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# Homology, Homotopy and Applications

## Volume 21 (2019)

### Number 2

### The $3 \times 3$ lemma in the $\Sigma$-Mal’tsev and $\Sigma$-protomodular settings: Applications to monoids and quandles

Pages: 305 – 332

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n2.a17

#### Authors

#### Abstract

We investigate what is remaining of the $3 \times 3$ lemma and of the denormalized $3 \times 3$ lemma, valid in a pointed protomodular and in a Mal’tsev category, respectively, in the context of partial pointed protomodular and partial Mal’tsev categories, relatively to a class $\Sigma$ of points (i.e. of split epimorphisms with a fixed section). The results apply, among other structures, to monoids, semirings, and quandles.

#### Keywords

$3 \times 3$ lemma, $\Sigma$-Mal’tsev category, $\Sigma$-protomodular category, monoid, semiring, quandle

#### 2010 Mathematics Subject Classification

18G50, 20M32, 20M50, 57M27

This work was partially supported by the Programma per Giovani Ricercatori “Rita Levi Montalcini”, funded by the Italian government through MIUR.

Received 25 December 2017

Published 3 April 2019