A sufficient condition for coinciding the Green graphs of semigroups
Mohammad Reza Sorouhesh, H. Doostie, Colin Matthew Campbell · Journal of Mathematics and Computer Science · 2017
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.