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

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

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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