Let P be a property of topological spaces. Let [P ] be the class of all varieties V having the property that any topological algebra in V has underlying space satisfying property P . We show that if P is preserved by finite products, and if ¬P is preserved by ultraproducts, then [P ] is a class of varieties that is definable by a Maltsev condition.The property that all T 0 topological algebras in V are j-step Hausdorff (H j ) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize the Maltsev condition associated to T 0 ⇒ H j by showing that this topological implication holds in every (2j + 1)-permutable variety, but not in every (2j + 2)-permutable variety.Finally, we show that the topological implication T 0 ⇒ T 2 holds in every k-permutable, congruence modular variety.

In the interpretability lattice, the varieties possessing a near unaninimity term constitute a filter. It is shown that this filter is the proper intersection of two larger filters. One of these two filters is shown to be modular.

The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning n-permutability for some n, and the satisfaction of nontrivial congruence identities. Our theorems require a technical result about the equational theory of linear varieties, which might be of independent interest.

A new condition of compatibility with projections, applicable to some Maltsev filters, is defined and shown to hold, among others, for the filter of congruence-modular varieties. As a consequence, it is shown that there exist no simple counterexamples (in a specified sense) to the modularity conjecture.

We show that every finite Abelian algebra A from congruencepermutable varieties admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure A ∼ of finite type. We give an explicit bound on the arities of the partial and total operations appearing in A ∼ . In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.Given an algebra A, we denote by A its underlying set, and by Sub(A) the set of subalgebras of A. The variety Var(A) (respectively, the quasivariety QVar(A)) generated by A is the smallest classes of algebras, with the signature of A, that is closed under taking products, subalgebras, and homomorphic images (respectively, products, subalgebras, and isomorphic algebras).Given X ⊆ A we denote by X the subalgebra of A generated by X. For an arbitrary set X and a variety V, we denote by F V (X) the algebra freely generated by X in V.A subproduct algebra A ≤ Π i A i is called a subdirect product if π i (A) = A i for each projection π i . An algebra is subdirectly irreducible if whenever it is isomorphic to a subdirect product, it is already isomorphic to one of its factors.An algebra A is affine if there exists an Abelian group structure A; +, 0, − such that t(x, y, z) = x − y + z is both a term function of A and a homomorphism from A 3 to A. A class of algebras C is affine if all of its algebras are. In the case of an affine variety V, it is easy to see that we may choose one term t that witnesses the affinity simultaneously for all members of V (e.g. we could take the term witnessing the affinity of F V (ω)).An algebra A is Abelian if [1 A , 1 A ] = 0 A , where 1 A and 0 A are the universal and trivial relations on A, and [·, ·] denotes the binary commutator on the congruences of A (we refer to [3] for the definition of the commutator). A class of algebras is Abelian if all of its members are. As usual, when dealing with commutator theoretic conditions, we restrict to algebras that generate congruence-modular varieties. With this condition, Abelian algebras and varieties coincide with affine algebras and varieties [3, Corollary 5.9], and we will use the two notations interchangeably throughout the paper. Our results will rely exclusively on the defining property of affine algebras.We repeat several results about congruences of Abelian algebras from [4].Definition 2.1 ([4], Definition 3.1). Let A be an Abelian algebra and B ∈ Sub(A). The congruence generated by B, denoted by Θ B is the smallest congruence of A containing B 2 .We remark that not every congruence of A can be written in the form Θ B for some subalgebra B, and that we might have Θ B = Θ C with B = C.Lemma 2.2 ([4], Lemma 3.3). Let A be an Abelian algebra, let B ∈ Sub(A), and let t be a term witnessing the affinity of A. ThenNote that this result implies that B is a congruence class of Θ B . Lemma 2.3 ([4], Corollary 3.7). Let A be an Abelian algebra and let B ∈ Sub(A) such that B is meet irreducible in the semilattice Sub(A);...