On equalizer-flat acts
Let \(S\) be a monoid and let \(A_S\) be a right \(S\)-act. \(A_S\) is said to satisfy condition (E) if, whenever \(a\in A_S\) and \(s,t\in S\) are such that \(as=at\), there exist \(a_1\in A\) and \(u\in S\) such that \(a=a_1u\), and \(us=ut\). An act \(A_S\) is called equalizer flat if the functor \(A_S\otimes-\) preserves equalizers. The main problem discussed is: for which monoids all \(S\)-acts satisfying condition (E) are equalizer flat? This problem is completely solved for completely simple and completely 0-simple semigroups (with 1 adjoined). For example, if \(S\) is a monoid such that \(S\setminus\{1\}\) is a completely 0-simple semigroup \({\mathcal M}^0(I,G,\Lambda,P)\), then every right \(S\)-act satisfying condition (E) is equalizer flat if and only if either (i) no two columns of the support of \(P\) are identical, or (ii) \(I=\{i,j\}\) is a two-element set, no entry of \(P\) is zero, and \(E_{i,j}(P)=G\), where \(E_{i,j}(P)\) denotes the subgroup of \(G\) generated by all matrix elements of the form \(p^{-1}_{\lambda,i}p_{\lambda,j}p^{-1}_{\mu,j}p_{\mu,i}\), \(\lambda,\mu\in\Lambda\).
scientific article; zbMATH DE number 1251157