A Study on Regular Semigroups and its Idempotents
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2018-10-02 https://doi.org/10.14419/ijet.v7i4.10.21214 -
Medial idempotent, Middle unit, Normal idempotent, Regular idempotent, Quasi-medial idempotent. -
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.
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References
[1] G.R. Baird, “Congruences on generalized inverse semigroupsâ€, 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 semigroupsâ€, 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 idempotentâ€, 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 semigroupsâ€, 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 idempotentsâ€, 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 transversalsâ€, 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 idempotentsâ€, 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 transversalâ€, 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 semigroupsâ€, J. Algebra, Vol.12, (1969), pp.1-9, available online: https://doi.org/10.1016/0021-8693(69)90013-1" target="_blank">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 semigroupsâ€, 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 idempotentâ€, Acta Mathematics Sinica, Vol.42, No.4, (1999), pp.683-690.
[13] T.E. Hall, “On regular semigroups whose idempotents form a subsemigroupâ€, Bull. Austral. Math. Soc., Vol.1, (1969), pp.195-208, available online: https://doi.org/10.1017/S0004972700041447
[14] T.E. Hall, “On regular semigroupsâ€, J. Algebra, Vol.24, (1973), pp.1-24, available online: https://doi.org/10.1016/0021-8693(73)90150-6" target="_blank">https://doi.org/10.1016/0021-8693(73)90150-6
[15] J.M. Howie, “Fundamentals of semigroup theoryâ€, Oxford University Press, New York, 1995.
[16] S. Hussain, T. Anwer & H. Chien, “Some properties of regular semigroups possess a medial idempotentâ€, Int. J. Algebra, Vol.4, No.9, (2010), pp.433-438.
[17] O. Hysa, “The construction of regular semigroups with medial idempotentâ€, 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 idempotentâ€, Int. J. Contemp. Math. Sciences, Vol.6, No. 12, (2011), pp.557-570.
[19] K. Indhira & V.M. Chandrasekaran, “Construction of generalized inverse semigroupsâ€, 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 idempotentâ€, Adv. Studies Theor. Phys., Vol.7, No.3, (2013), pp.107-114.
[21] Janet E. Ault, “Semigroups with midunitsâ€, 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 semigroupsâ€, 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 idempotentâ€, Semigroup Forum, Vol.36, (1987), pp.69-74, available online: https://doi.org/10.1007/BF02575006
[24] S. Madhavan, “Some results on generalized inverse semigroupsâ€, 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 idealsâ€, Trans. Amer. Math. Soc., Vol.170, (1972), pp.245-267.
[26] D.B. McAlister, “Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroupsâ€, 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 semigroupsâ€, 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 involutionâ€, Rocky Mountain J. Math., Vol.12, No.2, (1982), pp.205-211.
[29] P.S. Venkatesan, “Right (left) inverse semigroupsâ€, 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 idempotentsâ€, Publicationes Mathematicae, Vol.89, No.4, (2016), pp.441-448.
[31] Xiangfei Ni & Haizhou Chao, “Regular semigroups with normal idempotentsâ€, 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 idempotentsâ€, Acta Mathematica Sinica, Vol.61, No.1, (2018), pp.107-122.
[33] M. Yamada, “Regular semigroups whose idempotents satisfy permutation identitiesâ€, Pacific. J. Math., Vol.21, (1967), pp.371-392.
[34] M. Yamada, “On a regular semigroup in which the idempotents form a bandâ€, Pacific J. Math., Vol.33, (1970), pp.251-272.
[35] M. Yamada, “Orthodox semigroups whose idempotents satisfy a certain identityâ€, Semigroup Forum, Vol.6, No.1, (1973), 113-128, available online: https://doi.org/10.1007/BF02389116
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How to Cite
Thomas, J., Indhira, K., & M. Chandrasekaran, V. (2018). A Study on Regular Semigroups and its Idempotents. International Journal of Engineering & Technology, 7(4.10), 511-513. https://doi.org/10.14419/ijet.v7i4.10.21214Received date: 2018-10-07
Accepted date: 2018-10-07
Published date: 2018-10-02