A Study on Regular Semigroups and its Idempotents

  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract

    An idempotent of a semigroup T is an element e in T such that  In many semigroups, idempotents can be recognized easily. Thus it plays an important role in the structure of semigroups especially on regular semigroups. This article reviews about some research work done about the structure of regular semigroups with a special emphasis on its idempotents.




  • Keywords

    Medial idempotent; Middle unit; Normal idempotent; Regular idempotent; Quasi-medial idempotent.

  • References

      [1] G.R. Baird, “Congruences on generalized inverse semigroups”, Semigroup Forum, Vol.4, (1972), pp.200-205, available online: https://doi.org/10.1007/BF02570786

      [2] T.S. Blyth, “On middle units in orthodox semigroups”, Semigroup Forum, Vol.13, (1977), pp.261-265, available online: https://doi.org/10.1007/BF02194944

      [3] T.S. Blyth & R. McFadden, “Naturally ordered regular semigroups with a greatest idempotent”, Proc. Roy. Soc. Edinburgh, Vol.91A, (1981), pp.107-122, available online: https://doi.org/10.1017/S0308210500012671

      [4] T.S. Blyth & R. McFadden, “On the construction of a class of regular semigroups”, J. Algebra, Vol.81, (1983), pp.1-22, available online: https://doi.org/10.1016/0021-8693(83)90205-3

      [5] T.S. Blyth & M.H. Almeida Santos, “On naturally ordered regular semigroups with biggest idempotents”, Comm. Algebra, Vol.21, No.5, (1993), pp.1761-1771, available online: https://doi.org/10.1080/00927879308824651

      [6] T.S. Blyth & M.H. Almeida Santos, “A classification of inverse transversals”, Comm. Algebra, Vol.29, No.2, (2001), pp.611-624, available online: https://doi.org/10.1081/AGB-100001527

      [7] T.S. Blyth & M.H. Almeida Santos, “Regular semigroups with skew pairs of idempotents”, Semigroup Forum, Vol.65, (2002), pp.264-274, available online: https://doi.org/10.1007/s002330010112

      [8] T.S. Blyth & M.H. Almeida Santos, “Naturally ordered regular semigroups with inverse monoid transversal”, Semigroup Forum, Vol.76, (2008), pp.71-86, available online: https://doi.org/10.1007/s00233-007-9002-z

      [9] J.L. Chrislock, “On medial semigroups”, J. Algebra, Vol.12, (1969), pp.1-9, available online: https://doi.org/10.1016/0021-8693(69)90013-1

      [10] A.H. Clifford & G.B. Preston, The Algebraic Theory of Semigroups Vol. 1, Math. Surveys of the American Math. Soc. 7, Providence, Rhode Island, 1961.

      [11] D.G. Fitz-Gerald, “On inverses of products of idempotents in regular semigroups”, J. Austral. Math. Soc., Vol.13, (1972), pp.335-337, available online: https://doi.org/10.1017/S1446788700013756

      [12] X.J. Guo, “The structure of abundant semigroups with a weak normal idempotent”, Acta Mathematics Sinica, Vol.42, No.4, (1999), pp.683-690.

      [13] T.E. Hall, “On regular semigroups whose idempotents form a subsemigroup”, Bull. Austral. Math. Soc., Vol.1, (1969), pp.195-208, available online: https://doi.org/10.1017/S0004972700041447

      [14] T.E. Hall, “On regular semigroups”, J. Algebra, Vol.24, (1973), pp.1-24, available online: https://doi.org/10.1016/0021-8693(73)90150-6

      [15] J.M. Howie, “Fundamentals of semigroup theory”, Oxford University Press, New York, 1995.

      [16] S. Hussain, T. Anwer & H. Chien, “Some properties of regular semigroups possess a medial idempotent”, Int. J. Algebra, Vol.4, No.9, (2010), pp.433-438.

      [17] O. Hysa, “The construction of regular semigroups with medial idempotent”, International Journal of Basic and Applied Sciences IJBAS-IJENS, Vol.12, No.6, (2012), pp.49-52.

      [18] K. Indhira & V.M. Chandrasekaran, “Structure of regular semigroups with a regular idempotent”, Int. J. Contemp. Math. Sciences, Vol.6, No. 12, (2011), pp.557-570.

      [19] K. Indhira & V.M. Chandrasekaran, “Construction of generalized inverse semigroups”, Int. J. lgebra, Vol.5, No.15, (2011), pp.739-746.

      [20] K. Indhira & V.M. Chandrasekaran, “Idempotent separating congruence on a regular semigroup with a regular idempotent”, Adv. Studies Theor. Phys., Vol.7, No.3, (2013), pp.107-114.

      [21] Janet E. Ault, “Semigroups with midunits”, Trans. Amer. Math. Soc., Vol.190, (1974), pp.375-384, available online: https://doi.org/10.1007/BF02389143

      [22] G. Kudryavtseva & M.V. Lawson, “The structure of generalized inverse semigroups”, Semigroup Forum, Vol.89, No.1, (2014), pp.199-216, available online: https://doi.org/10.1007/s00233-013-9518-3

      [23] M. Loganathan, “Regular semigroup with a medial idempotent”, Semigroup Forum, Vol.36, (1987), pp.69-74, available online: https://doi.org/10.1007/BF02575006

      [24] S. Madhavan, “Some results on generalized inverse semigroups”, Semigroup Forum, Vol.16, (1978), pp.355-367, available online: https://doi.org/10.1007/BF02194635

      [25] Mario Petrich, “Regular semigroups satisfying certain conditions on idempotents and ideals”, Trans. Amer. Math. Soc., Vol.170, (1972), pp.245-267.

      [26] D.B. McAlister, “Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups”, J. Australian Math. Soc., Vol.31, (1981), pp.325-336, available online: https://doi.org/10.1017/S1446788700019467

      [27] D.B. McAlister & R. McFadden, “Maximum idempotents in naturally ordered regular semigroups”, Proc. Edinburgh Math. Soc., Vol.26, (1983), pp.213-220, available online: https://doi.org/10.1017/S0013091500016916

      [28] H.E. Scheiblich, “Generalized inverse semigroups with involution”, Rocky Mountain J. Math., Vol.12, No.2, (1982), pp.205-211.

      [29] P.S. Venkatesan, “Right (left) inverse semigroups”, J. Algebra, Vol.31, (1974), pp.209-217, available online: https://doi.org/10.1016/0021-8693(74)90064-7

      [30] Xiangfei Ni & Haizhou Chao, “A note on normal idempotents”, Publicationes Mathematicae, Vol.89, No.4, (2016), pp.441-448.

      [31] Xiangfei Ni & Haizhou Chao, “Regular semigroups with normal idempotents”, J. Australian Math. Soc., Vol.103, No.1, (2017), pp.116-125, available online: https://doi.org/10.1017/S1446788717000088

      [32] Xiangfei Ni & Xiao Jiang Guo, “Regular semigroups with weak medial idempotents”, Acta Mathematica Sinica, Vol.61, No.1, (2018), pp.107-122.

      [33] M. Yamada, “Regular semigroups whose idempotents satisfy permutation identities”, Pacific. J. Math., Vol.21, (1967), pp.371-392.

      [34] M. Yamada, “On a regular semigroup in which the idempotents form a band”, Pacific J. Math., Vol.33, (1970), pp.251-272.

      [35] M. Yamada, “Orthodox semigroups whose idempotents satisfy a certain identity”, Semigroup Forum, Vol.6, No.1, (1973), 113-128, available online: https://doi.org/10.1007/BF02389116




Article ID: 21214
DOI: 10.14419/ijet.v7i4.10.21214

Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.