# Abundant semigroups with medial idempotents

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Science, University of Tripoli, Tripoli, Libya

Abstract

The effect of the existence of a medial or related idempotent in any abundant semigroup is the subject of this paper. The aim is to naturally order any abundant semigroup $S$ which contains an ample multiplicative medial idempotent $u$ in a way that $\mathcal{L}^*$ and $\mathcal{R}^*$ are compatible  with the natural order and $u$ is a maximum idempotent. The structure of an abundant semigroup containing an ample normal medial idempotent studied in \cite{item6} will be revisited.

Keywords

#### References

[1] Benzaken, C. and Mayr, H.C., Notion de demi-bande demi-bandes de type deux, Semigroup Forum 10(1) (1975).
[2] Blyth, T.S., The structure of certain ordered regular semigroups, Proc. Roy. Soc. Ed. Sect. A: Math. 75(3) (1976), 235-257.
[3] Blyth, T.S. and Janowitz, M.F., Residuation Theory", International Ser. Monogr. Pure Appl. Math. 102 (1972).
[4] Blyth, T.S. and McFadden, R., Naturally ordered regular semigroups with a greatest idempotent, Proc. Roy. Soc. Ed. Sect A 91 (1981), 107-122.
[5] Blyth, T.S. and McFadden, R., On the construction of a class of regular semigroups, J. Algebra 81 (1983), 1-22.
[6] El-Qallali, A., On the construction of a class of abundant semigroups, Acta Math. Hung. 56(1-2) (1990), 77-91.
[7] El-Qallali, A., Congruences on Ample Semigroups, Semigroup Forum, Volume 99, (2019), 607-631.
[8] El-Qallali, A. and Fountain, J.B., Idempotent-connected abundant semigroups, Proc. Roy. Soc. Ed. Sect A 91 (1981), 79-90.
[9] El-Qallali, A. and Fountain, J.B., Quasi-adequate semigroups, Proc. Roy. Soc. Ed. Sect A 91 (1981), 91-99.
[10] Fountain, J.B., Adequate semigroups, Proc. Ed. Math. Soc. 22 (1979), 113-125.
[11] Fountain, J.B., Abundant semigroups, Proc. London. Math. Soc. 44 (1982), 103-125.
[12] Guo, X.J. and Shum, K.P., Naturally ordered abundant semigroups for which each idempotent has a greatest inverse, Eur. J. Pure Apl. Math. 4(3) (2011), 210-220.
[13] Guo, X.J. and Luo, Y.F., The natural partial orders on abundant semigroups, Adv. Math. (China) 34 (2005), 197-304.
[14] Guo, X.J. and Xie, X.Y., Naturally ordered semigroups in which each idempotent has a greatest inverse, Comm. Alg. 35 (2007), 2324-2334.
[15] Howie, J.M., An Introduction to Semigroup Theory, Academic Press (1976).
[16] Hou, Y.L. Guo, J.Y., and Guo, X.J., Abundant Semigroups with a *-normal idem- potent, Math. Slovaca 67 (2017), 863-874.
[17] Chen, H., Construction of a kind of abundant semigroups, Math. Commun. 11 (2006), 165-171.
[18] Lawson, M.W. The natural partial order on an abundant semigroup, Proc. Ed. Math. Soc. 30 (1987), 169-186.
[19] McAlister, D.B., and McFadden, R., Maximum idempotents in naturally ordered regular semigroups, Proc. Ed. Math. Soc. 26 (1983), 213-220.
[20] Ni, X., Abundant semigroups with quasi-medial idempotents, Internat. J. Pure Apl. Math. 72(1) (2011), 81-91.