Constructing inverse semigroups from category actions
Let \(C\) be a category regarded as a set equipped with a partial binary operation and let \(C_0\) be the set of identities of \(C\). Left and right identities of an element \(x\in C\) are denoted by \(\mathbf{r}(x)\) and \(\mathbf{d}(x)\), respectively. Let \(X\) be a set and \(\mathbf{p}\colon X\to C_0\) a function. Denote \(C*X=\{(a,x)\in C\times X:\mathbf{d}(a)=\text\textbf{p}(x)\}\) and suppose there is defined an action \(C*X\to X\). Then \(X\) is said to be a left \(C\)-system. A left \(C\)-system satisfies the orbit condition if \(C\cdot x\cap C\cdot y\not=\emptyset\) implies that \(C\cdot x\cap C\cdot y=C\cdot z\) for some \(z\in X\). The author proves that every inverse semigroup can be constructed from a category acting on a set satisfying the orbit condition. This is a generalization of both \textit{D. B. McAlister}'s and \textit{J. Leech}'s works describing inverse monoids [in J. Algebra 43, 231-251 (1976; Zbl 0349.20025) and Semigroup Forum 36, 89-116 (1987; Zbl 0634.18002), respectively]. A number of special cases and examples is discussed. For example, natural characterizations of \(0\)-simple and \(0\)-\(E\)-unitary inverse semigroups in terms of systems are obtained.
scientific article; zbMATH DE number 1308331