Right-cancellable protomodular algebras

TL;DR

This paper introduces right-cancellable protomodular algebras and establishes their role in characterizing the existence of group terms in universal algebra varieties, linking algebraic properties to topological loop regularity.

Contribution

It presents a new criterion for group term existence using right-cancellable protomodular algebras, extending classical algebraic theory and addressing open problems in loop theory.

Findings

Characterization of right-cancellable protomodular algebras as sets with principal group actions.

Equivalence between the presence of group terms and protomodular terms with right-cancellability.

Partial solution to the open problem on the regularity of Hausdorff topological loops.

Abstract

A new protomodular analog of the classical criterion for the existence of a group term in the algebraic theory of a variety of universal algebras is given. To this end, the notion of a right-cancellable protomodular algebra is introduced. It is proved that the algebraic theory of a variety of universal algebras contains a group term if and only if it contains protomodular terms with respect to which all algebras from the variety are right-cancellable. This, in particular, gives a partial answer to the extended version of an open problem from loop theory whether any Hausdorff topological (semi-)loop is completely regular. Moreover, the right-cancellable algebras from the simplest protomodular varieties are characterized as sets with principal group actions as well as groups with simple additional structures.