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On a class of completely join prime J-trivial semigroups with unique involution

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Abstract

A finite algebra is completely join prime if whenever it belongs to the complete join of some collection of pseudovarieties, then it belongs to one of the pseudovarieties. An infinite class of completely join prime J-trivial semigroups with unique involution is introduced to demonstrate the incompatibility between the lattice of pseudovarieties of involution semigroups and the lattice of pseudovarieties of semigroups. Examples are also exhibited to show that a finite involution semigroup and its semigroup reduct need not be simultaneously completely join prime.

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References

  1. Almeida J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  2. Dolinka I.: All varieties of normal bands with involution. Period. Math. Hungar. 40, 109–122 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Egri-Nagy A., Jackson M., Rhodes J., Steinberg B.: On the atoms of algebraic lattices arising in q-theory. Internat. J. Algebra Comput. 27, 157–188 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fajtlowicz S.: Equationally complete semigroups with involution. Algebra Universalis 1, 355–358 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kublanovsky S.I., Lee E.W.H., Reilly N.R.: Some conditions related to the exactness of Rees–Sushkevich varieties. Semigroup Forum 76, 87–94 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lee, E.W.H.: On the Lattice of Rees–Sushkevich Varieties. PhD thesis, Simon Fraser University (2002)

  8. Lee E.W.H.: A class of finite semigroups without irredundant bases of identities. Yokohama Math. J. 61, 1–28 (2015)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Lee, E.W.H.: A sufficient condition for the absence of irredundant bases. Houston J. Math. (in press)

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

    MathSciNet  Google Scholar 

  12. Reilly N.R.: Complete congruences on the lattice of Rees–Sushkevich varieties. Comm. Algebra 35, 3624–3659 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rhodes J., Steinberg B.: The q-Theory of Finite Semigroups. Springer Monographs in Mathematics. Springer, New York (2009)

    Book  MATH  Google Scholar 

  14. Sapir O.: Non-finitely based monoids. Semigroup Forum 90, 557–586 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Volkov M.V.: On a question by Edmond W. H. Lee. Izv. Ural. Gos. Univ. Mat. Mekh. No. 7(36), 167–178 (2005)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Edmond W. H. Lee

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  1. Edmond W. H. Lee
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Correspondence to Edmond W. H. Lee.

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Presented by M. Jackson.

Dedicated to the 85th birthday of Mario Petrich

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Lee, E.W.H. On a class of completely join prime J-trivial semigroups with unique involution. Algebra Univers. 78, 131–145 (2017). https://doi.org/10.1007/s00012-017-0442-3

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  • DOI: https://doi.org/10.1007/s00012-017-0442-3

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