Regularity in algebraic frames

This article considers algebraic frames in which the meet of two compact elements is compact, and, in that context, when the subframe of all regular elements is itself regular. Motivated by the study of a frame of convex ℓ-subgroups of a lattice-ordered group, a number of relevant sufficient conditions are given for this subframe to be regular. An example is given of a frame of convex ℓ-subgroups for which the subframe of regular elements is not regular.