Abstract. If for every elements x and y of an associative algebraic structure (S, ·) there exists a positive integer r such that ab = b r a, then S is called quasi-commutative. Evidently, every abelian group or commutative semigroup is quasi-commutative. Also every finite Hamiltonian group that may be considered as a semigroup, is quasi-commutative however, there are quasi-commutative semigroups which are non-group and non commutative. In this paper, we provide three finitely presented noncommutative semigroups which are quasi-commutative. These are the first given concrete examples of finite semigroups of this type.

A necessary condition for coinciding the Green graphs Γ L (S), Γ R (S), Γ J (S), Γ D (S) and Γ 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.

A Latin square graph [Formula: see text] is a simple graph associated with a Latin square [Formula: see text]. In this paper, we consider a Latin square graph [Formula: see text] in which [Formula: see text] and improve the upper bound of the domination number [Formula: see text] by showing that [Formula: see text]. Moreover, we study certain domination numbers like medium domination, accurate domination, independent transversal domination and vertex covering transversal domination for Latin square graphs of order [Formula: see text] to give useful facts.

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We consider certain properties of the semigroup S defined by the presentation