Regular semilattice of semigroups and its applications. (Q382932)

Let \(Y\) be a semilattice and \(\{S_\alpha:\alpha\in Y\}\) be a family of pairwise disjoint semigroups. If there are band congruences \(\rho_{\alpha,\beta}\) on \(S_\beta\) for every pair \(\alpha\geq\beta\) ( \(\alpha,\beta\in Y\)) and mappings \(\Phi_{\alpha,\beta}\colon S_\alpha\to 2^{S_\beta}\) satisfying some conditions, then a multiplication ``\(\circ\)'' can be defined on \(S=\bigcup _{\alpha\in Y}S_\alpha\). Under some additional conditions, \((S,\circ)\) is a semigroup which is called a regular semilattice \(Y\) of semigroups \(S_\alpha\), \(\alpha\in Y\). It is shown that a regular semilattice of semigroups is a generalization of a refined semilattice of semigroups satisfying some conditions and thus a generalization of a strong semilattice of semigroups. From this point of view, regular cryptogroups are considered.