Universal expansions and determinacy for finitely generated varieties of dp -algebras

category 𝒦 is α -determined for some cardinal α if any class of non-isomorphic 𝒦 -objects having isomorphic endomorphism monoids is a set with fewer than α elements. An α -expansion 𝒦 α is the category whose objects are all 𝒦 -objects augmented by α new constants and whose morphisms are exactly the 𝒦 -morphisms preserving these constants. And a category is alguniversal 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 α -determined for some cardinal α as well as those having α -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.