The 3x3 lemma in the ${\Sigma}$-Maltsev and ${\Sigma}$-protomodular settings. Applications to monoids and quandles

TL;DR

This paper explores how classical 3x3 lemmas adapt to partial pointed protomodular and Maltsev categories, with applications to algebraic structures like monoids, semirings, and quandles.

Contribution

It extends the 3x3 lemma framework to partial categories relative to a class of points, broadening its applicability to new algebraic contexts.

Findings

The 3x3 lemma partially holds in partial pointed protomodular categories.

The denormalized 3x3 lemma extends to partial Maltsev categories.

Applications include structures such as monoids, semirings, and quandles.

Abstract

We investigate what is remaining of the 3x3 lemma and of the denormalized 3x3 lemma, respectively valid in a pointed protomodular and in a Maltsev category, in the context of partial pointed protomodular and partial Maltsev categories, relatively to a class ${\Sigma}$ of points. The results apply, among other structures, to monoids, semirings and quandles