Abstract: | LetL
q(M) be a lattice of quasivarieties contained in a quasivarietyM. The quasivariety is closed under direct wreath Z-products if together with a group G, it contains its wreath product G ≀
Z with an infinite cyclic group Z. We prove the following: (a) ifM is closed under direct wreath Z-products then every quasivariety, which is a coatom inL
q(M), is likewise closed under these; (b) ifM is closed under direct wreath products thenL
q(M) has at most one coatom. An example of a quasivariety is furnished which is closed under direct wreath Z-products and whose
subquasivariety lattice contains exactly one coatom. Also, it turns out that the set of quasivarieties closed under direct
wreath Z-products form a complete sublatttice of the lattice of quasivarieties of groups.
Supported by RFFR grant No. 96-01-00088, and by the RF Committee of Higher Education.
Translated fromAlgebra is Logika, Vol. 38, No. 3, pp. 257–268, May–June, 1999. |