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Morita equivalence of finite semigroups

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Abstract

Two semigroups are called Morita equivalent if the categories of firm right acts over them are equivalent. We prove that every semigroup is Morita equivalent to its subsemigroup consisting of all products of n factors. Using this we show that a finite semigroup is Morita equivalent to its largest factorisable subsemigroup. It follows that two finite semigroups are Morita equivalent if and only if their Cauchy completions are equivalent categories. Since these categories are finite, the problem of Morita equivalence of finite semigroups is decidable.

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References

  1. Almeida, J., Margolis, S., Steinberg, B., Volkov, M.: Representation theory of finite semigroups, semigroup radicals and formal language theory. Trans. Am. Math. Soc. 361(3), 1429–1461 (2009). https://doi.org/10.1090/S0002-9947-08-04712-0

    Article  MathSciNet  MATH  Google Scholar 

  2. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. vol. I. Mathematical Surveys, No. 7. American Mathematical Society, Providence (1961)

  3. Eilenberg, S.: Automata, Languages, and Machines (vol. B). With two chapters (“Depth decomposition theorem” and “Complexity of semigroups and morphisms”) by Bret Tilson, Pure and Applied Mathematics, Vol. 59. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1976)

  4. Fountain, J.: Perfect semigroups. Proc. Edinburgh Math. Soc. (2) 20(2), 87–93 (1976). https://doi.org/10.1017/S0013091500010592

    Article  MathSciNet  MATH  Google Scholar 

  5. García, J.L., Marín, L.: Some properties of tensor-idempotent rings. In: Algebra and its Applications (Athens, OH, 1999), Contemp. Math., vol. 259, pp. 223–235. American Mathematical Society, Providence (2000). https://doi.org/10.1090/conm/259/04097

  6. Laan, V.: Context equivalence of semigroups. Period. Math. Hungar. 60(1), 81–94 (2010). https://doi.org/10.1007/s10998-010-1081-z

    Article  MathSciNet  MATH  Google Scholar 

  7. Laan, V.: Acceptable Morita contexts for semigroups. ISRN Algebra 5, 725627 (2012). https://doi.org/10.5402/2012/725627

    Article  MathSciNet  MATH  Google Scholar 

  8. Laan, V., Márki, L.: Fair semigroups and Morita equivalence. Semigroup Forum 92(3), 633–644 (2016). https://doi.org/10.1007/s00233-015-9723-3

    Article  MathSciNet  MATH  Google Scholar 

  9. Laan, V., Márki, L., Reimaa, Ü.: Morita equivalence of semigroups revisited: firm semigroups. J. Algebra 505, 247–270 (2018). https://doi.org/10.1016/j.jalgebra.2018.02.018

    Article  MathSciNet  MATH  Google Scholar 

  10. Laan, V., Márki, L., Reimaa, Ü.: Lattices and quantales of ideals of semigroups and their preservation under Morita contexts. Algebra Univ. 81(2), (2020). https://doi.org/10.1007/s00012-020-0650-0

  11. Laan, V., Reimaa, Ü.: Morita equivalence of factorizable semigroups. Internat. J. Algebra Comput. 29(4), 723–741 (2019). https://doi.org/10.1142/S0218196719500243

    Article  MathSciNet  MATH  Google Scholar 

  12. Lawson, M.V.: Morita equivalence of semigroups with local units. J. Pure Appl. Algebra 215(4), 455–470 (2011). https://doi.org/10.1016/j.jpaa.2010.04.030

    Article  MathSciNet  MATH  Google Scholar 

  13. Marín, L.: Morita equivalence based on contexts for various categories of modules over associative rings. J. Pure Appl. Algebra 133(1–2), 219–232 (1998). https://doi.org/10.1016/S0022-4049(97)00197-7

    Article  MathSciNet  MATH  Google Scholar 

  14. Talwar, S.: Strong Morita equivalence and a generalisation of the Rees theorem. J. Algebra 181(2), 371–394 (1996). https://doi.org/10.1006/jabr.1996.0125

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Ülo Reimaa.

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Communicated by Mark V. Lawson.

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Research of Ü. Reimaa and V. Laan was partially supported by the Estonian Research Council grant PUT1519, research of L. Tart was supported by the Estonian Institutional Research Project IUT20-57. The work of the first author was supported in part by the Estonian Research Council Grants PRG49 and PSG114.

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Reimaa, Ü., Laan, V. & Tart, L. Morita equivalence of finite semigroups. Semigroup Forum 102, 842–860 (2021). https://doi.org/10.1007/s00233-020-10153-y

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