Direct products of cyclic semigroups and unions of groups (Q1346305)

It is proved that if \(S\times\bigcup_{e\in E} G_e\) is isomorphic to \(T\times\bigcup_{f\in F} H_f\), where \(S\) and \(T\) are direct products of non-group finite cyclic semigroups, \(\bigcup_{e\in E} G_e\) and \(\bigcup_{f\in F} H_f\) are unions of finite groups with \(E\) and \(F\) finite sets of idempotents and \(G_e\), \(H_f\) groups for each \(e\in E\), \(f\in F\), then \(S\cong T\) and \(\bigcup_{e\in E} G_e\cong\bigcup_{f\in F} H_f\). Conditions under which a finite semigroup has such a decomposition are given. Then an algorithm for finding the direct factors is presented.