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Identity Problems, Solvability of Equations and Unification in Varieties of Semigroups Related to Varieties of Groups

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Algebraic Foundations in Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7020))

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Abstract

In this survey we consider three kinds of algorithmic questions concerning varieties of semigroups. We are interested in identity problems, in the solvability of a system of equations and in the structure of all solutions of a given system. We study them in significant varieties of semigroups, monoids, groups, completely simple semigroups, completely regular semigroups (in particular semigroups satisfying the identity x r ≈ x) and involutary semigroups (in particular star regular semigroups and inverse semigroups).

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References

  1. Adian, S.I.: The Burnside problem and identities in groups. Springer, New York (1979)

    Book  Google Scholar 

  2. Albert, M.H., Lawrence, J.: Unification in varieties of groups: nilpotent varieties. Canadian Journal of Mathematics 46(6), 1135–1149 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baader, F.: Unification in varieties of completely regular semigroups. In: Schulz, K.U. (ed.) IWWERT 1990. LNCS, vol. 572, pp. 210–230. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  4. Baader, F., Siekmann, J.H.: Unification Theory. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A., Siekmann, J.H. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 2, pp. 41–125. Oxford University Press, Oxford (1994)

    Google Scholar 

  5. Baader, F., Snyder, W.: Unification Theory. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 446–533. Elsevier, MIT Press (2001)

    Google Scholar 

  6. Burke, E.K., Truss, J.K.: Unification for nilpotent groups of class 2. Forum Math. 7, 435–437 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. O’Connor, J.J., Robertson, E.F.: A history of the Burnside problem, www-history.mcs.st-and.ac.uk/HistTopics/Burnside_problem.html

  8. Gerhard, J.A., Petrich, M.: Word problem for free objects in certain varieties of completely regular semigroups. Pacific J. Math. 104, 351–360 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Green, J.A., Rees, D.: On semigroups in which x r = x. Proc. of the Cambridge Phil. Soc. 48, 35–40 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deis, T., Meakin, J., Sénizergues, G.: Equations in free inverse monoids. International Journal of Algebra and Computation 17(4), 761–795 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Diekert, V.: Makanin’s algorithm. In: Lothaire, M. (ed.) Algebraic Combinatorics on Words, pp. 342–390. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  12. Hall Jr., M.: Theory of Groups. Macmillan, Basingstoke (1959)

    MATH  Google Scholar 

  13. Kharlampovich, O., Sapir, M.: Algorithmic problems in varieties. International Journal of Algebra and Computation 12, 379–602 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kisielewicz, A.: Unification in commutative semigroups. J. Algebra 200, 246–257 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kad’ourek, J., Polák, L.: On the word problem for free completely regular semigroups. Semigroup Forum 34, 127–138 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kad’ourek, J., Polák, L.: On free semigroups satisfying x r = x. Simon Stevin 64(1), 3–19 (1990)

    MathSciNet  MATH  Google Scholar 

  17. Klíma, O.: On the solvability of equations in semigroups with x r = x. Contributions to General Algebra 12, 237–245 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Klíma, O., Tesson, P., Thérien, D.: Dichotomies in the Complexity of Solving Systems of Equations over Finite Semigroups. Theory of Computing Systems 40(3), 263–297 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lankford, D.S., Butler, G., Brady, B.: Abelian group unification algorithms for elementary terms. Contemporary Mathematics 29, 193–199 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Makanin, G.S.: The problem of solvability of equations in free semigroups. Math. Sbornik 103, 147–236 (1977)

    MathSciNet  MATH  Google Scholar 

  21. Makanin, G.S.: Equations in a free group. Izv. Akad. Nauk SSSR, Ser. Mat. 46, 1199–1273 (1982)

    MathSciNet  MATH  Google Scholar 

  22. Mashevitzky, G.: Unification Types of Completely Regular Semigroups. J. Autom. Reasoning 29(2), 171–182 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Munn, W.D.: Free inverse semigroups. Proc. London Math. Soc. 30, 385–404 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  24. Plotkin, G.: Building-in Equational Theories. Machine Intelligence 7, 73–90 (1972)

    MATH  Google Scholar 

  25. Polák, L.: Unification in varieties of completely simple semigroups. Contributions to General Algebra 12, 337–348 (2000)

    MathSciNet  MATH  Google Scholar 

  26. Polák, L.: A solution of the word problem for free *-regular semigroups. Journal of Pure and Applied Algebra 157, 107–114 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Razborov, A.A.: On systems of equations in a free group. Izv. Akad. Nauk SSSR, Ser. Mat. 48(4), 779–832 (1984)

    MathSciNet  Google Scholar 

  28. Romankov, V.A.: Unsolvability of the endomorphic reducibility problem in free nilpotent groups and in free rings. Algebra and Logic 16, 310–320 (1977)

    Article  Google Scholar 

  29. Rozenblat, B.V.: Diophantine theories of free inverse semigroups. Sibirskii Matematicheskii Zhurnal 26(6), 101–107 (1986)

    MathSciNet  MATH  Google Scholar 

  30. Scheiblich, H.E.: Free inverse semigroups. Proc. Amer. Math. Soc. 38, 1–7 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  31. Siekmann, J.H., Szabó, P.: A Noetherian and Confluent Rewrite System for Idempotent Semigroups. Semigroup Forum 25(1), 83–110 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  32. Truss, J.K.: Equation-Solving in Free Nilpotent groups Class 2 and 3. Bull. London Math. Soc. 27(1), 39–45 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Ondřej Klíma & Libor Polák

Authors
  1. Ondřej Klíma
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  2. Libor Polák
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Editors and Affiliations

  1. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria

    Werner Kuich

  2. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    George Rahonis

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Klíma, O., Polák, L. (2011). Identity Problems, Solvability of Equations and Unification in Varieties of Semigroups Related to Varieties of Groups. In: Kuich, W., Rahonis, G. (eds) Algebraic Foundations in Computer Science. Lecture Notes in Computer Science, vol 7020. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24897-9_10

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  • DOI: https://doi.org/10.1007/978-3-642-24897-9_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24896-2

  • Online ISBN: 978-3-642-24897-9

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