A method for finding new sets of axioms for classes of semigroups

We introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices.

[1]  Marlow Sholander Postulates for Commutative Groups , 1959 .

[2]  Jeremy George Peterson Shortest single axioms for the classical equivalential calculus , 1976, Notre Dame J. Formal Log..

[3]  John Bacon Review: C. A. Meredith, A. N. Prior, Notes on the Axiomatics of the Propositional Calculus , 1968 .

[4]  Larry Wos,et al.  Short Single Axioms for Boolean Algebra , 2002, Journal of Automated Reasoning.

[5]  J. Howie Fundamentals of semigroup theory , 1995 .

[6]  Janusz Konieczny,et al.  Automorphism groups of centralizers of idempotents , 2003 .

[7]  William McCune,et al.  Automated discovery of new axiomatizations of the left group and right group calculi , 1992, Journal of Automated Reasoning.

[8]  Thomas W. Scharle Axiomatization of propositional calculus with Sheffer functors , 1965, Notre Dame J. Formal Log..

[9]  Branden Fitelson,et al.  Vanquishing the XCB Question: The Methodological Discovery of the Last Shortest Single Axiom for the Equivalential Calculus , 2004, Journal of Automated Reasoning.

[10]  Alfred Tarski,et al.  Equational Logic and Equational Theories of Algebras , 1968 .

[11]  William McCune,et al.  Single Axioms for the Left Group and the Right Group Calculi , 1992, Notre Dame J. Formal Log..

[12]  John A. Kalman A shortest single axiom for the classical equivalential calculus , 1978, Notre Dame J. Formal Log..

[13]  William McCune,et al.  Solution of the Robbins Problem , 1997, Journal of Automated Reasoning.

[14]  Lawrence J. Henschen,et al.  Questions concerning possible shortest single axioms for the equivalential calculus: an application of automated theorem proving to infinite domains , 1983, Notre Dame J. Formal Log..

[15]  Janusz Konieczny,et al.  Semigroups of Transformations Preserving an Equivalence Relation and a Cross-Section , 2004 .

[16]  A. P. Bowran A Boolean Algebra , 1965 .

[17]  Kenneth Kunen,et al.  Single axioms for odd exponent groups , 2004, Journal of Automated Reasoning.

[18]  J Konieczny,et al.  CENTRALIZERS IN THE SEMIGROUP OF PARTIAL TRANSFORMATIONS , 1998 .

[19]  William McCune,et al.  Single axioms for groups and Abelian groups with various operations , 1993, Journal of Automated Reasoning.

[20]  Henry M. Sheffer A set of five independent postulates for Boolean algebras, with application to logical constants , 1913 .

[21]  B. H. Neumann Another single law for groups , 1981, Bulletin of the Australian Mathematical Society.

[22]  A. Church Review: Carew A. Meredith, Single Axioms for the Systems $(C, N), (C, 0)$ and $(A, N)$ of the Two- Valued Propositional Calculus , 1954 .

[23]  E. V. Huntington A New Set of Independent Postulates for the Algebra of Logic with Special Reference to Whitehead and Russell's Principia Mathematica. , 1932, Proceedings of the National Academy of Sciences of the United States of America.

[24]  R. Padmanabhan,et al.  Equational theories of algebras with distributive congruences , 1973 .

[25]  S Winker Absorption and idempotency criteria for a problem in near-Boolean algebras , 1992 .

[26]  Larry Wos,et al.  Application of Automated Deduction to the Search for Single Axioms for Exponent Groups , 1992, LPAR.

[27]  Christopher Hollings,et al.  The Early Development of the Algebraic Theory of Semigroups , 2009 .

[28]  W. Taylor Equational logic , 1979 .

[29]  Vladimir Tasić On single-law definitions of groups , 1988, Bulletin of the Australian Mathematical Society.

[30]  Peter M. Neumann,et al.  What groups were: A study of the development of the axiomatics of group theory , 1999, Bulletin of the Australian Mathematical Society.

[31]  William McCune,et al.  Single identities for lattice theory and for weakly associative lattices , 1995 .

[32]  A. N. Prior,et al.  Equational logic , 1968, Notre Dame J. Formal Log..

[33]  E. V. Huntington Boolean algebra. A correction to: “New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell’s Principia mathematica” [Trans. Amer. Math. Soc. 35 (1933), no. 1, 274–304; 1501684] , 1933 .

[34]  A. N. Prior,et al.  Notes on the axiomatics of the propositional calculus , 1963, Notre Dame J. Formal Log..

[35]  R. Goodstein Boolean algebra , 1963 .

[36]  K. Kunen,et al.  A Generalization of Moufang and Steiner Loops , 2001, math/0105015.

[37]  C. A. Meredith Equational postulates for the Sheffer stroke , 1969, Notre Dame J. Formal Log..

[38]  William McCune,et al.  Computer Solutions of Problems in Inverse Semigroups , 2010 .

[39]  Kenneth Kunen Single axioms for groups , 2004, Journal of Automated Reasoning.

[40]  E. V. Huntington Sets of independent postulates for the algebra of logic , 1904 .

[41]  William McCune,et al.  Computer and Human Reasoning: Single Implicative Axioms for Groups and for Abelian Groups , 1996 .

[42]  Kenneth Kunen,et al.  The shortest single axioms for groups of exponent 4 , 1995 .

[43]  Bengt Stolt,et al.  Über Axiomensysteme die eine abstrakte Gruppe bestimmen , 1953 .