Almost Schreier monoids.
The notion almost-Schreier monoid is introduced and investigated. Here, a monoid means a (multiplicative) commutative semigroup with \(1\) and \(0\), such that the semigroup without \(0\) is cancellative; an example to keep in mind is a domain when only multiplication is considered as operation. Indeed, earlier the authors considered almost-Schreier domains, and in the present paper generalize these investigations to a purely multiplicative setting. A monoid \(H\), in the above sense, is called an almost-Schreier monoid if for non-zero \(a,b_1,b_2\) in \(H\) with \(a\mid b_1b_2\) one has that there exists some integer \(k\) such that \(a^k=a_1a_2\) and \(a_1\mid b_1^k\) and \(a_2\mid b_2^k\). First, several characterizations of almost-Schreier monoids are obtained, and some of there basic properties established. Then, more detailed investigations are undertaken. For example, it is shown that the \(t\)-class group of an almost-Schreier monoid is a torsion group; and, that among \(t\)-Prüfer monoids, almost-Schreier monoids are characterized by this property.
scientific article; zbMATH DE number 6010372