Department of Mathematics
Server

Varieties are classes of algebras axiomatized by a set of identities. Thus for any ring R, R-Mod, the class of all left R-modules, is a variety; the set L_v(K) of varieties contained in a variety K is a lattice under inclusion. Quasivarieties are classes of algebra axiomatized by sets of implications between equations, so every variety is a quasivariety. The class of all M in R-Mod such that the underlying abelian group of M is torsion-free is a quasivariety which is not always a variety; it is axiomitized relative to R-Mod by the set of implications {px = 0 implies x = 0 | p is a prime number}. The set L_q(K) of quasivarieties contained in a quasivariety K is a lattice under inclusion. If K is a variety, then L_v(K) is a meet-subsemilattice of L_q(K) since in both cases meet is just class intersection, but the quasivariety join of two varieties is often properly contained in the variety join of the two varieties.

This talk is based on a paper by Keith Kearnes; it examines the connection between the following two questions:

(A) When is L_v(R_Mod) a sublattice of L_q(R_Mod)?
(B) When is L_q(R-Mod) distributive?