Algebraic Asp ects of Orthomo dular Lattices

Gunter Bruns

McMaster University Hamilton Ontario LS K Canada

John Harding

New Mexico State University Las Cruces NM USA

In this pap er we try to give an uptodate account of certain asp ects of the the

ory of ortholattices abbreviated OLs orthomo dular lattices abbreviated OMLs

and mo dular ortholattices abbreviated MOLs not hiding our own research in

terests

Since most of the questions we deal with have their origin in Universal Alge

bra we start with a chapter discussing the basic concepts and results of Universal

Algebra without pro ofs In the next three chapters we discuss mostly with pro ofs

the basic results and standard techniques of the theory of OMLs In the remaining

ve chapters we work our way to the b order of present day research mostly no or

only sketchy pro ofs Chapter deals with pro ducts and sub direct pro ducts chap

ter with free structures and chapter with classes of OLs dened by equations

In chapter we discuss emb eddings of OLs into complete ones The last chap

ter deals with questions originating in Category Theory mainly amalgamation

epimorphisms and monomorphisms

The later chapters of this pap er contain an abundance of op en problems We

hop e that this will initiate further research

Basic Universal Algebra

We assume a denition of natural numb ers which makes every natural numb er n

the of all natural numb ers k n An nary op eration on a set A is a map f of

n n

A into A An element a A gives rise to a sequence a a an of

A more commonly written with indices a a a The numb er n is called

0 1 n1

0

the arity of the op eration It is imp ortant to allow the case n Since A fg

a ary or nullary op eration is a map of fg into A Since such a map is completely

determined by the element f we usually sp ecify ary op erations also called

constants by giving the element f A similar remark applies to unary ary

op erations We identify them with maps from A to itself

A closure system over a set A is a set C of of A satisfying

T

If B C then B C

Here we make the convention that the intersection of the empty of C is A

so that A C If C is a closure system over A and if X A we dene the closure

T

X of X with resp ect to the closure system C by X fC jX C C g X

is obviously the smallest element of C containing X as a subset If for B C we

W S V T

dene B B and B B then C b ecomes a complete is the

smallest and A is the largest element of this lattice

An algebraic closure system is a closure system C satisfying

S

If K C is a chain ie totally ordered by then K C

This denition one of many equivalent ones is particularly useful in pro ofs in

volving Zorns lemma

An algebra is a pair A A f where A is a set called the underlying

i iI

set or the universe of the algebra and each f is for some n an n ary op eration

i i i

on A The family n is called the typ e of the algebra Whereas every n

i iI i

is nite it is necessary in order to cover imp ortant examples to admit innitely

many op erations f

i

A subuniverse of an algebra A A f is a set B A which is closed

i iI

under all op erations f ie satises

i

n

i

then f a B If a B

i

or in more common and more cumb ersome notation

If a a a B then f a a a B

0 1 n 1 i 0 1 n 1

i i

Let S ubA b e the set of all subuniverses of A The most imp ortant and in fact

characteristic prop erty is

Prop osition If A is an algebra then S ubA is an algebraic closure system

As a consequence of this we obtain that for any X A there exists a smallest

subuniverse X containing X X is said to b e the subuniverse generated by X

A subalgebra of an algebra A A f is an algebra B B g

i iI i iI

n

i

ie such that B A and g is the restriction of f to B

i i

n

i

then g a f a If a B

i i

Clearly this requires that B is a subuniverse of A Conversely if B is a subuniverse

n

i

then B g b ecomes a subalgebra of of A and g is the restriction of f to B

i iI i i

A Because of this oneone corresp ondence b etween subuniverses and subalgebras

we will not b e to o fussy ab out distinguishing them Thus we refer to S ubA

as the closure system or lattice of subalgebras of A Many authors require the

denition of algebra and subalgebra that the underlying set b e not empty We see

no reason for this

Let A A f and B B g b e algebras of the same typ e A

i iI i iI

homomorphism of A into B is a map A B satisfying for every i I and

n

i

a A

f a g a

i i

or in cumb ersome notation

f a a a g a a a

i 0 1 n 1 i 0 1 n 1

i 1

B is said to b e a homomorphic image of A i there exists a homomorphism of A

onto B An emb edding of A into B is a oneone homomorphism of A into B An

isomorphism is an emb edding which is onto

Closely related to homomorphisms are congruence relations A congruence

relation short congruence on an algebra A A f is an equivalence

i iI

relation on A ie a reexive symmetric and transitive relation R satisfying

n

i

and ak R bk k n then f aR f b If a b A

i i i

We express this by saying that R is compatible with the op eration f If R is an

i

on A and a A dene the equivalence congruence if R is

a congruence of a mo dulo R by aR fbjaR bg and the quotient set of A

mo dulo R by AR faR ja Ag

n

i

If R is a congruence in A we may dene op erations g in AR If a A

i

n

i

by aR k ak R Then dene g by dene aR AR

i

n

i

then g aR f aR if a A

i i

in cumb ersome notation

g a R a R a R f a a a R

i 0 1 n 1 i 0 1 n 1

i i

With these op erations AR g b ecomes an algebra of the same typ e as A

i iI

the so called quotient algebra AR of A mo dulo R The map A AR

R

dened by a aR b ecomes a homomorphism of A onto AR called the

R

canonical homomorphism of A onto AR

We are now in a p osition to establish the basic relationship b etween homo

morphisms and congruences known under various names

Homomorphism Theorem Let A B be a homomorphism of A into B

Dene a relation R ker the kernel of by aR b i a b Then

R is a congruence in A and there exists a unique map AR B such that

This map is a oneone homomorphism of AR into B

R

A B

R

AR

If is onto then clearly is onto hence an isomorphism Thus every homo

morphic image of A is isomorphic with a quotient algebra of A

It is of some imp ortance that the set C onA of all congruences in A is an

algebraic closure system over A A

Q

A of the family consists Let A b e a family of sets The pro duct

k k k K

k K

of all choice functions on the family ie all maps with domain K such that

Q

A there come maps pr of k A for all k K With every pro duct

k k k

k K

the pro duct onto A dened by pr k The map pr is called the k th

k k k

k

pro jection If the A A f are algebras we dene op erations f in the

k k iI i

i

k

pro duct comp onentwise ie by f k f pr The resulting algebra

i k

i

Q

A is called the pro duct of the family A In case of a pro duct of

k k k K

k K

algebras the pr are homomorphisms

k

The cartesian pro duct A B of two sets is dened in basic set theory as

fa bja A b B g If A A f B B g we may dene op era

i iI i iI

tions h in A B by h a f pr a g pr a This gives a new algebra

i i i A i B

A B This can b e subsumed under the pro duct dened b efore We leave out the

details

An imp ortant prop erty of pro ducts which in more general settings can b e

used to dene pro ducts is the

Extension Prop erty of Pro ducts If A A k K are algebras and for every

k

k K is a homomorphism of A into A then there exists a unique homomor

k k

Q

A satisfying pr for every k phism of A into

k k k

k K

A sub direct pro duct of a family A of algebras is a subalgebra A of the

k k K

Q

A such that every pr maps A onto A A sub direct representation pro duct

k k k

k K

Q

A such that the of an algebra A is an emb edding of A into a pro duct

k

k K

maps pr map A onto A An algebra A is said to b e sub directly irreducible i

k k

Q

for every sub direct representation A A one of the maps pr is one

k k

k K

one hence an isomorphism The fundamental result ab out sub directly irreducibles

is

Birkho s Sub direct Representation Theorem Every algebra is isomorphic

with a subdirect product of subdirectly irreducible algebras

Sub direct irreducibility of an algebra has a neat and very useful character

ization in terms of congruences The smallest congruence of an algebra A

A f is obviously the diagonal fa aja Ag An algebra A

i iI A

A f is sub directly irreducible i there is a smallest congruence dierent

i iI

from

A

Let K b e a class of algebras of the same typ e We dene H K to b e the class

of all homomorphic images of algebras in K S K the class of all subalgebras of

algebras in K and P K the class of all pro ducts of algebras in K An equational

class or variety of algebras is a class K with H K K S K K and P K K

If K is any class of algebras of the same typ e then HSP K is the smallest variety

containing K

In our general considerations so far we have distinguished corresp onding op

erations in algebras of the same typ e by giving them the same index This is prac

tically never done in concrete cases In these cases one identies corresp onding

op erations by denoting them by the same symb ol For illustration let us consider

the case of groups In order to get subgroups as a sp ecial case of our general

notion of subalgebras we have to consider a as an algebra with the group

multiplication as a binary op eration the forming of inverses as a unary op eration

and the unit as a constant In order to make this t our general development so

far we would have to intro duce op erations say f f f Instead of doing this

0 1 2

1

we intro duce sp ecial symb ols for the group multiplication for the forming of

1

inverses and e for the unit We then say a group is an algebra G e of typ e

indicating the arities of the op erations in the given order

The basics of ortholattices and orthomo dular

lattices

We assume that the reader is familiar with the basics of lattice theory its descrip

tion as a and its representation by Hasse diagrams

A b ounded lattice is an algebra L where L is a lattice

is a lower b ound of L and is an upp er b ound of L An ortho complementation

on a b ounded lattice is a unary op eration satisfying

a a a a

a b b a

a a

An easy consequence of this are the DeMorgan laws

a b a b a b a b

Each of these can replace the second condition in the description thus dening

an ortho complementation by equations An ortholattice abbreviated OL is an

algebra L where L is a b ounded lattice and is an

ortho complementation on it If one interchanges the binary op erations and

the constants one again obtains an OL called the dual of the original OL If

the op erations are clear we sp eak simply of the OL L

An orthomo dular lattice abbreviated OML is an OL satisfying the ortho

mo dular law

if a b then a a b b

This law can again b e relaced by the equation

a a a b a b

The prime example of an OL which is not orthomo dular is the b enzene ring see

diagram

1

0

a

b

0

a

b

0

In fact we have

Prop osition Let L be an OL These are equivalent

L is an OML

if a b and a b then a b

the benzene ring is not a subalgebra of L

The pro of of this is straightforward

A fundamental concept in OLs is commutativity An element a is said to

commute with an element b in symb ols aC b if a a b a b Clearly a b

implies aC b and aC b implies aC b The commutator a b of elements a b of an

OL is dened by a b a b a b a b a b

Prop osition In an OL L the fol lowing are equivalent

L is an OML

aC b bC a

aC b a C b

aC b a a b a b

aC b a b

Pro of Assume aC b Then a b a b a b a b a b

a b a b a b b a b a b a b b a b a

b b a b a b b b b using the dual of the orthomo dular

law Assume conversely that a b Clearly a a b a b But

a a b a b a b which gives aC b by condition of

aC b a b b a bC a

aC b bC a bC a a C b

a a b a b a a bC a b a a b C a b a a b

a a b a b a a b a b a a b a b a b

a a a b a b Thus a a b a b

Obvious since a b implies aC b

In an OML aC b holds i a a b a b If this last equation holds we

have a b a b a a b a b a hence aC b

Prop osition Let L be an OML a x L i I If aC x i I and if

i i

W W W W

a x and dual ly x a x exists and a x exists then

i i i i

iI iI iI I

W

x is an upp er b ound of fa x ji I g Let v b e any upp er Pro of Clearly a

i i

I

W W

x x Then a x u a b ound of this set and put u v a

i i j

iI iI

W W W

x j I hence x a x x a x a and u a

i i i

j j

iI I iI

W W V W

x by of x hence u a x x a u a

i i i

i

iI iI iI iI

W

It follows that a x is the least upp er b ound of fa x ji I g

i i

iI

If a is an element of an OML L we dene C a fx LjaC xg If A L we

dene C A fx LjaC x holds for all a Ag

Prop osition If a is an element of an OML L if x C a i I and if

i

W W

x C a and dual ly In particular C a is a subalgebra x exists then

i i

iI iI

of L

W W W W

x a x a x x x a Pro of We have a

i i i i i

iI iI iI iI

W W

x C a x C a hence hence

i i

iI iI

An element c of an OML is said to b e central if it commutes with all elements

of L and the set of all central elements C L is called the centre of L

For elements a b in an OL L we dene a b fxja x bg and sp eak of

the interval a b

Prop osition Let a b be central in an OML L a b a b is

an OL where is dened by x a b x The map f L a b dened

by f x a b x is an onto homomorphism

Pro of Let a x b Then x x x a b x x b x

and as x b orthomo dularity gives x x b As xC a b x yields

x x x a b x x a x b x a Note x a b a b x

a b a b x Then as b x the dual of the orthomo dular law gives

x a a x and as a x the orthomo dular law gives x x Therefore

is an ortho complementation Obviously f x x In view of the DeMorgan

laws it suces to show f x y f x f y As b is central provides

f x y a b x y a b x b y a b x a b y f x f y

Therefore f is a homomorphism and trivially onto

Corollary Every interval a b in an OML L is an OML under the ortho

complementation x a b x Further this OML is a homomorphic image of

a subalgebra of L

Pro of Supp ose a b is an interval in L Consider the subalgebra S of L consisting

of all elements which commute with b oth a b Then S contains the interval a b

and a b are central in the OML S From the previous prop osition a b is an OL

under the ortho complementation x a b x and this OL is a homomorphic

image of the OML S hence is orthomo dular

Lemma If c is central in an OML L then f L c c dened by

f x x c x c is an isomorphism Conversely if f L A B is an

isomorphism then there is a central element c in L with A c and B c

Pro of Assume c is central in L By the map f L c c is

a homomorphism As x c x c x f is oneone For x c y c

commutativity gives f x y x y hence f is onto For the converse take c in

L with f c

The most imp ortant to ol for computations in OMLs is the

FoulisHolland Theorem If one of the elements a b c of an OML commutes

with the other two then the sublattice not subalgebra generated by fa b cg is

distributive

Pro of Assume a bC c Then c is central in fa b cg and hence the map x

x c x c is an isomorphism of fa b cg with the pro duct c c Clearly

fa c b c cg generate a distributive sublattice of c and fa c b c c g

generate a distributive sublattice of c The sublattice generated by fa b cg is

clearly isomorphic with a sublattice of the pro duct of these two sublattices

Prop osition Let X be a subset of an OML L The subalgebra X generated

by X is Boolean i any two elements of X commute

Pro of Clearly in a Bo olean algebra any two elements commute Assume that

any two elements of X commute Then X C X hence X C X hence

X C X hence X C X Thus any two elements of X commute which

implies that X is Bo olean by the FoulisHolland Theorem

Subalgebras

We b egin with Bo olean subalgebras Note that every element x of an OL L is

contained in a Bo olean subalgebra of L namely the subalgebra f x x g Also

as the of a chain of Bo olean subalgebras is again a Bo olean subalgebra

a direct application of Zorns lemma yields that each Bo olean subalgebra of L

is contained in a maximal Bo olean subalgebra of L also termed a blo ck of L

Therefore

Prop osition Every OL is the union of its blocks

Supp ose L is an OL If L is an OML then for any a b in L we have aC b hence

by fa bg is Bo olean so a b are elements of some blo ck of L Conversely

if L is not an OML then by there are a b in L which generate a b enzene

ring hence are not elements of any Bo olean subalgebra of L Therefore

Prop osition An OL L is an OML i the partial ordering on L is the union

of the partial ordering on the blocks of L

This shows that each OML is determined by its blo cks We remark that there

are a numb er of results such as Greechies Lo op Lemma Dichtls astroids

and Kalmbachs bundle lemma describing conditions on a family of Bo olean

algebras B which ensure their union is an OML See for a complete account

i I

We next collect a few basic and well known prop erties of blo cks all of which are

easily proved

Prop osition Let L be an OML The centre C L is the intersection of

the blocks of L If M is a subalgebra of L then the blocks of M are exactly

fB M jB is a block of Lg If L is a family of OMLs then the blocks of

i I

Q Q

B jB is a block of L g L are f

i i i i

I I

For OMLs L M and a homomorphism L M the image B of any blo ck

of L is obviously a Bo olean subalgebra of M but need not b e a blo ck consider

2

the identical emb edding of into We are not aware of such an example with

the homomorphism onto

An imp ortant class of OLs are those of height two ie in which the maximal

numb er of elements in a chain is three Clearly such an OL is determined by

sp ecifying its cardinality which if nite must b e some even numb er at least four

Dene MO for to b e the OL of height two having cardinality

k

Extend this by setting MO to b e a two element Bo olean algebra Clearly each

0

MO is a mo dular ortholattice abbreviated MOL A diagram of MO is given

2

b elow

1

0

0

a

a

b

b

0

It is a simple but useful observation that for any elements x y in an OL L

fx y g is MO i x y and L is nontrivial This yields

2

Lemma Let L be an OML generated by the elements x y Then x y is

either trivial or MO x y is Boolean and L x y x y

2

Pro of As x y commutes with b oth x y it is central in L So there are

homomorphisms f L x y and g L x y dened by setting

f z z x y and g z z x y Note f x f y f x y is the unit

of x y so this interval is either MO or trivial Similarly g x g y

2

so g x commutes with g y and by the interval x y is Bo olean

As any nitely generated Bo olean algebra is nite we then have

Corollary For x y elements of an OML fx y g is nite

An as an OML is Bo olean i all elements commute we have

Corollary If L is a nonBoolean OML then MO is a homomorphic image

2

of a subalgebra of L

A variety V is called lo cally nite if for every A V and every nite subset

S A the subalgebra S generated by S is nite The rst of these two corollaries

may raise some false exp ectationsnone of the varieties of MOLs OMLs or OLs

is lo cally nite To see this consider the MOL L of all subspaces of a three

dimensional vector space over the reals with ortho complementation b eing given

by orthogonal subspaces One can easily nd three elements of L that generate an

innite subalgebra

There is a useful weakening of the notion of lo cally nite

Denition A variety V has the nite embedding property fep if for every

A V and every nite S A there is a nite B V and a oneone set mapping

S B such that for each basic operation f and each s s S if

i 0 n 1

i

f s s S then f s s f s s

i 0 n 1 i 0 n 1 i 0 n 1

i i i

It follows from consideration of MacNeille completions of ortho complemented

p osets section that the variety of OLs has the fep and it follows from results

on nite MOLs section that the variety of MOLs do es not have fep There is a

natural hop e that one can combine the fact that Bo olean algebras are lo cally nite

with various techniques to pro duce OMLs by gluing together Bo olean algebras

to show that the variety of OMLs has the fep Such attempts have not so far

b een successful Due to the connection to the word problem the following is one

of the basic outstanding problems in the theory of OMLs

Problem Do es the variety of OMLs have the fep

Congruences and homomorphisms

Given an OL L one calls the b ounded lattice L the

b ounded lattice reduct of the OL Clearly any homomorphism b etween two OLs is

also a homomorphism b etween their b ounded lattice reducts For Bo olean algebras

B and C it is well known that a map f B C is

a homomorphism b etween the Bo olean algebras i it is a homomorphism b etween

their b ounded lattice reducts But it is a simple matter to nd an automorphism

of the b ounded lattice reduct of MO that is not an automorphism of MO

2 2

However

Prop osition If L is an OML then every congruence on the bounded lattice

reduct of L is a congruence on L

Pro of Let R b e a congruence on the lattice reduct of L We must show that

aR b implies a R b We rst show this in the sp ecial case that a b In this case

orthomo dularity gives a b b a So a b b a R b a a b In

the general case aR b implies a bR a b hence a b R a b and as a b

lie b etween a b and a b the result follows

We next examine more closely the structure of congruences on an OML

Lemma Let R be a congruence on an OML L For a b L these are equiv

alent aR b a b a b R a x b x for some x with xR

Pro of Assuming a b a b R a a a a Assuming set

x a b a b Assuming a a R a x b xR b b

An algebra A is called congruence regular if every congruence on A is de

termined by any one of its equivalence classes In other words A is congruence

regular if for any a in A and any congruences R S on A we have aR aS

implies that R S Groups and Bo olean algebras are examples of congruence

regular algebras while distributive lattices are not The following is a sp ecial case

of a well known result that applies to any lattice with where each interval x

is complemented

Prop osition Every OML is congruence regular

Pro of Supp ose R S are congruences on an OML L and that aR aS for some

a in L If xR then for y a complement of x in the interval a x we have that

xR implies y R a x hence y S a x giving xS Thus aR aS implies

R S Supp ose then that cR d Let e b e a complement of c in the interval

c d As cR c d we have eR hence eS so cS c d and cS d

As is customary with congruence regular algebras we often cho ose to work

with a particular of a congruence For groups we cho ose the

equivalence class containing the group identity which is a normal subgroup of the

group For OMLs we cho ose the equivalence class containing The following

formula for recovering a congruence from its equivalence class is implicit in the

previous pro of

Prop osition If R is a congruence on an OML L then aR b i a x b x

for some xR

An algebra A is called congruence p ermutable if for any congruences R S

on A we have R S S R Note that groups rings and Bo olean algebras

are congruence p ermutable while distributive lattices are not The following is a

sp ecial case of the well known result that any relatively complemented lattice is

congruence p ermutable pg

Prop osition Every OML is congruence permutable

Recall for an OL L the collection C onL of all congruences on L is an alge

braic closure system over L L hence forms under set inclusion

Meets in this lattice are given by set intersection and upwardly directed joins are

given by unions If L is an OML then as L is congruence p ermutable binary joins

in this lattice are given by relational pro duct The collection I dL of all ideals

of the lattice L also forms a complete lattice under set inclusion Meets in this

lattice are given by set intersection Upwardly directed joins are given by unions

and binary joins are given by I J fxjx a b for some a I b J g

Prop osition For an OML L the map F C onL I dL dened by F R

R is a bounded lattice embedding which preserves arbitrary joins and meets

Pro of For a congruence R clearly R is an ideal As L is congruence regular

F is oneone As meets in b oth C onL and I dL are given by intersections F

preserves arbitrary meets and similarly F preserves upwardly directed joins It

remains only to show that F preserves binary joins Let R S b e congruences

Note that their join in C onL is R S If z b elongs to F R S there is some xR

and some y S with x y x z hence z x y Conversely if xR and y S

then x y R S as R S is a congruence So z x y implies that z R S

This shows that F R S F R F S

It is of interest to characterize those ideals that arise as the zero equivalence

classes of congruences on an OML L much the way we distinguish normal sub

groups of a group

Prop osition Let I be an ideal of an OML L Then I is the zero equivalence

class of some congruence on L i x I and y L implies y y x I

Pro of Obviously if I R for some congruence R then x I y L implies

y y xR y y so y y x b elongs to I Conversely assume that I is

closed under the given condition Set R fa bja x b x for some x I g

As is in I R is reexive By the symmetry of the denition R is symmetric

As I is closed under nite joins it follows that R is transitive and further that

R is compatible with joins It remains only to show that R is compatible with

ortho complementation Supp ose aR b Then a x b x for some x in I So

a a x and b b x b elong to I hence a a b x and b a b x

b elong to I and as I is a downset a a b and b a b b elong to I As I is

closed under joins and a b commutes with b oth a b we have a b a b

b elongs to I But a a b a b a b and b a b a b a b

Thus a R b

Among the most useful results ab out the congruence lattice of an OL follows

b elow This will op en the do or to such p owerful techniques as Jonssons Theorem

pg

Prop osition For an OL L C onL is distributive

Pro of It is well known that the congruence lattice of any lattice is distributive

pg Our result then follows from the fact that the congruence lattice of

an algebra A is a sublattice of the congruence lattice of any reduct of A

To summarize we have shown that OMLs are congruence regular congru

ence p ermutable and congruence distributive It seems to b e an op en question

to completely characterize those lattices which are isomorphic to the congruence

lattice of some OML As a nal remark we note that matters are much worse in the

absence of orthomo dularity Ortholattices are not in general congruence regular

or congruence p ermutable but as shown ab ove are congruence distributive

Pro ducts directly and subdirectly irreducibles

A congruence R on an OML L is called a factor congruence if R has a complement

in the congruence lattice of L Note as C onL is distributive R will then have

exactly one complement which we denote by R For readers familiar with the

denition of factor congruences for general algebras pg we recall that

every OML is congruence p ermutable

Lemma If R is a factor congruence on an OML L then the natural map

f L LR LR is an isomorphism Conversely if f L A B is an

isomorphism then the kernels of pr f and pr f are complementary factor

1 2

congruences

Pro of By general considerations f is a homomorphism If f x f y then

x y b elongs to b oth R and R hence x y Given x y in L the fact that R R

p ermute and join to the largest congruence of L gives the existence of z with xR z

and z R y Then f z xR y R Therefore the map f is oneone and onto

Conversely if R and S are the kernels of the natural pro jections of A B onto A

and B then R S intersect to the identical relation on A B and for any a b

c d in A B a bR a d and a dS c d Thus R S is the universal relation

on A B

The exact nature of the corresp ondence b etween central elements and factor

congruences on an orthomo dular lattice L is made precise by the following result

We leave the pro of to the reader

Prop osition The map c fx y jx c y cg is a lattice isomorphism

between C L and the Boolean subalgebra of C onL of factor congruences

Denition An ortholattice L is cal led directly irreducible if for every iso

morphism f L L L there is an index k so that the projection

1 n

pr f L L is an isomorphism

k k

In view of the ab ove remarks we have the following result

Prop osition For an OML L these are equivalent L is directly irre

ducible C L consists of exactly two elements L has exactly two factor

congruences

Lemma If R is a congruence on an OML L and R has a largest element

c then c is central in L and R is a factor congruence

Pro of As cR it follows that x x c b elongs to R for each x in L Hence

x x c c so x x c x c for all x in L showing that c is central

Therefore c is central and there is a congruence R with R equal to c Then

R R are complements in the congruence lattice of L

Recall an OL L is sub directly irreducible if it has a least nonzero congruence

and simple if it has exactly two congruences Obviously any simple OL is sub

directly irreducible and every sub directly irreducible OL is directly irreducible

From the preceding lemma we have the following

Prop osition A nite OML is directly irreducible i it is simple Therefore

every nite OML is isomorphic to a nite direct product of simple OMLs

Pro of Every nite OML is isomorphic to a nite direct pro duct of directly

irreducible OMLs

This result can b e easily generalized to hold for any OML in which all chains

are nite More generally it is known to hold for any chain nite relatively com

plemented lattice pg The rst step towards a dierent generalization of

is given by the following

Theorem The notions of directly irreducible and simple coincide in any

variety V generated by a class of OMLs with a nite upper bound on the lengths

of their chains

It is hop eless to exp ect that each OML in a variety such as V will b e iso

morphic to a direct pro duct of simple algebras In the Bo olean case this would

X

amount to having each Bo olean algebra B isomorphic to a p ower for some set

X and by cardinality considerations alone this is imp ossible for any countable

Bo olean algebra However Stones theorem provides that any Bo olean algebra B

X

is isomorphic to the collection of all continuous functions in for some Bo olean

space X where is given the discrete top ology We obtain a weaker but useful

analogue of Stones theorem

Theorem Let L be in a variety V generated by a class of OMLs with a nite

upper bound on the lengths of their chains Then there is a family of OMLs L

x

S

indexed by the elements x of a Boolean space X and a topology on fL jx X g

x

such that i the subspace topology on each L discrete ii L simple for al l x in

x x

Q

a dense open subset of X and iii L ff L jf is continuousg

x

While this result might seem ungainly there are eective to ols for working

with such a representationone can essentially lift many rst order prop erties

from the L to L For further details of this result see We remark that

x

the reader familiar with the notion of discriminator varieties pg will

have seen representation theorems very similar to the one ab ove However

Prop osition The only varieties of OMLs which are discriminator varieties

are the trivial variety and the variety of Boolean algebras

Pro of Let V b e a nonBo olean variety of OMLs Then V contains MO

2

which is simple and has a subalgebra which is not simple So by lemma

V is not a discriminator variety

We mention a generalization of in another direction It seems not

unreasonable to hop e that the following result might b e extended to a variety

generated by OMLs having at most n blo cks

Theorem An OML with nitely many blocks is directly irreducible i it is

simple Therefore an OML with nitely many blocks is isomorphic to a nite direct

product of a Boolean algebra and simple OMLs

Likely the most useful result concerning representations of OMLs by direct

pro ducts remains Birkho s sub direct representation theorem which states that

every algebra is isomorphic to a sub direct pro duct of sub directly irreducible alge

bras Unfortunately when working with the full variety of OMLs the sub directly

irreducibles are dicult to narrow down In fact

Prop osition Every OML is a subalgebra of a simple hence subdirectly

irreducible OML

Pro of Given an OML L construct an OML M by gluing L and a four element

Bo olean at their b ounds

Still there are many varieties of OMLs where one has very go o d control over

the sub directly irreducibles As ortholattices are congruence distributive

one may apply Jonssons theorem pg and Los Theorem pg to

any variety V generated by a class K of OMLs to gain insight into the rst order

prop erties of the sub directly irreducibles in V For example if every memb er of

K has at most n elements in each of its chains then the same is true of every

sub directly irreducible and in view of of every directly irreducible memb er

of V

Free ortholattices

Denition Given a class K of algebras of the same type F K is K freely

generated by a set X if i X F ii X generates F and iii every set map

f X A with A K extends uniquely to a homomorphism f F A

By a standard argument two algebras K freely generated by X are isomorphic

We next show the existence of a K freely generated algebra over X where K is the

class of all algebras of a given typ e Such algebras are called absolutely freely

generated

Denition Given a set X and a type n let be the set of al l nite

i I

strings of symbols from X I Dene the set of terms of type over X to be the

smal lest subset S of such that i X S and ii if i I and p p S

0 n 1

i

then the string ip p is in S

0 n 1

1

We use T X to denote the set of terms of typ e over X and use the com

mon convention of writing f p p in place of the string ip p

i 0 n 1 0 n 1

1 i

For each index i I let f b e the n ary op eration on T X dened by setting

i i

f p p f p p Then T X f is an algebra of typ e

i 0 n 1 i 0 n 1 i iI

i i

called the term algebra of typ e over X The following result is well known

pg and easily proved

Prop osition The term algebra T X is absolutely freely generated by X

Next we show the existence of K freely generated algebras over a set X at

least under mild assumptions on K For any set X we dene X to b e the in

K

tersection of all congruences C onT X such that T X b elongs to IS K

Theorem If K is closed under I S P then the algebra T X X is K

K

freely generated by X X

K

Note if V is a variety containing an algebra with more than one element one

can easily show that X X is in bijective corresp ondence with X and it follows

V

that there is an algebra V freely generated by X We denote this essentially

unique algebra by F X For V the variety of one element algebras we let

V

F X b e a one element algebra In either case there is an obvious homomorphism

V

T X F X The reader should consult pg for a pro of of ab ove

V

result

Denition An equation or identity of type over X is an ordered pair

p q where p q T X An algebra A satises the equation written A j p q

if f p f q for every homomorphism f T X A and a class of algebras K

satises the equation written K j p q if A j p q for each A K

For example the pair x y y is an equation in the typ e of OLs over the set

X fx y z g This equation will b e valid in some algebras in any one element

algebra for instance but is not valid in any nontrivial ortholattice The following

result is well known pg

Prop osition For a variety V and terms p q in T X the fol lowing are equiv

alent i V j p q ii F X j p q iii p q X iv p q

V V

Recall T X F X is the natural homomorphism

V

Denition A variety V has a solvable free word problem over X if there is

an algorithm to determine for any terms p q in T X whether p q

In view of the ab ove prop osition a solvable free word problem over X gives

an algorithm to determine whether an equation p q holds for all algebras in V

Theorem The variety of lattices has solvable free word problem over any set

Pro of While we do not provide a complete pro of of this well known theorem

pg it is worthwhile to sketch its features Dene to b e the smallest

binary relation on T X satisfying i x x for all x in X ii a c and b c

implies a b c iii a b and a c implies a b c iv a b or a c

implies a b c and v a c or b c implies a b c One can show that is

a quasiorder on T X Setting to b e the usual equivalence relation asso ciated

with a quasiorder one then shows T X is freely generated in the variety of

lattices by X As can b e eectively computed the word free word problem

for lattices is solvable

Theorem The variety of OLs has solvable free word problem over any set

Pro of Again the reader is directed to for a complete pro of but we sketch the

details Given a set X take another set X in bijective corresp ondence with X

and disjoint from X Consider the term algebra T X X of the typ e of lattices

and dene the relation on T X X as ab ove As T X X is absolutely

free the obvious map X X X X extends to a homomorphism from

T X X to its dual Dene R to b e the smallest subset of T X X satisfying

i X X is contained in R ii a b R and a b a b implies a b R and

iii a b R and a b a b implies a b R One can show that adding a

top and b ottom element to R yields an ortholattice freely generated by X

Various useful results ab out free lattices and free ortholattices are collected in

the following Here Whitmans condition refers to the prop erty that a b c d

i one of a b c a b d a c d b c d

Prop osition Every free lattice satises Whitmans condition A

lattice freely generated by a three element set contains a sublattice freely generated

by a Every free ortholattice satises Whitmans condition

An ortholattice freely generated by a two element set contains a subalgebra freely

generated by a countable set

The rst two statements can b e found in pg The third is easily

seen from the ab ove construction of free ortholattices The fourth is found in

Another very useful fact easily proved along the lines of is the following

4

Prop osition MO is freely generated by a two element set in the

2

variety of OMLs and the variety of MOLs Therefore the free word problem on

two generators is solvable in the variety of OMLs and the variety of MOLs

This gives an extremely simple pro cedure to determine if an equation involving

only two variables is valid in every OMLone simply checks to see if it is valid

in MO See for a discussion of how this simple observation could greatly

2

simplify many pro ofs in the literature For more than two generators the situation

is nearly completely op en Some of the few known facts are collected b elow

Prop osition If X has at least three elements then an OML freely generated

by X contains a free lattice on countably many generators as a sublattice of its

lattice reduct

Pro of Kalmbach has shown that any lattice L can b e emb edded into the

lattice reduct of some OML K L If L is a lattice freely generated by X then

there is an OL homomorphism from the free OML F on X onto K L So there

is a lattice homomorphism from the lattice reduct of F onto K L and as L is

pro jective in the variety of lattices see section L is isomorphic to a sublattice

of the lattice reduct of F The result then follows as L contains a sublattice freely

generated by a countable set

Prop osition Let X be a set with at least three elements and let L be freely

generated by X in the variety of OMLs or MOLs L contains an innite chain

and has innitely many blocks L does not contain an uncountable chain

Pro of The previous result shows a free orthomo dular lattice over X has an

innite chain There is an example in of a generated MOL with innite

chains and innitely many blo cks This provides the other assertions in this claim

As noticed by several authors this is generally true of free algebras in any

variety of algebras having a reduct

While we do not wish to develop the notion of word problems for nitely

presented algebras we do want to mention one of the very signicant results in

the area The reader is directed to for general background and the pro of of

the following result

Theorem There is a nitely presented MOL with unsolvable word problem

There remain many unsolved problems in this area The rst is of paramount

imp ortance the others less imp ortant but still of considerable interest

Problems Is the free word problem for OMLs MOLs on three or more

generators solvable Can a freely generated OML have an uncountable blo ck

If a b are complements in a freely generated ortholattice are b a and b a

complements of a Characterize the nite subalgebras of a freely generated

ortholattice OML

Varieties of ortholattices

For an ortholattice L let L b e the variety of ortholattices generated by L and

for a class K of ortholattices let K b e the variety generated by K Note that the

class of all one element OLs is a variety often called the trivial variety

Prop osition The trivial variety is the smal lest variety of OLs Every

nontrivial variety of OLs contains the variety of Boolean algebras Every

nonBoolean variety of OLs contains either MO or B enz ene

Pro of Obvious Every ortholattice with more than one element contains

a two element Bo olean algebra as a subalgebra and the two element Bo olean

algebra generates the variety of Bo olean algebras By every ortholattice

which is not orthomo dular contains a subalgebra isomorphic to Benzene and in

we showed that every nonBo olean variety of OMLs contains MO

2

For varieties of OMLs somewhat more is known

Prop osition Let V be a variety of OMLs that is generated by its nite mem

bers If V is not contained in MO then V contains a variety generated by one

of the four OMLs shown below





In each of these gures ortho complementary elements are directly ab ove and

b elow one another or directly b eside one another for the middle elements In the

nal gure the two elements on the left end are to b e identied with the two on

the right end

For varieties of MOLs the situation b ecomes very interesting We remind

the reader that a sub directly irreducible ortho complemented mo dular lattice of

height three is called a ortho complemented pro jective plane

Theorem The varieties of MOLs generated by their nite members are ex

actly the MO where is a cardinal

Pro of This is a dicult theorem but we can outline the steps in the pro of Sup

p ose L is a nite sub directly irreducible MOL If L is of height two or less then L is

equal to MO for some n Otherwise L contains an element a of height By

n

a theorem of Bruns the interval a of L is an ortho complemented pro jective

plane But Baer showed that every involution on a nite pro jective plane has

a xed p oint hence no nite pro jective plane admits an ortho complementation

Thus every nite sub directly irreducible MOL is an MO for some n

n

Supp ose V is a variety generated by a class K of nite MOLs As every nite

MOL is a direct pro duct of simple hence sub directly MOLs we may assume

each memb er of K is sub directly irreducible hence equal to MO for some n

n

If fmjMO K g is nite then it has a maximum n and clearly V MO

m n

Supp ose that fmjMO K g is innite We claim that MO b elongs to V hence

m

V MO But this follows as V is an equational class and any equation in n

variables failing in MO must fail in some n generated subalgebra of MO hence

in MO Finally note that MO MO for each innite cardinal as MO

n

and MO satisfy the same equations

Note that it is an easy consequence of Jonssons theorem that MO is cov

n

ered by MO for each n hence the varieties MO form a chain of order

n+1

typ e But these are not the only varieties of MOLs Let P b e an ortho com

plemented pro jective plane such as the lattice of subspaces of a three dimensional

vector space over the reals with the ortho complement of a subspace S b eing its

orthogonal subspace S Clearly there are equations valid in all MO such as

x y z which are not valid in P Thus P is distinct from all MO

However it is a simple matter to show that MO is a subalgebra of an interval of

P hence P contains MO The following two theorems summarize the remain

ing facts known of varieties of MOLs The rst is due to Bruns and the second

to Ro ddy

Theorem If L is a subdirectly irreducible MOL containing an atom then

either L MO for some cardinal or L contains P for some orthocomple

mented projective plane P

Theorem Every variety of MOLs distinct from MO for al l cardinals

contains MO

We are left with the following op en problem sometimes referred to as Bruns

conjecture We consider it a basic op en problem in the theory of OMLs

Problem Do es every variety of MOLs which is dierent from MO for all

cardinals contain an ortho complemented pro jective plane

Completions

A lattice L is called complete if every subset of L has a greatest lower b ound and a

least upp er b ound A completion of L is a lattice emb edding of L into a complete

lattice C A completion of L is called regular if the emb edding preserves all existing

joins and meets from L and is called join meet dense if every element of C is

the join meet of images of elements of L It is well known that an emb edding

that is b oth dense is regular

Theorem Every lattice L can be join densely embedded into a complete lattice

C which satises exactly the same equations as L

This well known theorem pg is proved by considering the mapping of

L into the ideal lattice I dL of L which takes an element a of L to the principal

ideal a generated by a One easily checks that this emb edding preserves all

existing meets but destroys all but essentially nite joins

Theorem Every lattice can be join and meet densely embedded hence regu

larly embedded into a complete lattice C

Pro of We provide a sketch for complete details see Given a lattice L let

P b e the p ower set of L Dene maps L U P P by setting for each A L

LA fxja A x ag and U A fxja A a xg One easily checks that

the comp osite LU is a closure op erator on P Therefore the closed sets form a

complete lattice C under set inclusion Consider the map L P dened by

setting a a Obviously is a lattice emb edding of L into C If A LU A

T S

it follows that A fu ju U Ag and as A is a downset A fa ja Ag

Therefore is b oth join and meet dense

In it was shown that up to isomorphism there is only one join and meet

dense completion of a lattice L We call this the MacNeille completion of L

Unfortunately MacNeille completions of lattices are p o orly b ehaved when it comes

to preserving identities In fact the variety of all lattices and the variety of one

element lattices are the only varieties of lattices which are closed under MacNeille

completions One might hop e to nd a completion which is b oth regular and

preserves identities This is not p ossible pg

Prop osition There is a distributive lattice which can not be regularly embed

ded into any complete distributive lattice

We next turn our attention to Bo olean algebras Recall the classic result of

Stone that for each Bo olean algebra B there is a zero dimensional compact Haus

dor space X called the Stone space of B with B isomorphic to the Bo olean

algebra of clop en subsets of X Stones representation theorem provides two nat

ural completions for Bo olean algebras

Theorem Let B be a Boolean algebra with Stone space X Then the col lection

R eg X of al l regular open subsets of X is a complete Boolean algebra and the

natural embedding of B into R eg X is both join and meet dense hence regular

For a pro of of this well known theorem see pg In view of the charac

terization of MacNeille completions of lattices as join and meet dense completions

we call this the MacNeille completion of B and denote it B Obviously taking

the full p ower set of X will also provide a completion of B which we call the

canonical completion of B and denote by B An abstract characterization of this

completion follows b elow

Theorem Up to isomorphism there is a unique embedding e B C of a

Boolean algebra B into a complete Boolean algebra C such that i each element of

C is a join of meets and a meet of joins of elements of eB and ii if S T B

V W V W

with eS eT then there are nite S S T T with eS eT

We remark that the most useful of lattice completions the ideal lattice can

not b e applied to Bo olean algebras as I dB is only complemented if B is nite

However the MacNeille completion B do es provide even a strengthening of

in the Bo olean case We next turn our attention to completions of ortholattices

Again we can not use ideal lattices to obtain completions as they will not b e or

tho complemented In fact it is not at rst apparent that there are any general

metho ds to complete ortholattices To the b est of our knowledge there are two

Theorem Up to isomorphism there is a unique embedding e L C of an

OL L into a complete OL C which is both join and meet dense hence regular

We call this the MacNeille completion L of L Existence was proved by

MacLaren by taking all subsets A L which are equal to the lower b ounds

of their upp er b ounds and dening the ortho complementation A fu u is an

upp er b ound of Ag Uniqueness follows from

Theorem Up to isomorphism there is a unique embedding e L C of an

OL L into a complete OL C such that i every element of C is a join of meets

V W

and a meet of joins of elements of eL and ii if S T B with eS eT

V W

then there are nite S S T T with eS eT

We call this the canonical completion L of L We remark that the corre

sp onding theorem holds for b ounded lattices as well Metho ds of obtaining such a

completion have b een around for some time For lattices one uses Urquharts

stable sets and for ortholattices Goldblatts lter space However the rst

abstract characterization and detailed study of this completion is in Unfor

tunately neither completion b ehaves well with resp ect to preserving equations

Prop osition There is an OML L with neither L nor L orthomodular

To pro duce an OML whose MacNeille completion is not orthomo dular take

an incomplete inner pro duct space E Let L b e the OML of all subspaces S of

E which are either nite dimensional or whose orthogonal subspace S is nite

dimensional Then the MacNeille completion of L is the ortholattice LE of

all subspaces S of E which satisfy S S But by a theorem of Amemiya and

Araki LE is orthomo dular i E is complete More elementary examples are

given in using a technique to construct an orthomo dular lattice from a given

lattice due to Kalmbach An example of an OML whose canonical completion

is not orthomo dular is given in This example is also based on the Kalmbach

construction Taking the direct pro duct of these two counterexamples yields an

OML with neither L nor L orthomo dular We remark that co mpletions can

b e found for these examples based on completing the underlying inner pro duct

space or the underlying lattice used in the Kalmbach construction but in general

the following remains one of the ma jor op en problems in the area

Problem Can every OML b e emb edded into a complete OML

There are partial results known In the presence of a niteness condition

we obtain the following generalization of the well known fact that the MacNeille

completion of a Bo olean algebra is Bo olean

Theorem Let V be a variety generated by a class of OMLs with a nite

upper bound on the lengths of their chains Then V is closed under MacNeil le

completions

The pro of of this theorem relies heavily on the representation theorem

for such varieties outlined in theorem It is a minor op en problem whether

this theorem would apply to a variety generated by a class K of OMLs with a

nite upp er b ound on the numb er of their commutators The following results are

useful in setting limits on what one can hop e to obtain in a completion See

for a pro of of the following

Prop osition Every regular completion of an OML factors through the Mac

Neil le completion Therefore there is an OML which cannot be regularly embedded

into a complete OML

Theorem There is a MOL which cannot be embedded into a complete MOL

Pro of A deep theorem of Kaplansky pg shows every complete MOL

is a continuous geometry and therefore has a dimension function Let M b e the

MOL of all subspaces S of a Hilb ert space H for which either S or S is nite

dimensional As M contains a countable set of pairwise p ersp ective atoms M

cannot admit a dimension function hence cannot b e emb edded into a complete

MOL

We conclude this section with a p ositive result which may eventually b e helpful

in solving the completion problem This result has a long history and we honestly

do not know who to credit for it See for an outline of a pro of and description

of the history

Prop osition Every OML can be embedded into an OML in which each ele

ment is a join of two or fewer atoms

Categorical prop erties

Every variety of algebras naturally forms a category whose ob jects are the alge

bras in the variety and whose morphisms are the homomorphisms b etween these

algebras not necessarily onto homomorphisms There are a large numb er of cat

egorical questions one can ask of such varieties We content ourselves with but a

few namely questions relating to monomorphisms epimorphisms injectives and

pro jectives The survey article is excellent source of information on categorical

issues relating to varieties of algebras

Denition Let V be a variety and h B C be a homomorphism between

members of V We say h is a monomorphism if for al l algebras A in V and al l

homomorphisms f g A B we have h f h g implies f g Similarly

h B C is an epimorphism if for al l algebras D in V and al l homomorphisms

f g C D we have f h g h implies f g

One easily sees that oneone homomorphisms are monomorphisms and onto

homomorphisms are epimorphisms The question arises whether there are any

others For monomorphisms the answer is easily found

Prop osition Let V be a variety of ortholattices Then the monomorphisms

in V are exactly the oneone homomorphisms

This is a well known result which holds for any variety of algebras The pro of

follows by noting that for any B in V and any x y in B there are homomorphisms

f g from the free algebra on one generator a four element Bo olean algebra in our

setting with f mapping the generator to x and g mapping the generator to y The

dual question whether every epimorphism is onto p oses much greater diculty

Before describing the known results we intro duce an additional notion pg

which is also of considerable interest

Denition Let K be a class of algebras of the same type A V formation in K

is a quintuplet B L L f f where B L L are algebras in K and f B L

1 2 1 2 1 2 i i

i are embeddings An amalgamation of the V formation in K is a triple

C g g where C is an algebra in K and g L C i are embeddings

1 2 i i

with g f g f The amalgamation is cal led strong if g L g L

1 1 2 2 1 1 2 2

g f B The class K is said to have the strong amalgamation property if every

1 1

V formation in K has a strong amalgamation

The connection b etween amalgamations and epimorphisms is given by the

following well known result

Lemma If a variety V has the strong amalgamation property then the epi

morphisms in V are exactly the onto homomorphisms

Pro of Supp ose h B C is not onto If D f g is a strong amalgamation of

the V formation hB C C id id then f h g h but f g

Prop osition The variety of OLs has the strong amalgamation property

therefore the epimorphisms in this variety are exactly the onto homomorphisms

Pro of A more detailed treatment is given in but we can outline the idea

Supp ose L and L are ortholattices and that B L L is a subalgebra of b oth

1 2 1 2

Dene a relation on L L by setting x y i one of the following o ccurs i x y

1 2

b elong to the same L i and x y or ii x b elongs to L y b elongs to L

i i i j

and there is some b in B with x b y One easily checks that L L is a

i j 1 2

partially ordered set and that the union of the ortho complementations on L L is

1 2

an ortho complementation on L L The result then follows from the well known

1 2

and easily proved fact that the MacNeille completion of a ortho complemented

p oset is an ortholattice

The following well known result is an interesting exercise

Prop osition The variety of Boolean algebras has the strong amalgamation

property so epimorphisms in this variety are exactly the onto homomorphisms

The situation for orthomo dular lattices is not so fortunate

Prop osition Neither the variety of OMLs nor the variety of MOLs have

the amalgamation property

There are however a numb er of sp ecial cases where V formations can b e amal

gamated in OML The rst result b elow was established in the second is a

reformulation of Greechies celebrated paste job

Theorem In the variety of OMLs any V formation B L L f f with

1 2 1 2

B Boolean has a strong amalgamation

Theorem Let B L L f f be a V formation in the variety of OMLs

1 2 1 2

such that there is an element a in B with f B the union of the principal ideal

i

f a and the principal lter f a in L Then there is a strong amalga

i i i

mation C g g of this V formation with L g L g L

1 2 1 1 2 2

Unfortunately the ab ove considerations have left the following op en questions

which we consider to b e basic op en problems in the area

Problem In the variety of OMLs MOLs are the epimorphisms exactly the onto

homomorphisms

We remark that in an eective pro cedure is given to determine if epimor

phisms coincide with onto homomorphisms in any variety generated by a nite

numb er of nite OMLs We next turn our attention to injective and pro jective

algebras

Denition An algebra C in a variety V is cal led injective if for every

monomorphism f A B and every homomorphism g A C there exists

a homomorphism h B C with h f g Dual ly C is cal led projective if for

every epimorphism f B A and every homomorphism g C A there exists

a homomorphism h C B with f h g The notions of weakly injective and

weakly projective are formed by replacing monomorphisms and epimorphisms with

oneone and onto homomorphisms

Note that injectives and weakly injectives coincide in any variety of algebras

as the monomorphisms in any variety are exactly the oneone homomorphisms

Characterizing injectives in certain varieties of OLs will p ose little diculty

Theorem In the variety of Boolean algebras an algebra is injective i it is

complete In the variety of OLs the variety of OMLs and the variety of MOLs

an algebra is injective i it has exactly one element

Pro of The result for Bo olean algebras is well known pg Supp ose V is

one of the varieties of OLs OMLs MOLs and C is a memb er of V having more

than one element As each of these varieties has simple algebras of arbitrarily

large cardinality MO there is a simple algebra B in V with cardinality greater

than C For f B and g C the obvious emb eddings there is no

homomorphism h B C with or without h f g

Note that pro jectives and weakly pro jectives need not coincide in a variety

where epimorphisms are not exactly the onto homomorphisms We consider only

weakly pro jectives The following well known result pg provides an abstract

characterization of the weakly pro jectives in any variety

Theorem For C an algebra in a variety V these are equivalent C is

weakly projective in V There is a free algebra F in V and homomorphisms

f F C and g C F with f g the identity on C

In particular any free algebra in V is weakly pro jective in V and any weakly

pro jective in V must b e a subalgebra of a free algebra However it can b e dicult

to provide a more direct characterization of weakly pro jectives Even for the

variety of Bo olean algebras no satisfactory description is known But we do have

the following sucient condition

Prop osition In the variety of Boolean algebras every at most countable

algebra with more than one element is weakly projective

It is p erhaps surprising that there are complete descriptions of the Bo olean

algebras that are weakly pro jective in the varieties of OLs and OMLs

Prop osition In the variety of ortholattices a Boolean algebra is weakly pro

jective i it has two four or eight elements

Pro of Let B b e a Bo olean algebra If B has one two or four elements the result

is trivial Kearnes established the result for an eight element Bo olean algebra If

B has more than eight elements then B do es not satisfy Whitmans condition By

every free ortholattice satises Whitmans condition Therefore B is not a

subalgebra of a free ortholattice hence is not weakly pro jective

Theorem In the variety of OMLs a Boolean algebra is weakly projective i

it has more than one element and is at most countable

A pro of of this result is found in There are a numb er of miscellaneous

results that may provide some feel for the topic First in the variety of OLs

b enzene is weakly pro jective More generally one obtains weakly pro jectives in

OLs by replacing the intervals on the sides of b enzene with a weakly pro jective

lattice and its dual Second MO is weakly pro jective in the variety of OMLs

2

Third MO is not weakly pro jective in the variety of MOLs The rst two

3

results are slight mo dications of well known and easily proved results The

third is much more dicult requiring in part the delicate construction of an

innite MOL with rather particular prop erties Also established in is the

following

Prop osition Let V be a variety of OMLs generated by a class of OMLs with

a nite upper bound on the lengths of their chains If A V is nite then A

is weakly projective in V

No doubt the reader is aware there are many op en questions in this area

Problems Characterize the weakly pro jectives in the variety of OLs OMLs and

MOLs

References

D H Adams The completion by cuts of an orthocomplemented modular lat

tice Bull Austral Math So c

I Amemiya and H Araki A remark on Pirons paper Pub Res Inst Math

Ser Kyoto Univ Ser A

R Baer Polarities in nite projective planes Bull Amer Math So c

D R Balb es and P Dwinger Distributive Lattices University of Missouri

Press Columbia Missouri

B Banaschewski Hul lensysteme und Erweiterungen von QuasiOrdnungen

Z Math Logik Grundl Math

L Beran Orthomodular Lattices Algebraic Approach Academia Prague

D Reidel Dordrecht

G Birkho Lattice Theory Third edition American Mathematical So ciety

Collo quium Publications Vol XXV American Mathematical So ciety Provi

dence RI

G Bruns Free ortholattices Can J Math no

G Bruns Varieties of modular ortholattices Houston J Math vol

G Bruns and J Harding Amalgamation of ortholattices Order

G Bruns and J Harding Epimorphisms in certain varieties of algebras Sub

mitted to Order Octob er

G Bruns and G Kalmbach Varieties of orthomodular lattices II Canad J

Math vol XXIV No

G Bruns and G Kalmbach Some remarks on free orthomodular lattices

Pro c Univ of Houston Lattice Theory Conf Houston

G Bruns and M Ro ddy A nitely generated modular ortholattice Canad

Math Bull Vol

G Bruns and M Ro ddy Projective orthomodular lattices Canad Math Bull

vol

G Bruns and M Ro ddy Projective orthomodular lattices II Algebra Univer

salis

S Burris and H P Sankappanavar A Course in Universal Algebra Springer

Verlag

P Crawley and R P Dilworth Algebraic Theory of Lattices Prentice Hall

G D Crown J Harding and M F Janowitz Boolean products of lattices

Order

M Dichtl Astroids and pastings Algebra Universalis

M Gehrke and J Harding Bounded lattice expansions Submitted to the

Trans Amer Math So c August

R I Goldblatt The Stone space of an ortholattice Bull London Math So c

G Gratzer General Lattice Theory Academic Press

R J Greechie On the structure of orthomodular lattices satisfying the chain

condition J Combin Theory

R J Greechie Orthomodular lattices admitting no states J Combin Theory

J Harding Orthomodular lattices whose MacNeil le completions are not or

thomodular Order

J Harding Irreducible orthomodular lattices which are simple Algebra Uni

versalis

J Harding Completions of orthomodular lattices II Order

J Harding Any lattice can be regularly embedded into the MacNeil le comple

tion of a distributive lattice The Houston Journal of Math

J Harding Canonical completions of lattices and ortholattices Tatra Mt

Math Publ

G Kalmbach Orthomodular lattices do not satisfy any special lattice equation

Arch Math Basel

G Kalmbach Orthomodular Lattices Academic Press

E W Kiss L Marki P Prohle and W Tholen Categorical algebraic prop

erties Compendium on amalgamation congruence extension epimorphisms

residual smal lness and injectivity Studia Sci Math Hungar

M D MacLaren Atomic orthocomplemented lattices Pac J Math

H M MacNeille Partial ly ordered sets Trans Amer Math So c

F Maeda and S Maeda Theory of Symmetric Lattices Springer

M Navara On generating nite orthomodular sublattices Tatra Mts Math

Publ

P Ptak and S Pulmannova Orthomodular Structures as Quantum Logics

Veda Bratislava

M Ro ddy An orthomodular analogue of the BirkhoMenger theorem Alge

bra Universalis

M Ro ddy Varieties of modular ortholattices Order

M Ro ddy On the word problem for orthocomplemented modular lattices

Canad J Math Vol XLI No

A Urquhart A topological representation theory for lattices Algebra Univer

salis