Classes of Algebras and closure operations

TL;DR

This paper explores the generalization of class and closure operation theories from group theory and specific algebra varieties to the broader context of all non-associative algebras.

Contribution

It investigates how existing theories in group and algebra varieties can be extended to the entire class of non-associative algebras.

Findings

Theories from group and algebra varieties can be partially generalized.

New closure operations applicable to non-associative algebras are identified.

Potential frameworks for further research in algebra classes are proposed.

Abstract

The calculus of classes and closure operations has proved to be a useful tool in group theory and has led to a deep theory in the study of finite soluble groups. More recently, parallel theories have started to be developed in various varieties of algebras, such as Lie, Leibniz, and Malcev algebras. This paper seeks to investigate the extent to which these later theories can be generalised to the variety of all non-associative algebras.