Abundant semigroups whose idempotents satisfy permutation identities
Let \(S\) be a semigroup and \(a,b\in S\). Put \(a{\mathcal L}^*b\) (respectively, \(a{\mathcal R}^*b\)) whenever \(a\) and \(b\) are related by Green's relation \(\mathcal L\) (respectively, \(\mathcal R\)) in some oversemigroup of \(S\). A semigroup is called abundant if each \({\mathcal L}^*\)-class and each \({\mathcal R}^*\)-class contains an idempotent. An abundant semigroup \(S\) is called quasi-adequate if its idempotents form a subsemigroup of \(S\); if, moreover, idempotents of \(S\) permute then \(S\) is called adequate. The set of idempotents of \(S\) is denoted by \(E(S)\). Theorem 2.1 states that, for an abundant semigroup \(S\), the following are equivalent: (1) \(E(S)\) satisfies a permutation identity; (2) \(E(S)\) is a normal band; (3) \(S\) is quasi-adequate and \(eSe\) is an adequate subsemigroup of \(S\) for any \(e\in E(S)\). Let \(T\) be an adequate semigroup with semilattice of idempotents \(Y\), and let \(L=S(Y;L_a;\varphi_{a,b})\) (respectively \(R=S(Y;R_a;\psi_{a,b})\)) be a strong semilattice of left (right) zero bands \(L_a\) (respectively, \(R_a\)). The author construct a new semigroup depending on \(L,R,T\) and \(Y\), call it the quasi-spined product of \(L,R\) and \(T\) on \(Y\) and denote it by \(QS(L,R,T;Y)\). According to Theorem 2.3, if \(S=QS(L,R,T;Y)\) then \(S\) is an abundant semigroup whose idempotents satisfy a permutation identity and \(S/\delta\cong T\) for some natural congruence \(\delta\) on \(S\). Conversely, Theorem 2.4 states that any abundant semigroup whose idempotents satisfy a permutation identity is isomorphic to some semigroup of the kind \(QS(L,R,T;Y)\). Theorem 3.4 states that, for an abundant semigroup \(S\), the following are equivalent: (1) \(S\) satisfies a permutation identity; (2) \(S\) satisfies the identity \(xyzt=xzyt\); (3) \(S\) is isomorphic to a spined product of a normal band and a commutative semigroup which is a semilattice of cancellative monoids. (The definition of the spined product see, for example, in \textit{M. Petrich} [Lectures in semigroups (Wiley, London, 1977; Zbl 0369.20036)].) This theorem gives also additional information about the semigroups under consideration whenever \(E(S)\) is either a left (right) normal band or a semilattice.
scientific article; zbMATH DE number 1022339