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Intervals of varieties of involution semigroups with contrasting reduct intervals

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Abstract

It is known that the lattice structure of an interval of varieties of involution semigroups can be very different from that of its reduct interval of varieties of semigroups, but it is uncertain how wide this difference can be. The present article exhibits intervals to show that the difference can be between being a two-element chain and having an infinite chain. These intervals are bounded by varieties generated by an involution semigroup of order between four and six. One of the involution semigroups of order four turns out to be a smallest involution semigroup to generate a non-Specht variety. In contrast, it is long known that every semigroup of order up to four generates a Specht variety.

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References

  1. Auinger, K., Dolinka, I., Volkov, M.V.: Equational theories of semigroups with involution. J. Algebra 369, 203–225 (2012)

    Article  MathSciNet  Google Scholar 

  2. Auinger, K., Dolinka, I., Volkov, M.V.: Matrix identities involving multiplication and transposition. J. Eur. Math. Soc. 14, 937–969 (2012)

    Article  MathSciNet  Google Scholar 

  3. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981)

    Book  Google Scholar 

  4. Distler, A., Kelsey, T.: The semigroups of order 9 and their automorphism groups. Semigroup Forum 88, 93–112 (2014)

    Article  MathSciNet  Google Scholar 

  5. Dolinka, I.: Remarks on varieties of involution bands. Commun. Algebra 28, 2837–2852 (2000)

    Article  MathSciNet  Google Scholar 

  6. Dolinka, I.: On the lattice of varieties of involution semigroups. Semigroup Forum 62, 438–459 (2001)

    Article  MathSciNet  Google Scholar 

  7. Edmunds, C.C.: Varieties generated by semigroups of order four. Semigroup Forum 21, 67–81 (1980)

    Article  MathSciNet  Google Scholar 

  8. Forsythe, G.E.: SWAC computes 126 distinct semigroups of order 4. Proc. Am. Math. Soc. 6, 443–447 (1955)

    Article  MathSciNet  Google Scholar 

  9. Gao, M., Zhang, W.T., Luo, Y.F.: A non-finitely based involution semigroup of order five. Algebra Univ. 81, Art. 31, 14 pp (2020)

  10. Jackson, M.: Finite semigroups whose varieties have uncountably many subvarieties. J. Algebra 228, 512–535 (2000)

    Article  MathSciNet  Google Scholar 

  11. Jackson, M., Volkov, M.V.: The algebra of adjacency patterns: Rees matrix semigroups with reversion. In: Fields of Logic and Computation, Lecture Notes in Comput. Sci., vol. 6300, pp. 414–443. Springer, Berlin (2010)

  12. Kleĭman, E.I.: On basis of identities of Brandt semigroups. Semigroup Forum 13, 209–218 (1977)

    Article  MathSciNet  Google Scholar 

  13. Kleĭman, E.I.: Bases of identities of varieties of inverse semigroups. Sib. Math. J. 20, 530–543 (1979) (Translation of Sibirsk. Mat. Zh. 20, 760–777 (1979))

  14. Lee, E.W.H.: Identity bases for some non-exact varieties. Semigroup Forum 68, 445–457 (2004)

    Article  MathSciNet  Google Scholar 

  15. Lee, E.W.H.: Combinatorial Rees–Sushkevich varieties are finitely based. Int. J. Algebra Comput. 18, 957–978 (2008)

    Article  MathSciNet  Google Scholar 

  16. Lee, E.W.H.: Combinatorial Rees–Sushkevich varieties that are Cross, finitely generated, or small. Bull. Aust. Math. Soc. 81, 64–84 (2010)

    Article  MathSciNet  Google Scholar 

  17. Lee, E.W.H.: Finite basis problem for semigroups of order five or less: generalization and revisitation. Stud. Log. 101, 95–115 (2013)

    Article  MathSciNet  Google Scholar 

  18. Lee, E.W.H.: Finite involution semigroups with infinite irredundant bases of identities. Forum Math. 28, 587–607 (2016)

    Article  MathSciNet  Google Scholar 

  19. Lee, E.W.H.: Finitely based finite involution semigroups with non-finitely based reducts. Quaest. Math. 39, 217–243 (2016)

    Article  MathSciNet  Google Scholar 

  20. Lee, E.W.H.: Equational theories of unstable involution semigroups. Electron. Res. Announc. Math. Sci. 24, 10–20 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Lee, E.W.H.: On a class of completely join prime \(J\)-trivial semigroups with unique involution. Algebra Univ. 78, 131–145 (2017)

    Article  MathSciNet  Google Scholar 

  22. Lee, E.W.H.: A sufficient condition for the absence of irredundant bases. Houst. J. Math. 44, 399–411 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Lee, E.W.H.: Varieties generated by unstable involution semigroups with continuum many subvarieties. C. R. Math. Acad. Sci. Paris 356, 44–51 (2018)

    Article  MathSciNet  Google Scholar 

  24. Lee, E.W.H.: Non-finitely based finite involution semigroups with finitely based semigroup reducts. Korean J. Math. 27, 53–62 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Lee, E.W.H.: Non-Specht variety generated by an involution semigroup of order five. Trans. Mosc. Math. Soc. 81, 87–95 (2020)

    Article  MathSciNet  Google Scholar 

  26. Perkins, P.: Bases for equational theories of semigroups. J. Algebra 11, 298–314 (1969)

    Article  MathSciNet  Google Scholar 

  27. Petrich, M.: Introduction to Semigroups. Merrill, Columbus (1973)

    MATH  Google Scholar 

  28. Specht, W.: Gesetze in Ringen. I. Math. Z. 52, 557–589 (1950). (German)

    Article  MathSciNet  Google Scholar 

  29. Trahtman, A.N.: The finite basis question for semigroups of order less than six. Semigroup Forum 27, 387–389 (1983)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author is indebted to the following colleagues: the anonymous reviewer for valuable suggestions, Wen Ting Zhang for helpful discussions on Lemma 3.2, Marcel Jackson for information on subvarieties of \(\mathrm {var}\,B_2^1\), and Mikhail Volkov for providing the proof of Proposition 3.1.

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  1. Department of Mathematics, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale, FL, 33314, USA

    Edmond W. H. Lee

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Lee, E.W.H. Intervals of varieties of involution semigroups with contrasting reduct intervals. Boll Unione Mat Ital 15, 527–540 (2022). https://doi.org/10.1007/s40574-022-00317-9

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