainder Theorem says that a system of nitely many congruences has a solu
tion provided that every subsystem consisting of two congruences has a solution
Now t x solves the subsystems I II I IV I I IV The choice t y
is a solution of I III and I I III and t z is a solution of I I I IV
Hence c is a term function this means c F Since a b mo d and
c a a a c a a b we get c b a a c b a b mo d which implies
that a b lies in
If A is nite we get the following result
Proposition Let A be a nite algebra that is contained in a congruence
permutable variety Then the fol lowing are equivalent
STRICTLY AFFINE COMPLETE ALGEBRAS
Every subalgebra of A is neutral
The variety generated by A is arithmetical
Proof Since A is nite Prop osition yields that there is a term t
such that t agrees with Pix A on dom Pix A This term t makes the whole
A
variety generated by A arithmetical
If the variety generated by A is arithmetical then so is the class
SP A Hence Prop osition yields that every subalgebra of A is neutral
f
Strictly a ne complete algebras
If one expands an algebras by adding further op erations one may destroy some
of the prop erties of the original algebra For example some congruences may
b e ruined Therefore it is interesting to study those functions that do not ruin
any congruences Polynomial functions are of that kind but sometimes there
are more of them We will call such functions congruence preserving or simply
compatible
Definition Let A b e a universal algebra let k N and let D b e a subset
k
of A Then a function f D A is a compatible or congruence preserving
function on A i for all a b D we have
f a f b mo d a b
A
where a b is the congruence generated by a b a b a b
A k k
Following Wer an algebra is to b e called a ne complete if every congruence
preserving function is p olynomial In the present thesis we consider the following
types of a ne completeness
Definition We call an algebra A k a ne complete i every congruence
k
preserving function from A to A is a p olynomial function
Definition We call an algebra A strictly k a ne complete i every k ary
partial congruence preserving function with nite domain is the restriction of a
p olynomial function
Remark Let us brie y investigate the relation b etween these two concepts If
A is a nite strictly k a ne complete algebra then A is clearly k a ne com
plete On the other hand there are examples of nite algebras that are a ne
complete but not strictly a ne complete the group Z Z is an example
of such an algebra If A is an in nite strictly k a ne complete algebra then
A is not necessarily k a ne complete As an example take the eld of the ra
tionals A connection b eween strict a ne completeness and a ne completeness
is established by K Kaarli s Extension Principle for compatible functions A
consequence of this principle is the following Prop osition
STRICTLY AFFINE COMPLETE ALGEBRAS
Proposition Kaa We assume that A is an arithmetical algebra k
k
is a natural number D is a nite subset of A c is a k ary partial compatible
k
function on A with domain D and d lies in A Then there exists a compatible
c D fdg A with cj c function
D
From this principle it follows that a nite or countable arithmetical k a ne com
plete algebra is also strictly k a ne complete
We want to relate this concept to the concept of polynomial completeness
Definition We call an algebra A k polynomially complete i every function
k
from A to A is a p olynomial function
Definition We call an algebra A locally k polynomially complete i every k
ary partial function with nite domain is the restriction of a p olynomial function
It is easy to see that an algebra is k p olynomially complete i it is simple and
k a ne complete It is lo cally k p olynomially complete i it is strictly k a ne
complete and simple Every p olynomially complete algebra is also lo cally p oly
nomially complete every in nite eld shows that the converse is not true in
general
For the case of A b eing a group results ab out a ne completeness have b een
obtained in N ob Kai Kaa Kaa Theorem of HH char
acterizes those algebras that are strictly k a ne complete for all k N Due to
the imp ortance of their result let us state and prove it here For an algebra A of
type F let A b e the algebra of a new type that we get from A by adding
all its elements as constant op erations
Proposition Let A be a universal algebra and let k Then the fol lowing
are equivalent
A is neutral and SP A is congruence permutable
f
SP A is arithmetical
f
A is strictly a ne complete
A is strictly k a ne complete
Before proving Prop osition we state the following lemma
Lemma Let A be a universal algebra We assume that for every pair
a b A A there is a binary polynomial function q with q a a q a b
q b a a and q b b b Then A is neutral
Proof of Lemma By Lemma it is su cient to show for
each congruence of A Let us therefore x Con A We will show
STRICTLY AFFINE COMPLETE ALGEBRAS
We know that for all a b c d A and p P A the conditions
a b mo d
c d mo d and p a c p a d
imply
p b c p b d mo d
Let a b and let q b e a binary p olynomial function on A satisfying
q a a q a b q b a a and q b b b Now we know a
b mo d and q a a q a b Hence we have
q b a q b b mo d
But this just means a b which we had to prove
Proof of Proposition If A is neutral then so is A Clearly the
k
algebra A contains no prop er subalgebras Let T b e any nite subset of A and
let c b e a compatible function from T to A Using Prop osition it is easy
to show that c can b e interpolated at any subset of T with no more than two
elements by a p olynomial function p P A Now we apply Prop osition for
the function algebra F from T to A with universe
F fpj j p P A g
T k
This prop osition yields that there exists a p olynomial function f F that agrees
with c on T
Obvious
If A is strictly a ne complete then it is also strictly a ne com
plete For every pair a b A A the function c a a c a b c b a
a c b b b is compatible and hence the restriction of a p olynomial function
So Lemma yields that A is neutral
For showing that SP A is congruence p ermutable we show that Mal A can
f
b e interpolated at every nite subset of its domain by a term function on A i e
a p olynomial function on A Then the implication of Prop osition
implies that SP A is congruence p ermutable Since A is strictly a ne com
f
plete it is su cient to show that Mal A is a compatible function But this is
obvious
This follows immediately from Prop osition
From this pro of we get the the following consequence
Proposition Let A be an algebra such that SP A is congruence per
f
mutable and let k Then the fol lowing are equivalent
A is neutral
A is strictly a ne complete
A is strictly k a ne complete
STRICTLY AFFINE COMPLETE ALGEBRAS
Proof The only part of this result that do es not follow from Prop osition is
that a strictly a ne complete algebra is neutral For each pair a b A A
the function c fa bg fa bg c a a c a b c b a a c b b b is
compatible The algebra A is strictly a ne complete and therefore c is the
restriction of a p olynomial function So Lemma makes A neutral
Let us give the following consequence for p olynomial completeness
Corollary Let A be a simple algebra for which SP A is congruence
f
permutable If A is not abelian then it is locally n polynomially complete for al l
n N
Proof Since a simple non ab elian algebra A is neutral the result follows from
of Prop osition
A ne complete algebras
It seems to b e much harder to determine which algebras are k a ne complete
Although the problem has b een settled completely for ab elian groups in N ob
and Kaa and also for hamiltonian groups in Sak no answer is known for
varieties of groups that contain nonab elian groups In her PhD thesis Dor
A Dorda constructed a a ne complete group of order p and nilp otency class
for every o dd prime p
What we can do now is to characterize all nite algebras in a congruence p er
mutable variety that have the prop erty that all of their homomorphic images are
k a ne complete We obtain the following result
Proposition Let A be a nite algebra A in a congruence permutable va
riety and let k Then the fol lowing are equivalent
A is neutral
A is strictly a ne complete
A is strictly k a ne complete
Every homomorphic image of A is a ne complete
Every homomorphic image of A is k a ne complete
Proof and are equivalent by Prop osition
Let B b e a homomorphic image of A It follows from Prop osition
of FM that B is neutral Thus we may apply Prop osition to B and get
that B is strictly k a ne complete But since B is nite this implies that B is
k a ne complete
SOME CONSEQUENCES FOR POLYNOMIAL INTERPOLATION
Let B b e a homomorphic image of A and let b e a total compatible
function from B to B The function de ned by
k
B B
b b b b b
k
is compatible hence by it is a p olynomial function that lies in P B
k
Now we de ne a p olynomial function q P B by q x y x y y y
The p olynomial function q is now the required representation of the compatible
function
We show that
for all Con A
It is elementary to see that this condition implies neutrality If Equation
holds we have
Now supp ose that Equation fails Then there are and in Con A such that
and covers For the notions of lattice theory consult MMT
p Since Con A is nite there is an element that is maximal with the
prop erty and By this maximality prop erty is meet irreducible
and therefore has a unique upp er cover Now the intervals I and I
are pro jective in other words and From this we get
But since and we get
Now we consider the factor B A and notice that this algebra is sub directly
irreducible with ab elian monolith Now let c c b e two elements in
B that lie in the same class of The function de ned by
B B
c if b b c
b b
c else
is clearly compatible since it maps into one class of the unique minimal con
gruence Since B is a ne complete we may assume that B is a p olyno
mial function Now we have c c c c mo d and therefore
by the de nition of commutators also c c c c mo d Since
we have c c which is a contradiction
B B
Some consequences for p olynomial interpolation
Let us rep eat some well known results ab out p olynomial completeness that we
will need in the sequel
STRICTLY AFFINE COMPLETE ALGEBRAS
Proposition LN Chapter Theorem Every polynomially
complete algebra is n polynomially complete for al l n N
Proposition Kai b Hilfssatz Every locally polynomially comp
lete algebra is locally n polynomially complete for al l n N
From Prop osition and Prop osition we immediately get the following
consequence
Proposition Let A be a universal algebra and let k Then the
fol lowing are equivalent
A is locally polynomially complete
A is locally k polynomially complete
A is simple and not abelian and SP A is congruence permutable
f
The well known example of a set S with all unary functions as op erations
shows that there can b e algebras that are p olynomially complete and not
p olynomially complete
H Hule and W N obauer HN have started to investigate lo cal p olynomial
functions The sets of lo cal p olynomial functions are just the sets that arise from
the set of p olynomial functions on an algebra using the construction given in
De nition This means that for k N and and any cardinal t we have
k k
L P A fl A A j S A jS j t f P A f j l j g
t k k S S
T
Furthermore they de ne LP A L P A In N ob W N obauer
k n k
n N
asked the following question
Given a class of algebras do es there exist a natural number t such
that L P A L P A implies L P A LP A for all alge
k t k k k
bras of the class
If SP A is congruence p ermutable we see that L P A is equal to C A
f k k
where C A denotes the set of all k ary compatible functions on A In
k
W N obauer knew that for simple algebras in congruence p ermutable varieties
the equality L P A L P A implies L P A LP A IKP A
k k k k
p ossible generalization of this statement for algebras that are not simple is the
following
Proposition Let A be an algebra such that SP A is congruence per
f
mutable and let k Assume that every k ary partial compatible function can
be interpolated at every subset of its domain with no more than four points by a
1
Note that if A is a simple algebra in a congruence p ermutable variety we have L P A
2 k
k
(A )
A
SOME CONSEQUENCES FOR POLYNOMIAL INTERPOLATION
polynomial function on A Then every k ary partial compatible function can be
interpolated at every nite subset of its domain by a polynomial function on A
Proof For every pair a b A A the function c a a c a b c b a a
c b b b is compatible Since by assumption every such c is the restriction of a
p olynomial function Lemma yields that A is neutral Now the result follows
by Prop osition
Using the Extension Principle for compatible functions given in Kaa we get
Corollary Let A be a countable arithmetical algebra in a congruence
permutable variety and let k If L P A L P A then L P A
k k k
LP A
k
Proof We show that the assumptions of Prop osition are satis ed To this
k
end let c b e a partial compatible function from some subset of A with no more
than four elements into A Then using the Extension Principle of Kaa c can
k
b e extended to a total compatible function c from A to A The function c lies in
L P A and therefore by assumption in L P A Hence it can b e interpolated
k k
k
at every four element subset of A and therefore in particular at the whole
domain of c by a p olynomial function Now we can apply Prop osition and
obtain that in particular every function in L P A can b e interpolated at each
k
k
nite subset of A by a p olynomial function
At this p oint we should like to put some problems concerning a ne completeness
that we would like to have answered
Does there exist an algebra A that is strictly a ne complete but not
strictly a ne complete By Prop osition for such an A the class
SP A is not congruence p ermutable Also the algebra A cannot b e simple
f
if it were simple then it would b e lo cally p olynomially complete and
hence by Kai b Hilfssatz lo cally n p olynomially complete for all
n N
For n does there exist an algebra A which is n a ne complete but not
n a ne complete For n we could take the group Z For n
we note that A cannot b e simple LN Chapter Prop osition
and LN Chapter Theorem give that for n a simple p oly
nomially n complete algebra is automatically n p olynomially complete
Let k N Is there an arithmetical algebra A such that there is a k ary
compatible function on A with nite domain that cannot be extended to a
k
total compatible function from A to A
Is there an algebra A that is polynomially complete has a surjective
binary operation and is not polynomially complete Such an A cannot
b e nite by a result of Slu Hence the algebra A has more than
countably many op erations
STRICTLY AFFINE COMPLETE ALGEBRAS
Bibliography
Adl I Adler Composition rings Duke Math J
Aic E Aichinger Interpolation with near rings of polynomial functions Master s thesis
University of Linz
Aic E Aichinger Local interpolation near rings as a frame work for the density the
orems Contributions to General Algebra vol Verlag H older Pichler Tempsky
Wien Verlag B G Teubner Stuttgart pp
Aic E Aichinger A note on simple composition rings Near rings near elds and K
lo ops G Saad and M J Thomsen eds Kluwer Acad Publisher pp
Aic E Aichinger Local polynomial functions on the integers Riv Mat Univ Parma
to app ear
Bet G Betsch Some structure theorems on primitive near rings
Coll Math So c J Bolyai North Holland Amsterdam
BS S Burris and H P Sankappanavar A course in universal algebra Springer New
York Heidelb erg Berlin
Che G Chen Interpolationseigenschaften von Halbgruppen und Halbgruppen Ph D
thesis Technische Universit at Wien
Dor A Dorda Uber Vol lst andigkeit bei end lichen Gruppen Ph D thesis Technische
Universit at Wien
FM R Freese and R McKenzie Commutator theory for congruence modular varieties
London Math So c Lecture Note Ser vol Cambridge University Press
Fuc P Fuchs On the structure of ideals in sandwich near rings Results in Mathematics
Gor D Gorenstein Finite groups nd ed Chelsea Publishing Company New York
GU H P Gumm and A Ursini Ideals in universal algebras Algebra Universalis
Gum H P Gumm Algebras in permutable varieties geometrical properties of a ne alge
bras Algebra Universalis
HH J Hagemann and C Herrmann Arithmetical locally equational classes and repre
sentation of partial functions Universal Algebra Esztergom Hungary vol
Collo q Math So c J anos Bolyai pp
Hig P J Higgins Groups with multiple operators Pro c London Math So c
HM D Hobby and R McKenzie The structure of nite algebras Contemporary math
ematics vol American Mathematical So ciety
HN H Hule and W N obauer Local polynomial functions on universal algebras Anais
da Acad Brasiliana de Ciencias
Ihr T Ihringer Al lgemeine Algebra B G Teubner Stuttgart
BIBLIOGRAPHY
IK M Istinger and H K Kaiser A characterization of polynomially complete algebras
Journal of Algebra
IKP M Istinger H K Kaiser and A F Pixley Interpolation in congruence permutable
algebras Collo q Math
Jac N Jacobson Structure of rings nd ed AMS Collo quium Publications vol
XXXVI I American Mathematical So ciety
Kaa K Kaarli On near rings generated by the endomorphisms of some groups Acta et
Commentationes Universitatis Tartuensis University of Tartu Estonia
Kaa K Kaarli A ne complete abelian groups Math Nachr
Kaa K Kaarli Compatible function extension property Algebra Universalis
Kaa K Kaarli On the structure of non zerosymmetric nearrings Invited talk at the
conference on near rings and near elds at Hamburg August
Kai a H Kaiser A class of locally complete universal algebras J London Math So c
Kai b H Kaiser Uber lokal polynomvollst andige universale Algebren Hbg Math Abh
Kai H K Kaiser Uber kompatible Funktionen in universalen Algebren Acta Mathemat
ica Academiae Scientiarum Hungaricae
Kur A G Kurosh Lectures on general algebra Chelsea New York
LN H Lausch and W N obauer Algebra of polynomials North Holland Amsterdam
London American Elsevier Publishing Company New York
Mli R Mlitz Jacobson density theorems in universal algebra Contributions to universal
algebra vol Collo qu Math So c Janos Bolyai pp
MMT R N McKenzie G F McNulty and W F Taylor Algebras lattices varieties vol
ume I Wadsworth Bro oks Cole Advanced Bo oks Software Monterey Califor
nia
MPS C J Maxson M R Pettet and K C Smith On semisimple rings that are centralizer
near rings Paci c J Math
MS C J Maxson and K C Smith The centralizer of a set of group automorphisms
Comm Alg
MvdW C J Maxson and A P J van der Walt Centralizer near rings over free ring modules
J Austral Math So c Series A
N ob W N obauer Uber die a n vol lst andigen end lich erzeugbaren Moduln Monatshefte
f ur Mathematik
N ob W N obauer Local polynomial functions results and problems Preprint of the
Techn Univ Wien Austria June
Pil G F Pilz Near rings nd ed North Holland Publishing Company Amsterdam
New York Oxford
Pil G F Pilz Algebra ein Reisef uhrer durch die sch onsten Gebiete Universit atsverlag
Rudolf Trauner Linz
Pol S V Polin Primitive m near rings over multioperator groups Mat USSR Sb ornik
no
PV Q N Petersen and S Veldsman Composition near rings Near rings near elds
and K lo ops G Saad and M J Thomsen eds Kluwer Acad Publisher
pp
Ram D Ramakotaiah Structure of primitive near rings Math Z
Rob D J S Robinson A course in the theory of groups Springer Velag
BIBLIOGRAPHY
Sak M Saksa Polynomiaalsed funktsioonid ryhmadel polynomial functions on groups
Master s thesis University of Tartu Estonia
Sco S D Scott Some interesting open problems Near ring newsletter number
Y Fong G Pilz A Oswald and K C Smith eds National Cheng Kung Uni
versity Taiwan January pp
Sco S D Scott The structure of groups Near rings near elds and K lo ops G Saad
and M J Thomsen eds Kluwer Acad Publisher pp
Slu J Slup ecki A criterion of completeness of many valued logic C R So c Sci Varsovie
Smi J D H Smith Mal cev varieties Lecture Notes in Math vol Springer Verlag
Berlin
Tho J G Thompson Finite groups with xed point free automorphisms of prime order
Pro c Nat Acad Sci U S A
Wer H Werner Produkte von Kongruenzklassengeometrien universel ler Algebren
Math Z
BIBLIOGRAPHY
Index
V comp osition algebra ab elian
congruence p ermutable ab elian algebras
congruence preserving Adler Irving
constantly b ehaving
a ne
countable
complete
C
function
strictly a ne complete
density
Aichinger
dom f
Bruno und Gertraud
domain
algebra
arity
element
constantly b ehaving
base of equality
e unital
expansion
centralizer
extension principle
comp osition rings
centralizer comp osition algebras
F
Chinese Remainder Theorem
+
F
F
commutator
+
F
of congruences
A
F
of ideals of groups
faithful
re invention by S D Scott
eld
compatible
of order
function
xed p oint free
near ring
functions
complete
induced by a term
a ne
p olynomial
p olynomially
fundamental op erations
strictly a ne
F x
complexity
F x
of an equation
comp osition algebra
I
over type
GF
comp osition algebras
tame
ideal
type of
identity
comp osition near rings
I S
V
comp osition ring
L F comp osition rings
n
L F comp osition type over
n
INDEX
lo cal interpolation algebras group of automorphisms
regular
M A
relation pro duct
Mal A
ring
Mal cev
of constants
function
Scott
op eration
Stuart
mappings
skew free
algebra of all mappings
SP A
algebra of blo ckwise constant
f
strictly a ne complete
M A
sub F mo dules
M V
C
subalgebra
M F
sub direct pro duct
mo dule
subuniverse
F mo dules
op eration
tame
with reduct in V
comp osition algebras
mo dules
near rings
faithful
tame
type of F mo dules
term
varieties of F mo dules
term functions
M A
S
a b
A
M R
S
T A
k
type
near rings
of F mo dules
tame
of algebras
neutral
of comp osition algebras
N groups
universe
group
op eration
variety
symbols
op erations
zero symmetric
fundamental
comp osition algebra
orbit contained
comp osition rings
p olynomials
P A
k
ZP V
k
Pilz
G unter
Pix A
Pixley
function
op eration
p olynomial functions
p olynomially complete
p olynomials
zero symmetric
problems in a ne completeness
reduct regular
Tabellarischer Leb enslauf
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