Examples of non-quasicommutative semigroups decomposed into unions of groups

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 groups. Our examples are the semigroups presented by the following presentations: π1=⟨a,b∣an+1=a,b3=b,ba=an−1b⟩, (n≥3), π2= ⟨a,b∣a1+pα=a,b1+pβ=b,ab=ba1+pα−γ⟩ where, p is an odd prime, α,β and γ are integers such that α≥2γ, β≥γ≥1 and α+β 3.