Examples of non-quasicommutative semigroups decomposed into unions of groups

Abstract

Decomposability of an algebraic structure into the union of its sub-structures goes back to G. Scorza's Theorem of 1926 for groups. An analogue of this theorem for rings has been recently studied by A. Lucchini in 2012. On the study of this problem for non-group semigroups, the first attempt is due to Clifford's work of 1961 for the regular semigroups. Since then, N.P. Mukherjee in 1972 studied the decomposition of quasicommutative semigroups where, he proved that: a regular quasicommutative semigroup is decomposable into the 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 <br />groups. Our examples are the semigroups presented by the following presentations: $$pi_1 =langle a,bmid a^{n+1}=a, b^3=b, ba=a^{n-1}brangle,~(ngeq 3),$$ <br />$$pi_2 =langle a,bmid a^{1+p^alpha}=a, b^{1+p^beta}=b, ab=ba^{1+p^{alpha-gamma}}rangle$$where, $p$ is an odd prime, $alpha, beta$ and $gamma$ are integers such that $alpha geq 2gamma$, $beta geq gamma geq 1$ and $alpha +beta > 3$.