Examples of non-quasicommutative semigroups decomposed into unions of groups

© 2016 Iranian Mathematical Society. Decomposability of an algebraic structure into a union of its sub-structures have been studied by many authors for groups, rings and non-group semigroups since 1926. A sub-class of non-group semigroups is the well known quasicommutative semigroups where it is known that a regular quasicommutative semigroup is decomposable into a union of groups. The converse of this result is a natural question. Obviously, if a semigroup S is decomposable into a union of groups then S is regular so, the aim of this paper is to give examples of non-quasicommutative semigroups which are decomposable into the disjoint unions of groups. Our examples are two infinite classes of finite semigroups.