On coatoms in lattices of quasivarieties of algebraic systems. (Q1771855)

Let \(qK\) stand for the quasivariety of algebraic systems generated by a class \(K\), let \(L_q(M)\) be the lattice of subquasivarieties contained in a quasivariety \(M\). Coatoms in the lattice \(L_q(M)\) for a finite set \(K\) of finite algebraic systems were studied by \textit{A.\ I.\ Budkin} and \textit{V.\ A.\ Gorbunov} [Algebra Logika 14, 123--142 (1975; Zbl 0317.08003)]. It was shown that this lattice has a finite set of coatoms and each proper subquasivariety of \(qK\) is contained in some atom. The aim of the present paper is to find a necessary condition for the lattice \(L_q(M)\) to have a finite set of coatoms. In particular, it is shown that \(L_q(M)\) has finitely many atoms for the quasivariety \(M\) generated by a finitely generated abelian-by-polycyclic-by-finite group or a totally ordered group.