Any finitely generated regular variety $𝕍$ of distributive double $p$-algebras is finitely determined, meaning that for some finite cardinal $n\left(𝕍\right)$, any subclass $S\subseteq 𝕍$ of algebras with isomorphic endomorphism monoids has fewer than $n\left(𝕍\right)$ pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double $p$-algebras...

We show that any finitely generated variety V of double Heyting algebras is finitely determined, meaning that for some finite cardinal n(V), any class $𝒮$ ⊆ V consisting of algebras with pairwise isomorphic endomorphism monoids has fewer than n(V) pairwise non-isomorphic members. This result complements the earlier established fact of categorical universality of the variety of all double Heyting algebras, and contrasts with categorical results concerning finitely generated varieties of distributive...