Single for Groups and Ab elian

Groups with Various Op erations

Wil liam W McCune

Mathematics and Computer Science Division

Argonne National Lab oratory

Argonne Illinois

USA

email mccunemcsanlgov

March

Abstract

This pap er summarizes the results of an investigation into single axioms for

groups b oth ordinary and Ab elian with each of following six sets of op erations

fpro duct inverseg fdivisiong fdouble division identityg fdouble division inverseg

fdivision identityg and fdivision inverseg In all but two of the twelve corresp onding

theories we present either the rst single axioms known to us or single axioms shorter

than those previously known to us The automated theoremproving program Otter

was used extensively to construct sets of candidate axioms and to search for and nd

pro ofs that given candidate axioms are in fact single axioms

Introduction

A single for an equational theory is an equality from which the entire theory can b e

derived For example each of the equalities

xxxy z xxxz y

1 1 1 1 1

x y x z u y u z

1

is a single axiom for ordinary groups Equation in terms of division

was shown to b e a single axiom by G Higman and B H Neumann in and was

given by Neumann in Each of and axiomatizes groups in the sense that

each generates a theory denitionally equivalent to standard axiomatizations for example

the triple

e x x

1

x x e

x y z x y z

where e is the identity



This work was supp orted by the Applied Mathematical Sciences subprogram of the Oce of Energy Research US Department of Energy under Contract WEng

The investigation summarized in this pap er fo cused on searching for simple single ax

ioms for groups and for Ab elian groups each in terms of each of the six sets of op era

tions fpro duct inverseg fdivisiong fdouble division identityg fdouble division inverseg

fdivision identityg and fdivision inverseg There is no single axiom in terms of

fpro duct inverse identityg New single axioms were found for each of the twelve

corresp onding theories In seven of the theories no single axioms were previously known to

us in three of the theories the new single axioms are shorter than those previously known

to us and in the remaining two cases the new single axioms are the same size as the ones

previously known

1

Operations Throughout the pap er we use for pro duct for inverse e for the

1 1 1

for division and k for double division Given a

single axiom in one of op erations it may seem trivial to obtain a single axiom in other

op erations by applying a simple transformation For example given xy z xy z

which is a single axiom for Ab elian groups and making the obvious transformation say

f g one obtains a single axiom in the sense that it is denitionally equivalent

to all other axiomatizations however f is not pro duct and g is not inverse

Mirror Images The mirror image of an equality with resp ect to a binary op erator is

obtained by reversing the arguments of all o ccurrences of the op erator The mirror image

of a single axiom in terms of pro duct and inverse or in terms of double division is also a

single axiom and the mirror image of a single axiom in terms of right division is a single

1

axiom in terms of left division

Axiom Type We considered length number of variable o ccurrences and number of

distinct variables as measures when searching for simple single axioms It is known that in

a single axiom say for any variety of groups either or must b e a variable

Assuming is the variable we say that a single axiom has type hL N D i if L is the

number of variable and op erator o ccurrences in N is the number of variable o ccurrences

in and D is the number of distinct variables in Kunen classies axioms by just

hN D i

The Otter automated theoremproving program was used extensively in two distinct

ways Section during the investigation as a symbolic calculator to construct sets

of candidate axioms and to search for pro ofs that given candidates are single axioms

Theoremproving programs have b een used in the past to verify known single axioms for

groups and to search for and nd new single axioms for nonequality theories of groups

Kunens goal in his recent study of single axioms for groups was to nd

precisely how small a single axiom for ordinary groups in terms of pro duct and inverse

can b e By giving non mo dels of all candidates he showed that no single axiom of

type hx i exists When trying to show that there are no single axioms of type h i

he found with help from Otter several of that type eg b elow

Previously Known Single Axioms

As far as we know the following are the simplest previously known single axioms for groups

and Ab elian groups The type and reference are given for each

Ordinary Groups

xxxy z xxxz y h i

1 1 1 1 1

z x y z y y y x h i

x k y k e k z k u k u k e k x k e k y z h i

Ab elian Groups

xy z xy z h i

1 1 1 1 1 1 1

x y y x z u z v w u w v h i

The preceding equalities except can b e shown to b e single axioms by the metho ds

presented in Section Kunen veried by using Otter with a nonstandard strategy

Neumann claims in that

1 1 1 1

x k y k z k y k u k z k x u

is a single axiom for ordinary groups but a twoelement mo del of a k a a b k b a

1 1

a k b b b k a b a b b a shows that it is not b ecause there is no element e

1

for which e e The counterexample was found by J Slaneys program Finder

Prior to the investigation we did not know of any single axioms for the remaining

theories Tarski states in p that single axioms exist for fdivision identityg and

fdivision inverseg but none is given there Neumann states p that it should b e

quite feasible to nd single axioms for ordinary groups in terms of fdivision identityg and

in terms of fdivision inverseg and for Ab elian groups in terms of fdouble division identityg

Neumann also conjectured that the simplest single axiom for ordinary groups in

terms of pro duct and inverse has type h i However Kunens axiom has type

h i and we present one of type h i in the following section

New Single Axioms

Tables and contain representatives of the single axioms that were found by the metho ds

summarized in Section Pro ofs for axioms and are given in Section Pro ofs

for the other single axioms listed in this section can b e found in

Table New Single Axioms for Ordinary Groups

Op erators Axiom Type Ref

1 1 1 1

and x y z z u y x u h i

xy y y z y y xz y h i

and e exy xxxz z y h i

1 1

and xxy z uy u z h i

k and e x k x k y k z k y k e k e k e z h i

1 1 1 1

k and x k x k y k z k u k y k u z h i

Table New Single Axioms for Ab elian Groups

Op erators Axiom Type Ref

1 1

and x y z x z y h i

xxy z y z h i

and e exy z xz y h i

1 1

and xy xz z y h i

k and e x k z k x k y k e k y k e k e z h i

1 1 1 1

k and x k x k y k z k y z h i

The axioms in division alone and are the same type as those previously

known The remaining axioms in Tables and are either the rst known to us or simpler

than those previously known to us

Metho dology

Otter is a computer program that searches for pro ofs of conjectures stated in rst

order logic with equality The user sp ecies inference rules search strategies and the way

that derived formulas are to b e pro cessed Inference rules are of two types resolution

rules which are based on a generalization of mo dus p onens and paramo dulation rules

which generalize equality substitution Search strategies include restricting application of

the inference rules and metho ds for selecting the next formula on which to fo cus Pro cessing

of derived formulas includes metho ds for discarding them and metho ds for turning derived

equalities into simplication rules to b e applied to subsequently andor previously derived

formulas

Trying to Prove That a Candidate Is a Single Axiom

Otter can b e directed to p erform a search based on the KnuthBendix completion pro ce

dure for equational theories Briey the KnuthBendix pro cedure attempts to convert

a set of equalities into a terminating and conuent set of rewrite rules which is a decision

pro cedure for the word problem for the theory The pro cedure derives new equalities by a

restricted form of paramo dulation using a usersupplied ordering on terms to orient new

equalities into rewrite rules and keeps everything fully simplied with resp ect to the set of

rewrite rules Success o ccurs if every derived equality can b e oriented and the pro cedure

terminates We used two wellknown extensions to the pro cedure turning it into

a pro of refutation search by including denials of known axiomatizations in the input and

allowing nonorientable equalities to enter the search The extended pro cedure is useful

even in cases when it do es not terminate

We typically started searches with a candidate and with denials of all single axioms

known to us In addition we input denials of other prop erties such as asso ciativity of

pro duct and the existence of an identity Otter was directed to output all pro ofs that

it found within a sp ecied time Although the precise settings of the Otter parameters

varied for the dierent theories we explored we remark on the general Otter

strategies we used

We set the kunthbendix ag which automatically sets several paramo dulation de

mo dulation and ordering parameters

We used the lexicographic recursive path ordering the ordering on op erators was

1 1 1

typically e k e and k with and k having

lefttoright status

We set an initial maximum of or on the weight symbol count of inferred

clauses then reduced the limit to to after given clauses

When more than one prop erty was required to show that a candidate is a single axiom

we input the denials as unit clauses rather than as a multiliteral clause and marked

success with multiple pro ofs rather than with a single pro of the reasons are fairly

technical It is sound to do so in this case b ecause paramo dulation and demo dulation

alone will not cause the unit denials to interact with one another

Constructing and Testing Sets of Candidates

Aside from automated theorem proving Otter can also serve as a symbolic calculator

for which the user programs his or her task with formulas and equalities that have a

pro cedural rather than a declarative interpretation Examples of four types of task are

given a set of equalities decide which are true in all groups given a string of

symbols generate all binary asso ciations of the symbols given a set of equalities insert

in a wellformed way sp ecied terms into each and given a set of equalities apply

paramo dulation with sp ecied equalities in a very constrained way Otter was used in

this programmed mo de to construct sets of prosp ective single axioms

The practical limit on the size of candidate sets was ab out members and most

candidate sets we used had fewer than members That way we could run a set

overnight or over a long weekend and allow seconds for each candidate Given a set

of candidates we ran a simple program that initiates a separate Otter search with each

candidate The search strategy and the list of denials of known single axioms and other

key prop erties were xed for the sequence of runs The time limit for each search varied

from to seconds dep ending on the size of the set Any pro ofs that were found were

collected in a le

Groups in terms of pro duct and inverse Neumanns axiom has type h i

We built the set of identities of type hx i for x corresp onding to

o ccurrences of inverse At that time we were not aware of Neumanns theorem that all

single axioms have an o dd number of o ccurrences of inverse which eliminates lengths

and The set has members Lo oking to Neumanns single axioms for

guidance we decided to consider candidates without inverse at the outermost level and with

inverse applied to pro duct in at least two places No single axioms were found for length

but many were found for length We later found others of the same type with inverse at

the outermost level A total of single axioms of type h i excluding mirror images

were found We list here ve representatives

1 1 1

x y z z u y x u

1 1 1

x y z x y u u z

1 1 1

x y z z u y x u

1 1 1

x y y z u x z u

1 1 1

x y z z u x y u

We then tried without success to nd axioms of type h i by taking instances of the

axioms All other attempts at nding simpler axioms failed Kunen later showed

that the only p ossibility for a simpler axiom is h i

Ab elian groups in terms of pro duct and inverse Neumanns axiom has type

h i The rst approach was to take axioms of type h i for ordinary groups and

commute pro ducts in all p ossible ways This led to many single axioms of type h i

6

Of the commuted variants of just the rst of the axioms were found to b e

single axioms for Ab elian groups To search for simpler axioms we extracted the identities

of type hx i for x from the member set mentioned in the

preceding paragraph and obtained all commuted variants of those This approach led to

three single axioms excluding mirror images of type h i

1

x y z x z y

1

x y z x z y

1

x x y z y z

Note that simpler single axioms are imp ossible b ecause three variables and at least

one o ccurrence of inverse are required

Groups in terms of division Higman and Neumanns axiom has type h i

We rst considered identities of types h i h i and h i and found single

axioms of type h i We then examined threevariable instances of the axioms and

found the following two new axioms of type h i

xy y y z y y xz y

xxxy xxxz z y

Ab elian groups in terms of division Two single axioms of type h i were previ

ously known to us Tarskis and Higman and Neumanns xy z y x z

No simpler axiom is p ossible b ecause three variables are required We examined identities

of the same type and found the following four additional single axioms

xxy z y z

xy z xy z

xxy z y z

xy xz y z

Groups in terms of division and identity No single axioms were previously known

to us We rst considered without success identities of type h i three o ccurrences

of identity We then to ok the single axioms in division alone type h i each has

two o ccurrences of for variable Considering the six candidates of type h i

obtained from those by replacing one o ccurrence of with the identity we found the

following four single axioms

exy xxxz z y

xxxy exz z y

xey z xxxz y

xxxy z exz y

We note that by replacing xx with e in a single axiom for groups is obtained with

ee as the identity shown by Otter but e is not the identity shown by Finder

Ab elian groups in terms of division and identity No single axioms were previously

known to us We started with the four single axioms for ordinary groups in terms of division

and identity and applied limited paramo dulation with the Ab elian identities xxy y

and xy z xz y to obtain a set of candidates of various types Several single

axioms were found the simplest b eing the following three of type h i

exy z xz y

exy y z x z

exy xz y z

Groups in terms of division and inverse No single axioms were previously known

to us We to ok the single axioms in division alone and applied paramo dulation

1

from the denition of inverse y y x x one time at each p ossible p osition obtaining

candidates of various type The simplest single axioms discovered were fteen of type

h i We list ve representatives here

1

xxy z uy u z

1

xy z u uz x y

1

xy z uxuz y

1

xy z uz y x u

1

xxy z y u u z

Ab elian groups in terms of division and inverse No single axioms were previously

known to us We rst to ok the single axioms from the preceding paragraph and applied

paramo dulation with the Ab elian identity xxy y obtaining candidates In that

set fourteen single axioms were found all of type h i We then turned to an exhaustive

search of the h i candidates The following three single axioms of that type were found

1

xy xz z y

1

xy z xz y

1

xy z xz y

Groups in terms of double division and identity The only single axiom known to

us was Neumanns of type h i We tried all candidates of type h i three

o ccurrences of identity and discovered single axioms These axioms are noteworthy

b ecause they are the simplest in terms of variable o ccurrences and distinct variables of all

known single axioms for ordinary groups We list here ve representatives

x k x k y k z k y k e k e k e z

x k e k x k e k y k z k y k e z

e k x k e k x k y k e k z k y z

e k x k y k z k e k e k x k z y

e k x k y k e k y k z k x k e z

Ab elian groups in terms of double division and identity No single axioms were

previously known to us We to ok the single axiom for ordinary groups applied com

mutativity of k in all combinations and deleted mirror images obtaining candidates

Eleven of those also of type h i were found to b e single axioms We list here ve

representatives

x k y k x k z k e k z k e k e y

x k y k x k z k z k e k e k e y

x k y k z k x k z k e k e k e y

x k y k x k z k y k e k e k e z

x k e k y k z k y k x k e k e z

This case double division and identity is noteworthy b ecause it is the only case in which

the currently known single axioms for Ab elian groups are not simpler than the single axioms

for ordinary groups

Groups in terms of double division and inverse No single axioms were previously

known to us We rst considered without success all identities of type hx i for x

corresp onding to three four and ve o ccurrences of inverse We then considered

1 1

identities of type h i without o ccurrences of k or k for variable

and discovered two single axioms

1 1 1

k x k y k z k u k y k u z x

1 1 1

x k y k z k y k u k x k u z

Ab elian groups in terms of double division and inverse No single axioms were

previously known to us We rst to ok the rst single axiom for ordinary groups and applied

commutativity of k in all combinations Six of the resulting candidates type h i

were found to b e single axioms We then considered all identities of types h i and

h i one and three o ccurrences of inverse and found the following eight single axioms

of type h i

1 1 1

x k x k y k z k y z

1 1 1

x k y k x k z k y z

1 1 1

x k y k x k z k y z

1 1 1

x k y k x k z k y z

1 1 1

x k y k x k z k z y

1 1 1

x k y k x k z k z y

1 1 1

x k y k z k x k z y

1 1 1

x k y k z k x k z y

Pro ofs for Pro ductInverse Axioms and

This section contains Otters pro ofs that and are single axioms in terms of

pro duct and inverse for groups and Ab elian groups resp ectively Pro ofs found by our most

successful Otter strategies usually are more complex than necessary and the following

pro ofs are the result of several tricks to co erce Otter into nding shorter pro ofs

The justication m n indicates paramo dulation from n into n substitution of an

instance of the left side of m for an instance of a term in the left side of n and i j

indicates simplication with i j The numbering of the equalities reects the sequence

of equalities retained by the program

Theorem The theory of groups can be dened by the single axiom

1 1 1

x y z z u y x u

Proof First Eq holds in groups a fact that can b e checked by straightforward

calculation Next consider the following derivation starting with

1 1 1

x y z z u y x u

1 1 1 1

x y y z u v v z x u

1 1 1 1 1

x y z z u x v v y u w w

1 1 1 1 1

y y z z u x u v v x

1 11 1

x x y z z y

1 1 1 1

x x y z u u y z

1 1 1 1

x x y z z y

1 1

u u v v

1 1 1 1

x x y y z z

1 1 1 1 1

x x y y z u u z

1 1 1

x y z z x y

1 11 1 1

x u u w x w

1 1 1 1 1

x x y z z y

1 1 11 1 1 1

x y y z z x

1 1 11 1 1

y y x z z x

1 11 1 1

x y y z z x

1 11 1 1

x x y y z z

1 1

x y y x

1 1 11

z z u u

1 1 1 11

x y y x

1 1 1 11

y z z x y x

1 1 1

x y y x

1 1 11

x x y y

1 1 1

u u y y

1

y u u y

1

y z z y

11

x y x y

1

x x z z

1

x y x y u u

11

y y

1 1 1

z x x z

1

x x u y u y

x y z x y z

1

By there exists a unique element e such that for all x x x e it follows from

that e is a right identity and shows the asso ciativity of pro duct

Theorem The theory of Abelian groups can be dened by the single axiom

1

x y z x z y

Proof First Eq holds in Ab elian groups a fact that can b e checked by straight

forward calculation Next consider the following derivation starting with

1

x y z x z y

1 1

x y z x u y z u

1 1

x y x y z z

1 1 1

x y z x y z

1 1

y z y z x x

1 1

x y x y z z

1

x y x y

1 1 1

z y z x x y

1 1

x y x z y z

1

x z x z

1 1

x x z z

1 1

x y x y

1 1 1

y z y z

1 1

x y z x y z

1 11

x y y x

1 11 11

x x y y

11

x x

1

x x y y

1

x y y x

1

x x y y

x y y x

1

x y y x

1 1

x x y y

1

z x y z x y

x y z z x y

x y z x y z

1

By there exists a unique element e such that for all x x x e it follows from

that e is a right identity and and resp ectively show the commutativity and

asso ciativity of pro duct

Conclusion

The remaining question is whether there exist single axioms simpler than the ones listed in

Tables and Our fo cus has b een on nding short axioms rather than on nding ones

with few variables The two cases that currently use four variables are fdivision inverseg

and fdouble division inverseg for ordinary groups and in Table Are there

single axioms for these cases with just three variables

With our type criteria there is currently no simplestknown single axiom for groups

in terms of pro duct and inverse the ones given here have type h i and Kunens

have type h i The only p ossibility for a simpler single axiom is h i b ecause a

single axiom must have an o dd number of o ccurrences of inverse at least seven variable

o ccurrences and at least three variables

The goal of the work rep orted here was to nd simple single axioms We made o ccasional

use of Finder to search for small counterexamples when we had a promising candidate

that Otter could not prove to b e a single axiom However further work in this area will

most likely b enet from extensive mo del searches for b oth simple mo dels with a program

like Finder and for more complex mo dels as in Kunens metho ds

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