ON MONOASSOCIATIVE GROUPOIDS

Abstract

The subject of this paper is the variety (denoted by \(\mathit{Mass}\)) of monoassociative groupoids, i.e. groupoids in which every cyclic subgroupoid is a subsemigroup. A description of free objects in \(\mathit{Mass}\) is given. Using a convenient definition of injective groupoids in \(\mathit{Mass}\), it is shown that a groupoid \(H\) is free in \(\mathit{Mass}\) iff \(H\) is injective in \(\mathit{Mass}\) and the set of prime elements in \(H\) generates \(H\). (This property is named Bruck Theorem for \(\mathit{Mass}\).) Neither of the classes \(\mathit{Massin}\) (injective objects in \(\mathit{Mass}\)) and \(\mathit{Massfr}\) (free objects in \(\mathit{Mass}\)) is hereditary. A characterization of free subgroupoids of a groupoid \(H \in \mathit{Massfr}\) is obtained. It is shown that every groupoid \(H \in \mathit{Massfr}\) with a two-element basis has a subgroupoid \(Q \in \mathit{Massfr}\) with an infinite basis.