On algebraic and more general categories whose split epimorphisms have underlying product projections

TL;DR

This paper characterizes algebraic varieties where split epimorphisms are product projections and provides new characterizations of several special classes of varieties.

Contribution

It introduces a characterization of varieties with split epimorphisms as product projections and offers new insights into protomodular, unital, subtractive, and other varieties.

Findings

Characterization of varieties with split epimorphisms as product projections

New characterizations of protomodular, unital, and subtractive varieties

Descriptions of varieties of right omega-loops and biternary systems

Abstract

We characterize those varieties of universal algebras where every split epimorphism considered as a map of sets is a product projection. In addition we obtain new characterizations of protomodular, unital and subtractive varieties as well as varieties of right omega-loops and biternary systems.