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.
We begin by introducing an algebraic structure with three constants and one ternary operation to which we call mobi algebra. This structure has been designed to capture the most relevant properties of the unit interval that are needed in the study of geodesic paths. Another algebraic structure, called involutive medial monoid (IMM), can be derived from a mobi algebra. We prove several results on the interplay between mobi algebras, IMM algebras and unitary rings. It turns out that every unitary ring with one half uniquely determines and is uniquely determined by a mobi algebra with one double. This paper is the second of a planned series of papers dedicated to the study of geodesic paths from an algebraic point of view, the first paper in the series is [2].
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