A sufficient condition for coinciding the Green graphs of semigroups


Authors

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.


Abstract

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.


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