We explore some properties of Schreier split epimorphisms between monoids, which correspond to monoid actions. In particular, we prove that the split short five lemma holds for monoids, when it is restricted to Schreier split epimorphisms, and that any Schreier reflexive relation is transitive, partially recovering in monoids a classical property of Mal'tsev varieties.

a b s t r a c tWe describe actions, semidirect products and crossed modules in categories of monoids with operations. Moreover we characterize, in this context, the internal categories corresponding to crossed modules. Concrete examples in the cases of monoids, semirings and distributive lattices are given.

We investigate the notion of pointed S-protomodular category, with respect to a suitable class S of points, and we prove that these categories satisfy, relatively to the class S, many partial aspects of the properties of Mal'tsev and protomodular categories, like the split short five lemma for S-split exact sequences, or the fact that a reflexive S-relation is transitive.The main examples of S-protomodular categories are the category of monoids and, more generally, any category of monoids with operations, where the class S is the class of Schreier points.

Abstract. The aim of this paper is to solve a problem proposed by Dominique Bourn: to provide a categorical-algebraic characterisation of groups amongst monoids and of rings amongst semirings. In the case of monoids, our solution is given by the following equivalent conditions:(i) G is a group; (ii) G is a Mal'tsev object, i.e., the category Pt G pMonq of points over G in the category of monoids is unital; (iii) G is a protomodular object, i.e., all points over G are stably strong, which means that any pullback of such a point along a morphism of monoids Y Ñ G determines a split extensionin which k and s are jointly strongly epimorphic. We similarly characterise rings in the category of semirings.On the way we develop a local or object-wise approach to certain important conditions occurring in categorical algebra. This leads to a basic theory involving what we call unital and strongly unital objects, subtractive objects, Mal'tsev objects and protomodular objects. We explore some of the connections between these new notions and give examples and counterexamples.