The 3x3 lemma in the Σ-Maltsev and Σ-protomodular settings. Applications to monoids and quandles

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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 Σ of points. The results apply, among other structures, to monoids, semirings and quandles