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Order structure of U-semiabundant semigroups and rings. Part I: Left Lawson’s order

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Abstract

In 1991, Lawson introduced three partial orders on reduced U-semiabundant semigroups. Their definitions are formally similar to recently discovered characteristics of the diamond, left star and right star orders respectively on Rickart *-rings; lattice properties of these orders have been studied by several authors. Motivated by these similarities, we turn to the lattice structure of U-semiabundant semigroups and rings under Lawson’s orders. In this paper, we deal with his order ≤l on (a version of) right U-semiabundant semigroups and rings. In particular, existence of meets is investigated, it is shown that (under some natural assumptions) every initial section of such a ring is an orthomodular lattice, and explicit descriptions of the corresponding lattice operations are given.

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References

  1. D.H. Adams, Equational classes of Foulis semigroups and orthomodular lattices, Proc. Univ. Hscouston, Lattice Theory Conf., Houston, (1973), 486–497.

    Google Scholar 

  2. L. Beran, Orthomodular lattices. Algebraic Approach, D. ReidelPubl. Co., Dordrecht e.a.; Academia, Prague, 1985.

    Book  Google Scholar 

  3. S.K. Berberian, Baer *-rings, 2nd ed., Springer, Berlin-Heidelberg-New York, 2011.

    MATH  Google Scholar 

  4. J. Cīrulis, On one-sided star partial orders on a Rickart *-ring, arXiv, 4693 (2014), 8 pp; arXiv: 1410.4693.

  5. J. Cīrulis, Quasi-orthomodular posets and weak BCK-algebras, Order, 31 (2014), 403–419.

    Article  MathSciNet  Google Scholar 

  6. J. Cīrulis, Lattice operations on Rickart *-rings under the star order, Linear Multilinear Algebra, 63 (2015), 497–508.

    Article  MathSciNet  Google Scholar 

  7. J. Cīrulis, Relatively orthocomplemented skew nearlattices in Rickart rings, Demonstr. Math., 48 (2015), 492–507.

    MathSciNet  MATH  Google Scholar 

  8. J. Cīrulis, Extending the star order to Rickart rings, Linear Multilinear Algebr., 64 (2016), 1498–1508.

    Article  MathSciNet  Google Scholar 

  9. J. Cīrulis, The diamond partial order for strong Rickart rings, Linear Multilinear Algebra, 65 (2017), 192–203.

    Article  MathSciNet  Google Scholar 

  10. J. Cīrulis, Focal Baer semigroups and a restricted star order, Acta Sci. Math. (Szeged), 85 (2019), 97–117.

    Article  MathSciNet  Google Scholar 

  11. W.H. Cornish, Rings and implicative BCK-algebras, Math. Semin. Notes, Kobe Univ., 8 (1980), 147–155.

    MathSciNet  MATH  Google Scholar 

  12. W.H. Cornish, Boolean skew algebras, Acta Math. Acad. Sci. Hung., 36 (1980), 281–291.

    Article  MathSciNet  Google Scholar 

  13. M.S. Djikić, D.S. Djordjević, Coherent and precoherent elements in Rickart rings, Linear Algebra Appl., 509 (2016), 64–81.

    Article  MathSciNet  Google Scholar 

  14. M.P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc., 11 (1960), 648–654.

    Google Scholar 

  15. D.J. Foulis, Baer *-semigroups, The Logico-Algebraic Approach to Quantum Mechanics, ed.: C.A. Hooker, The University of Western Ontario Series in Philosophy of Science (A Series of Books on Philosophy of Science, Methodology, and Epistemology Published in Connection with the University of Western Ontario Philosophy of Science Programme), vol 5a. Springer, Dordrecht, 1975, 648–654.

    Google Scholar 

  16. D.J. Foulis, Relative inverses in Baer *-semigroups, Michigan Math. J., 10 (1963), 65–84.

    Article  MathSciNet  Google Scholar 

  17. Ch. Hollings, From right PP monoids to restriction semigroups: a survey, Eur. J. Pure Appl. Math., 2 (2009), 21–57.

    MathSciNet  MATH  Google Scholar 

  18. J.M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995.

    MATH  Google Scholar 

  19. J. Janin, On quasi-inverse monoids (and premorphisms), HAL, hal-00673123v3 (2012), available at https://hal.archives-ouvertes.fr/hal-00673123/.

  20. M.F. Janowitz, Baer semigroups, Duke Math. J., 32 (1965), 83–96.

    Article  MathSciNet  Google Scholar 

  21. M.F. Janowitz, A semigroup approach to lattices, Canad. J. Math., 18 (1966), 1212–1223.

    Article  MathSciNet  Google Scholar 

  22. M.F. Janowitz, A note on generalized orthomodular lattices, J. Natur. Sci. Math., 8 (1968), 89–94.

    MathSciNet  MATH  Google Scholar 

  23. M.F. Janowitz, On the *-order for Rickart *-rings, Algebra Universalis, 16 (1983), 360–369.

    Article  MathSciNet  Google Scholar 

  24. I. Krēmere, Left-star order structure of Rickart *-rings, Linear Multilinear Algebra, 64 (2016), 341–352.

    Article  MathSciNet  Google Scholar 

  25. T.Y. Lam, Lectures on Modules and Rings, Springer, Berlin — Heidelberg — New York, 1999.

    Book  Google Scholar 

  26. M.V. Lawson, Rees matrix semigroups, Proc. Edinb. Math. Soc, 33 (1990), 33–37.

    Article  MathSciNet  Google Scholar 

  27. M.V. Lawson, Semigroups and ordered categories. I. The reduced case, J. Algebra, 141 (1991), 422–462.

    Article  MathSciNet  Google Scholar 

  28. M.V. Lebtahi, P. Patricio and Th. Néstor, The diamond order in rings, Linear-Multilinear Algebra, 62 (2013), 386–395.

    Article  MathSciNet  Google Scholar 

  29. J. Marovt, On partial orders in Rickart rings, Linear Multilinear Algebra, 63 (2015), 1707–1723.

    Article  MathSciNet  Google Scholar 

  30. J. Marovt, D.S. Rakić, D.S. Djordjević, Star, left-star, and right-star partial orders in Rickart *-rings, Linear Multilinear Algebra, 63 (2015), 343–365.

    Article  MathSciNet  Google Scholar 

  31. A. Meyet-Ippolito, Generalized orthomodular posets, Demonstr. Math., 24 (1991), 263–274.

    MathSciNet  MATH  Google Scholar 

  32. C.J. Penning, Minimal duplicator rings, Indag. Math., 25 (1963), 295–312.

    Article  MathSciNet  Google Scholar 

  33. T.P. Speed and M.W. Evans, A note on commutative Baer rings, J. Aust. Math. Soc., 13 (1972), 1–6.

    Article  MathSciNet  Google Scholar 

  34. T. Stokes, Domain and range operations in semigroups and rings, Comm. Algebra, 43 (2015), 3979–4007.

    Article  MathSciNet  Google Scholar 

  35. T. Stokes, D-semigroups and constellations, Semigroup Forum, 94 (2017), 442–462.

    Article  MathSciNet  Google Scholar 

  36. T. Stokes, Generalized domain and E-inverse semigroups, Semigroup Forum, 97 (2018), 38–52.

    Article  Google Scholar 

  37. D. Saracino and V. Weispfenning, On algebraic curves over commutative regular rings, Model Theor. Algebra, Mem. Tribute Abraham Robinson, Lect. Notes Math. 498, Springer, 1975, 306–383.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Computer Science, University of Latvia, Riga, Latvia

    Jānis Cīrulis

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  1. Jānis Cīrulis
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Correspondence to Jānis Cīrulis.

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Communicated by M. B. Szendrei

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Cīrulis, J. Order structure of U-semiabundant semigroups and rings. Part I: Left Lawson’s order. ActaSci.Math. 86, 359–403 (2020). https://doi.org/10.14232/actasm-019-426-3

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  • DOI: https://doi.org/10.14232/actasm-019-426-3

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