The five-element Brandt semigroup as a forbidden divisor (Q1590064)

If a semigroup \(T\) is a homomorphic image of some subsemigroup \(S'\) of a semigroup \(S\), then we say the semigroup \(T\) divides the semigroup \(S\), or \(T\) is a divisor of \(S\). For a subset \(X\) of a semigroup \(S\), the radical \(\sqrt X\) of \(X\) is defined by \(\sqrt X=\{a\in S\mid(\exists n\in\mathbb{N})\;a^n\in X\}\). Let \(RIS\) denote the class of all semigroups having the property that the radical of every ideal is a subsemigroup, and let \(Her(RIS)\) be the class of all semigroups having the property that every subsemigroup belongs to \(RIS\), also let \(\mathbb{B}_2\) denote the Brandt semigroup of five elements. In this paper all semigroups which are not divided by \(\mathbb{B}_2\) are characterized in two ways: First characterization. The following are equivalent. (1) \(S\in Her(RIS)\), (2) \(\mathbb{B}_2\) is not a divisor of \(S\), (3) \((\forall a,b\in S)(\exists n\in\mathbb{N})\;(ab)^n\in\langle a,b\rangle a^2\langle a,b\rangle\cup\langle a,b\rangle b^2\langle a,b\rangle\) where \(\mathbb{N}\) is the set of positive integers, \(\langle a,b\rangle\) is the subsemigroup generated by \(a\) and \(b\). Next, let \(A^+\) denote the free semigroup over a set \(A\). Any pair \((u,v)\) of words from \(A^+\) is called an identity, and denoted by \(u=v\). The authors define the notion of a solution of the equation \(u\varphi=v\varphi\) for a homomorphism \(\varphi\) of \(A^+\) into a semigroup \(S\). Then \(Her(RIS)\) is characterized in terms of identities and solutions of equations. It is interesting that all identities over \(A=\{x,y\}\) are determined, all homomorphisms of \(A^+\) into \(\mathbb{B}_2\) are considered and the partially ordered set of kernels of homomorphisms is exhibited.