The universal covering of an inverse semigroup.
Considerable part of the article is devoted to presheaves on a small category. For example, transition injective presheaves (that is, its transition maps are injective) as well as transition bijective presheaves on a small category are described. The results obtained are applied to the category \(L(T)\), \(T\) an inverse semigroup, whose objects are the idempotents of \(T\), and morphisms \(t\colon e\to f\) are elements of \(T\) such that \(e=t^*t\) and \(t=ft\). Denoting by \(\mathcal B(T)\) the topos of presheaves on the category \(L(T)\) and by \(\mathbf S(e)=\{t\mid t^*t=s\}\) the Schützenberger presheaf, it follows, that \(\mathbf S\) is a torsion-free generator of \(\mathcal B(T)\) and splits all locally constant objects. Moreover, it is proved that the underlying set of the maximum group image of \(T\) is isomorphic to the set of connected components of \(\mathbf S\times\mathbf S\). The final two sections of the article are mainly devoted to revealing some properties of E-unitary inverse semigroups (an inverse semigroup is said to be `E-unitary' if the compatibility relation on it is transitive; elements \(s\) and \(t\) are said to be `compatible' if \(st^*\) and \(s^*t\) are idempotents). For example, an inverse monoid \(T\) is E-unitary if and only if \(\mathcal B(T)\) is locally decidable. Some facts the article is based on (the `localic groupoid' definition of the classifying topos of an inverse semigroup \(T\) and of an étale right \(T\)-set; some information about topoi etc.) are presented in the appendices.
scientific article; zbMATH DE number 5705491
scientific article