Remarks on ``Identities and quasi-identities in lattices of pseudovarieties''
Theorem 2 of our paper [Russ. Math. 37, No. 3, 38-48 (1993); translation from Izv. Vyssh. Uchebn. Zaved. Mat. 1993, No. 3(370), 40-50 (1993; Zbl 0832.08005)] asserts that the following conditions concerning an arbitrary pseudovariety of semigroups \(\mathcal X\) are equivalent: a) the lattice \(L_p({\mathcal X})\) is weakly semimodular downwards; b) the lattice \(L_p({\mathcal X})\) is weakly semimodular upwards; c) the lattice \(L_p({\mathcal X})\) is semimodular downwards; d) the lattice \(L_p({\mathcal X})\) is semimodular upwards; e) the lattice \(L_p({\mathcal X})\) is modular; f) the lattice \(L_v(\text{var }A)\) is modular for any semigroup \(A\in{\mathcal X}\); g) \({\mathcal X}\not\supseteq{\mathcal V}_{\text{fin}}\), where \(\mathcal V\) is the arbitrary locally finite almost modular variety of semigroup. But recently, it became clear that on the whole this theorem fails. Indeed, there are equivalent: conditions b) and d)--g) on the one hand, and conditions a) and c) on the other hand, but the conditions a) and b) are in general not equivalent. Nevertheless, the equivalence of conditions a)--g) is valid for a rather wide class of pseudovarieties, which is described in this paper.
scientific article; zbMATH DE number 1023845