Abstract. A category ${\cal K}$ is $\alpha $-determined for some cardinal $\alpha $ if any class of non-isomorphic ${\cal K}$-objects having isomorphic endomorphism monoids is a set with fewer than $\alpha $ elements. An $\alpha $-expansion ${\cal K}_{\alpha }$ is the category whose objects are all ${\cal K}$-objects augmented by $\alpha $ new constants and whose morphisms are exactly the ${\cal K}$-morphisms preserving these constants. And a category is alg-universal if it contains an isomorphic copy of any category of algebras as a full subcategory. This paper characterizes the finitely generated varieties of distributive double $p$-algebras which are $\alpha $-determined for some cardinal $\alpha $ as well as those having $\alpha $-expansions which are alg-universal. Results of this paper complete the project of a structural classification of finitely generated varieties of distributive double $p$-algebras according to their categorical properties.

AMS Subject Classification (1991): 18B15

Keyword(s): distributive $dp$-algebra, finitely generated variety of $dp$-algebras, relatively full embedding, relative alg-universality, determinacy, Priestley duality, expansion by nullary operations

Received August 2, 2011, and in revised form January 23, 2012. (Registered under 39/2011.)