**Volume 17, Issue 2, pp 216-219**

**Publication Date**: 2017-06-15

http://dx.doi.org/10.22436/jmcs.017.02.03

Mohammad Reza Sorouhesh - Department of Mathematics, Tehran Science and Research Branch Islamic Azad University, Tehran, 14515/1775, Iran.

Hossein Doostie - Department of Mathematics, Tehran Science and Research Branch Islamic Azad University, Tehran, 14515/1775, Iran.

Colin M. Campbell - School of Mathematics and Statistics, University of St. Andrews, North Haugh, St. Andrews, Fife, KY16 9SS Scotland, UK.

A necessary condition for coinciding the Green graphs \(\Gamma_{\textit{L}}(S), \Gamma_{\Re}(S), \Gamma_{\jmath}(S), \Gamma_{D}(S)\) and \(\Gamma_{H}(S)\) of a finite semigroup S has been studied by Gharibkhajeh [A. Gharibkhajeh, H. Dosstie, Bull. Iranian Math. Soc., 40 (2014), 413–421]. Gharibkhajeh et al. proved that the coinciding of Green graphs of a finite semigroup S implies the regularity of S. However, the converse is not true because of certain well-known examples of finite regular semigroups. We look for a sufficient condition on non-group semigroups that implies the coinciding of the Green graphs. Indeed, in this paper we prove that for every non-group quasi-commutative finite semigroup, all of the Green graphs are isomorphic.

Quasi-commutativity, finitely presented semigroups, Green relations, Green graphs.

[1] E. Adan-Bante, Conjugacy classes and finite p-groups, Arch. Math. (Basel), 85 (2005), 297–303.

[2] A. Cherubini, A. Varisco, Quasi-commutative semigroups and \(\sigma\)-reflexive semigroups, Semigroup Forum, 19 (1980), 313–321.

[3] A. H. Clifford, G. B. Preston, The algebraic theory of semigroups I, Amer. Math. Soc., Providence, (1961).

[4] A. Gharibkhajeh, H. Dosstie, A graphical difference between the inverse and regular semigroups, Bull. Iranian Math. Soc., 40 (2014), 413–421.

[5] A. Nagy, Special classes of Semigroups, Kluwer Academic Publishers, Dordrecht, (2001).

[6] N. P. Mukherjee, Quasi-commutative semigroups I, Czechoslovak Math. J., 22 (1972), 449–453.

[7] M. R. Sorouhesh, H. Dosstie, Quasi-commutative semigroups of finite order related to Hamiltonian groups, Bull. Korean Math. Soc., 52 (2015), 239–246.