### A concentration of categories with non-injective monomorphisms

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A variety 𝕍 of algebras of a finite type is almost ff-universal if there is a finiteness-preserving faithful functor F: 𝔾 → 𝕍 from the category 𝔾 of all graphs and their compatible maps such that Fγ is nonconstant for every γ and every nonconstant homomorphism h: FG → FG' has the form h = Fγ for some γ: G → G'. A variety 𝕍 is Q-universal if its lattice of subquasivarieties has the lattice of subquasivarieties of any quasivariety of algebras of a finite type as the quotient of its sublattice....

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) universal if any category of algebras has a full embedding into $\mathbb{K}$, and $\mathbb{K}$ is almost universal 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.