Advertisement

Archive for Mathematical Logic

, Volume 51, Issue 5–6, pp 461–474 | Cite as

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

  • João AraújoEmail author
  • Janusz Konieczny
Article

Abstract

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.

Keywords

Equational logic Groups of finite exponent Bands Semilattices Inverse semigroups 

Mathematics Subject Classification

08B05 03C05 20A05 20M20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Araújo J., Konieczny J.: Automorphism groups of centralizers of idempotents. J. Algebra 269, 227–239 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Araújo J., Konieczny J.: Semigroups of transformations preserving an equivalence relation and a cross-section. Comm. Algebra 32, 1917–1935 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Araújo J., McCune W.: Computer solutions of problems in inverse semigroups. Comm. Algebra 38(3), 1104–1121 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Croisot R.: Demi-groupes et axiomatique des groupes. C. R. Acad. Sci. Paris 237, 778–780 (1953)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hart J., Kunen K.: Single axioms for odd exponent groups. J. Automat. Reason. 14, 383–412 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Higman G., Neumann B.H.: Groups as groupoids with one law. Publ. Math. Debrecen 2, 215–221 (1952)MathSciNetGoogle Scholar
  7. 7.
    Hollings C.: The early development of the algebraic theory of semigroups. Arch. Hist. Exact Sci. 63, 497–536 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Howie J.M.: Fundamentals of Semigroup Theory. Oxford University Press, New York, NY (1995)zbMATHGoogle Scholar
  9. 9.
    Huntington E.V.: New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell’s principia mathematica. Trans. Am. Math. Soc. 35, 274–304 (1933)MathSciNetGoogle Scholar
  10. 10.
    Huntington E.V.: 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. Am. Math. Soc. 35, 557–558 (1933)MathSciNetGoogle Scholar
  11. 11.
    Kalman J.A.: A shortest single axiom for the classical equivalential calculus. Notre Dame J. Form. Log. 19, 141–144 (1978)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kinyon M.K., Kunen K., Phillips J.D.: A generalization of Moufang and Steiner loops. Algebra Universalis 48, 81–101 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Konieczny J., Lipscomb S.: Centralizers in the semigroup of partial transformations. Math. Japon. 48, 367–376 (1998)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kunen K.: Single axioms for groups. J. Automat. Reason. 9, 291–308 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kunen K.: The shortest single axioms for groups of exponent 4. Comput. Math. Appl. 29, 1–12 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Łukasiewicz J.: Selected works. In: Borkowski, L. (eds) Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, London (1970)Google Scholar
  17. 17.
    McCune W.: Automated discovery of new axiomatizations of the left group and right group calculi. J. Automat. Reason. 9, 1–24 (1992)MathSciNetzbMATHGoogle Scholar
  18. 18.
    McCune W.: Single axioms for groups and abelian groups with various operations. J. Automat. Reason. 10, 1–13 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    McCune W.: Single axioms for the left group and right group calculi. Notre Dame J. Form. Log. 34, 132–139 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    McCune W.: Solution of the Robbins problem. J. Automat. Reason. 19, 263–276 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    McCune W., Padmanabhan R.: Single identities for lattice theory and for weakly associative lattices. Algebra Universalis 36, 436–449 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    McCune W., Sands A.D.: Computer and human reasoning: single implicative axioms for groups and for abelian groups. Am. Math. Mon. 103, 888–892 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    McCune W., Veroff R., Fitelson B., Harris K., Feist A., Wos L.: Short single axioms for Boolean algebra. J. Automat. Reason. 29, 1–16 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    McCune, W., Wos, L.: Applications of automated deduction to the search for single axioms for exponent groups, Logic programming and automated reasoning (St. Petersburg, (1992), 131–136, Lecture Notes in Comput. Sci., 624, Springer, Berlin (1992)Google Scholar
  25. 25.
    Meredith C.A.: Single axioms for the systems (C, N), (C, O) and (A, N) of the two-valued propositional calculus. J. Comput. Syst. 1, 155–164 (1953)MathSciNetGoogle Scholar
  26. 26.
    Meredith C.A.: Equational postulates for the Sheffer stroke. Notre Dame J. Form. Log. 10, 266–270 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Meredith C.A., Prior A.N.: Notes on the axiomatics of the propositional calculus. Notre Dame J. Form. Log. 4, 171–187 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Meredith C.A., Prior A.N.: Equational logic. Notre Dame J. Form. Log. 9, 212–226 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Neumann B.H.: Another single law for groups. Bull. Austral. Math. Soc. 23, 81–102 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Neumann P.M.: What groups were: a study of the development of the axiomatics of group theory. Bull. Austral. Math. Soc. 60, 285–301 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Padmanabhan R., Quackenbush R.W.: Equational theories of algebras with distributive congruences. Proc. Am. Math. Soc. 41, 373–377 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Peterson J.G.: Shortest single axioms for the classical equivalential calculus. Notre Dame J. Form. Log. 17, 267–271 (1976)zbMATHCrossRefGoogle Scholar
  33. 33.
    Rezus A.: On a theorem of Tarski. Libertas Math. 2, 63–97 (1982)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Scharle T.W.: Axiomatization of propositional calculus with Sheffer functors. Notre Dame J. Form. Log. 6, 209–217 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Sheffer H.M.: A set of five independent postulates for Boolean algebras, with application to logical constants. Trans. Am. Math. Soc. 14, 481–488 (1913)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Sholander M.: Postulates for commutative groups. Am. Math. Mon. 66, 93–95 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Sobociński B.: Axiomatization of a partial system of three-value calculus of propositions. J. Comput. Syst. 1, 23–55 (1952)Google Scholar
  38. 38.
    Stolt B.: Über Axiomensysteme die eine abstrakte Gruppe bestimmen, (Thesis), University of Uppsala, Almqvist & Wiksells, Uppsala (1953)Google Scholar
  39. 39.
    Tarski, A.: Equational logic and equational theories of algebras. Contributions to Math. Logic (Colloquium, Hannover, 1966), North-Holland, Amsterdam (1968)Google Scholar
  40. 40.
    Tasić V.: On single-law definitions of groups. Bull. Austral. Math. Soc. 37, 101–106 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Winker S.: Absorption and idempotency criteria for a problem in near-Boolean algebras. J. Algebra 153, 414–423 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Wos L., Ulrich D., Fitelson B.: Vanquishing the XCB question: The methodological discovery of the last shortest single axiom for the equivalential calculus. J. Automat. Reason. 29, 107–124 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Wos L., Winker S., Veroff R., Smith B., Henschen L.: Questions concerning possible shortest single axioms for the equivalential calculus: an application of automated theorem proving to infinite domains. Notre Dame J. Form. Log. 24, 205–223 (1983)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Universidade AbertaLisbonPortugal
  2. 2.Centro de ÁlgebraUniversidade de LisboaLisbonPortugal
  3. 3.Department of MathematicsUniversity of Mary WashingtonFredericksburgUSA

Personalised recommendations