### Concerning endomorphisms of finite algebra

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Any finitely generated regular variety $\mathbb{V}$ of distributive double $p$-algebras is finitely determined, meaning that for some finite cardinal $n\left(\mathbb{V}\right)$, any subclass $S\subseteq \mathbb{V}$ of algebras with isomorphic endomorphism monoids has fewer than $n\left(\mathbb{V}\right)$ 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...

A concrete category $\mathbb{K}$ is (algebraically) if any category of algebras has a full embedding into $\mathbb{K}$, and $\mathbb{K}$ is if there is a class $\mathcal{C}$ of $\mathbb{K}$-objects such that all non-constant homomorphisms between them form a universal category. The main result of this paper fully characterizes the finitely generated varieties of $0$-lattices which are almost universal.

We show that prohibiting a combinatorial tree in the Priestley duals determines an axiomatizable class of distributive lattices. On the other hand, prohibiting $n$-crowns with $n\ge 3$ does not. Given what is known about the diamond, this is another strong indication that this fact characterizes combinatorial trees. We also discuss varieties of 2-Heyting algebras in this context.

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