Cohomology and extensions of inverse semigroups
The concept of idempotent separating extensions of inverse semigroups was studied by \textit{D'Alarco} and a cohomological Interpretation was given by \textit{Lausch}. Our definition of extension is a particular case of \textit{D'Alarco's} definition. In this paper, we define \(\theta\)-extension of a semilattice of abelian groups \(A\) by a semilattice of groups \(B\) and we introduce a multiplication in the set \(\mathrm{Ext}(B,A)\) of equivalence classes of \(\theta\)-extensions of \(A\) by \(B\), similar to the Baer multiplication in the case of group extensions. We show that \(\mathrm{Ext}(B,A)\) is an abelian group and study the functional properties of \(\mathrm{Ext}(+ , -)\). We prove that \(\mathrm{Ext}(B,A)\) and \(H_{B^{2}}^{2}(A^{0})\) are naturally isomorphic.
scientific article; zbMATH DE number 3558114