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Single Axioms for Groups and Abelian Groups with Various Operations 1 Single Axioms 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 axiom 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 identity element for division and k for double division Given a single axiom in one set 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 nongroup 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
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