Flatness properties of diagonal acts over monoids.
Let \(S\) be a monoid. The right \(S\)-act \(S\times S\), denoted by \(D(S)\), equipped with componentwise \(S\)-action, is called the diagonal act of \(S\). In this paper, some kinds of flatness properties for \(D(S)\) are discussed. Conditions (including some sufficient and necessary conditions) for \(D(S)\) to be principally weakly flat, weakly flat, flat, strongly flat, and satisfying condition (P), condition (E) are obtained in succession. Characterizations of projective, free, flat, and other diagonal acts of some special kinds of semigroups, such as right cancellative monoids, commutative monoids, and complete simple semigroups, complete 0-simple semigroups, are studied simultaneously.
scientific article; zbMATH DE number 5614944