Abstract
Normal cryptogroups represented as strong semilattices of completely simple semigroups admit a detailed study of their quasivarieties; in particular of their lattices and groupoids under Malcev product. In this context, we construct the lattice of quasivarieties generated by semilattices \({\mathcal{S}}\), groups \({\mathcal{G}}\), and rectangular bands \({\mathcal{RB}}\) ; it is a 17-element distributive lattice. We compute the groupoids of quasivarieties generated by pairs \({(\mathcal{S}, \mathcal{G}), (\mathcal{S},\mathcal{RB})}\), and \({(\mathcal{G},\mathcal{RB})}\), as well as find the list of quasivarieties which can be expressed by Malcev products with words of length at most equal to 3. This analysis presupposes multiple characterizations of each quasivariety involved in order to establish their relationship, and in particular to prove that all the joins are obtained by means of subdirect products.
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