On the homological classification of semigroups with local units
The branch of investigations of semigroups studying connections between properties of the category of right (or left) acts over a semigroup \(S\) and those of the semigroup \(S\) itself is called the homological classification of semigroups. The paper under review concerns with the homological classification of semigroups with local units. A semigroup \(S\) is said to have local units if for every \(s\in S\) there exist idempotents \(e, f \in S\) such that \(es = s = sf\). A right \(S\)-act \(A\) is called unitary if \(AS = A\). When \(S\) is a semigroup with local units it is evident that \(A\) is unitary if and only if for any \(a\in A\) there exists an idempotent \(e\in S\) such that \(a= ae.\) The full subcategory of right \(S\)-acts consisting of unitary right \(S\)-acts is denoted by \(\textbf{UAct}\)-\(S\). For a semigroup \(S\), a right \(S\)-act \(A\) is called firm if the mapping \(\mu_A : A\otimes S\to A, a\otimes s\mapsto as\) is bijective. The full subcategory of all firm right S-acts of the category \(\textbf{UAct}\)-\(S\) is denoted by \(\textbf{FAct}\)-\(S\). In the main Section 4 of this paper, the authors investigate semigroups with local units over which all firm (right) acts being principally weakly flat, weakly flat, flat or torsion-free. For example see the theorem below on page 2911: Theorem 32: Let \(S\) be a semigroup with local units. In the category \(\textbf{FAct}\)-\(S\), the following statements are equivalent: \begin{itemize} \item[(1)] All \(S\)-acts are torsion free. \item[(2)] All cyclic \(S\)-acts are torsion free. \item[(3)] All Rees factor acts of \(S\) are torsion free. \item[(4)] Every right cancellable element of \(S\) is right \(e\)-invertible for some idempotent \(e\in S\). \end{itemize} The last section is devoted to study Rees short exact sequences of acts over semigroups with local units.
scientific article; zbMATH DE number 7384655
scientific article