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Single axioms for groups and Abelian groups with various operations
 Preprint MCSP2701091, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 1991
"... This paper summarizes the results of an investigation into single axioms for groups, both ordinary and Abelian, with each of following six sets of operations: fproduct, inverseg, fdivisiong, fdouble division, identityg, fdouble division, inverseg, fdivision, identityg, and fdivision, inverseg. In al ..."
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This paper summarizes the results of an investigation into single axioms for groups, both ordinary and Abelian, with each of following six sets of operations: fproduct, inverseg, fdivisiong, fdouble division, identityg, fdouble division, inverseg, fdivision, identityg, and fdivision, inverseg. In all but two of the twelve corresponding 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 proofs that given candidate axioms are in fact single axioms. 1
Single Axioms for Odd Exponent Groups
 J. Automated Reasoning
, 1995
"... With the aid of automated reasoning techniques, we show that all previously known short single axioms for odd exponent groups are special cases of one general schema. We also demonstrate how to convert the proofs generated by an automated reasoning system into proofs understandable by a human. x0. I ..."
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Cited by 13 (6 self)
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With the aid of automated reasoning techniques, we show that all previously known short single axioms for odd exponent groups are special cases of one general schema. We also demonstrate how to convert the proofs generated by an automated reasoning system into proofs understandable by a human. x0. Introduction. There are two eras in the history of single axioms for groups and varieties of groups. The early results, by Neumann and others [7], often produced single axioms which were larger than the minimal possible size, but which were constructed via some scheme which made them easy to verify by hand. Although not optimal, these results had the virtue that a person could conceptually grasp their proofs. The second era began with the advent of McCune's automated reasoning system OTTER [4]; now one could produce shorter and simpler single axioms, which often required much more complex verifications. Short single axioms for groups and some varieties of groups were found by McCune and Wos [...
Single Identities for Lattice Theory and for Weakly Associative Lattices
 Algebra Universalis
, 1995
"... . We present a single identity for the variety of all lattices that is much simpler than those previously known to us. We also show that the variety of weakly associative lattices is onebased, and we present a generalized onebased theorem for subvarieties of weakly associative lattices that can be ..."
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Cited by 11 (10 self)
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. We present a single identity for the variety of all lattices that is much simpler than those previously known to us. We also show that the variety of weakly associative lattices is onebased, and we present a generalized onebased theorem for subvarieties of weakly associative lattices that can be defined with absorption laws. The automated theoremproving program Otter was used in a substantial way to obtain the results. 1 Introduction Equational identities are, perhaps, the simplest form of sentences expressing many basic properties of algebras. Several familiar classes of algebras, such as semigroups, groups, rings, lattices, and Boolean algebras, are defined by equational identities. Such a class of algebras is known as an equational Supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W31109Eng38. y Supported by an operating grant from NSERC of Canada (#A8215). class of algebras or a variety of algebras (for mathematical properti...
Single Identities for Ternary Boolean Algebras
 Computers and Mathematics with Applications
, 1993
"... this paper, we show that the equational theory of TBAs is onebased. Our methods for finding a single identity for the theory of TBAs are interesting from two distinct points of view. First, from the algebraic, since TBAs enjoy both permutable and distributive congruences, they admit a single ternar ..."
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Cited by 9 (6 self)
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this paper, we show that the equational theory of TBAs is onebased. Our methods for finding a single identity for the theory of TBAs are interesting from two distinct points of view. First, from the algebraic, since TBAs enjoy both permutable and distributive congruences, they admit a single ternary polynomial p(x; y; z), the socalled Pixley polynomial [1, p. 405]. We first find such a polynomial p(x; y; z) and use a technique of R. Padmanabhan and R. W. Quackenbush [7] to construct a single identity for the equational theory in question. This is done in Section 2. Second, from the viewpoint of automated reasoning, we use the program Otter to discover new single identities based upon the results of the algebraic view. Actually we obtain here three new identities shorter in length than those obtained by the formal algebraic process of Section 2each characterizing the equational theory of TBAs. The relevant Otter proofs are also included. 2 The Algebraic View
Tarski’s finite basis problem via A(T
 Trans. Amer. Math. Soc
, 1997
"... Abstract. R. McKenzie [7] associates to each Turing machine T a finite algebra A(T) having some remarkable properties. We add to the list of properties, by proving that the equational theory of A(T) is finitely axiomatizable if T halts on the empty input. This completes an alternate (and simpler) pr ..."
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Cited by 9 (3 self)
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Abstract. R. McKenzie [7] associates to each Turing machine T a finite algebra A(T) having some remarkable properties. We add to the list of properties, by proving that the equational theory of A(T) is finitely axiomatizable if T halts on the empty input. This completes an alternate (and simpler) proof of McKenzie’s negative answer to A. Tarski’s finite basis problem [8]. It also removes the possibility (raised in [6]) of using A(T) to answer an old question of B. Jónsson. An algebra A = 〈A, (Fi)i∈I 〉 is finite if its universe A is a finite set, and is of finite type if its fundamental operations Fi are finite in number. All algebras to be considered in this paper are assumed to be of finite type. A variety is any nonempty class of algebras (all having the same type of fundamental operations) which is closed under the formation of arbitrary direct products, subalgebras, and homomorphic images. We say that a variety V is residually very finite and write κ(V) < ω if there exists a finite cardinal n such that every member of V can be embedded in a product of algebras all having fewer than n elements. V
Computer and Human Reasoning: Single Implicative Axioms for Groups and for Abelian Groups
, 1996
"... single axiom, but then \Delta is not product, and \Gamma1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and Supported ..."
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Cited by 3 (1 self)
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single axiom, but then \Delta is not product, and \Gamma1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W31109Eng38. the identity element [8]. Single equational axioms in terms of product and inverse have been reported by Neumann [5] and others [3, 2]. In this note we consider single implicative axioms, that is, axioms of the form ff = fi ) fl = ffi. For Abelian groups, an axiom of this type with five variables was given by Sholander [6]. If we allow one of f
Automated Equational Deduction with Otter
, 1995
"... Contents 1 Introduction 1 2 Otter and MACE 3 2.1 Otter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.1.1 Notes on Otter Proof Notation : : : : : : : : : : : : : : : 3 2.2 MACE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Test Chapter 3 3 Lattices a ..."
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Contents 1 Introduction 1 2 Otter and MACE 3 2.1 Otter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.1.1 Notes on Otter Proof Notation : : : : : : : : : : : : : : : 3 2.2 MACE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Test Chapter 3 3 Lattices and Latticelike Structures 9 4 The Rule (gL) 23 4.1 Problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 4.2 Sample Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 5 Quasigroups 51 6 Semigroups 57 6.1 A Conjecture of Padmanabhan : : : : : : : : : : : : : : : : : : : 57 7 Groups 69 7.1 SelfDual Bases for Group Theory : : : : : : : : : : : : : : : : : 69 8 TC and RC 73 9 Problems not yet placed in the proper chapter 83 iii iv CONTENTS List