\({\mathcal K}G\)-strong semilattice structure of regular orthocryptosemigroups.
A semigroup \(S\) is called 1) `semisuperabundant' if each of its \(\widetilde{\mathcal H}\)-classes contains an idempotent, where \(\widetilde{\mathcal H}\) is Green's \(\sim\)-relation; 2) `orthosemigroup' if \(E(S)\) is a subsemigroup of \(S\); 3) `cryptosemigroup' if \(\widetilde{\mathcal H}\) is a congruence; 4) `u-semigroup' if \(S\) has a unique idempotent, which is its identity. The authors introduce the concept of \(\widetilde{\mathcal K}G\)-strong semilattice decomposition of a semigroup \(S\) where \(\widetilde{\mathcal K}\) is an equivalence relation on a semilattice decomposition of \(S\). Let \(S\) be a semisuperabundant orthosemigroup. Then: 1) \(S\) is a regular orthocryptosemigroup if and only if \(S\) is an \(\widetilde{\mathcal H}G\)-strong semilattice of rectangular u-semigroups; 2) \(S\) is a quasinormal orthocryptosemigroup if and only if \(S\) is an \(\widetilde{\mathcal L}G\)-strong semilattice of rectangular u-semigroups; 3) \(S\) is a normal orthocryptosemigroup if and only if \(S\) is a \(\widetilde{\mathcal J}G\)-strong semilattice of rectangular u-semigroups.
scientific article; zbMATH DE number 5118110