Computer Solutions of Problems in Inverse Semigroups

In 1981, Tamura posed a number of problems regarding the axiomatic definition of inverse semigroups. The main goal of this article is to use automated reasoning to solve these problems. In the process, we find some new defining sets of identities for the class of inverse semigroups and provide a single identity for groups in terms of two binary operations.

[1]  D. McNair.  Duals of ideals in rings with zero divisors , 2011 .

[2]  George F. McNulty.  Minimum bases for equational theories of groups and rings: the work of Alfred Tarski and Thomas Green , 2004, Ann. Pure Appl. Log..

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

[4]  Dolph Ulrich.  A Legacy Recalled and a Tradition Continued , 2004, Journal of Automated Reasoning.

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

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

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

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

[9]  William McCune,et al.  OTTER 3.3 Reference Manual , 2003, ArXiv.

[10]  William McCune,et al.  Mace4 Reference Manual and Guide , 2003, ArXiv.

[11]  William McCune,et al.  Ivy: a preprocessor and proof checker for first-order logic , 2000 .

[12]  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.

[13]  Bernd I. Dahn.  Robbins Algebras Are Boolean: A Revision of McCune's Computer-Generated Solution of Robbins Problem , 1998 .

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

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

[16]  William McCune,et al.  Single identities for ternary Boolean algebras , 1995 .

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

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

[19]  George F. McNulty,et al.  A Field Guide to Equational Logic , 1992, J. Symb. Comput..

[20]  A. Tarski,et al.  Cylindric Algebras. Part II , 1988 .

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

[22]  Ewing Lusk,et al.  Using automated reasoning tools: A study of the semigroupF2B2 , 1987 .

[23]  B. H. Neumann.  Yet another single law for groups , 1986 .

[24]  Mario Petrich,et al.  Inverse semigroups , 1985 .

[25]  Günter Pilz,et al.  Near-rings : the theory and its applications , 1983 .

[26]  E. L. Lusk,et al.  Semigroups, antiautomorphisms, and involutions: a computer solution to an open problem. I , 1981 .

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

[28]  On the Theory of Inverse Semigroups and Generalised Grouds , 1979 .

[29]  Don Pigozzi,et al.  Equational Logic and Equational Theories of Algebras , 1975 .

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

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

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

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

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

[35]  Alfred Tarski.  Ein Beitrag zur Axiomatik der Abelschen Gruppen , 1938 .

[36]  E. V. Huntington.  Boolean Algebra. A Correction , 1933 .

[37]  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 .

[38]  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.

[39]  B. A. Bernstein.  A set of four independent postulates for Boolean algebras , 1916 .

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