We present a description of the (non-modular) commutator, inspired by that of Kearnes in [Kea95, p. 930], that provides a simple recipe for computing the commutator.

We consider the following practical question: given a finite algebra [Formula: see text] in a finite language, can we efficiently decide whether the variety generated by [Formula: see text] has a difference term? We answer this question (positively) in the idempotent case and then describe algorithms for constructing difference term operations.

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We present a novel approach to the construction of new finite algebras and describe the congruence lattices of these algebras. Given a finite algebra $(B_0, \dots)$, let $B_1, B_2, \dots, B_K$ be sets that either intersect $B_0$ or intersect each other at certain points. We construct an \emph{overalgebra} $(A, F_A)$, by which we mean an expansion of $(B_0, \dots)$ with universe $A = B_0 \cup B_1 \cup \cdots \cup B_K$, and a certain set $F_A$ of unary operations that includes mappings $e_i$ satisfying $e_i^2 = e_i$ and $e_i(A) = B_i$, for $0\leq i \leq K$. We explore two such constructions and prove results about the shape of the new congruence lattices $Con(A, F_A)$ that result. Thus, descriptions of some new classes of finitely representable lattices is one contribution of this paper. Another, perhaps more significant contribution is the announcement of a novel approach to the discovery of new classes of representable lattices, the full potential of which we have only begun to explore.Comment: 22 pages, 10 figure

We present pairs of isotopic algebras with congruence lattices of different sizes, thus answering negatively the question of whether isotopic algebras must have isomorphic congruence lattices.It is well known that two algebras in a congruence modular variety that are isotopic have isomorphic congruence lattices. In fact, this result holds more generally, but to date the most general result of this kind (recalled below) assumes some form of congruence modularity. It is natural to ask to what extent the congruence modularity hypothesis could be relaxed and whether it is possible to prove that all isotopic algebras have isomorphic congruence lattices. In this note, we show that the full generalization is not possible; we construct a class of counter-examples involving pairs of algebras whose congruence lattices are obviously not isomorphic, and then prove that these pairs of algebras are isotopic.If A, B, and C are algebras of the same signature, we say that A and B are isotopic over C, and we write A ∼ C B, if there exists an isomorphism ϕ : A × C → B × C such that for all a ∈ A and c ∈ C, the second coordinate of ϕ(a, c) is c; that is, ϕ(a, c) = (b, c) for some b ∈ B. We say that A and B are isotopic, and we write A ∼ B, provided A ∼ C B for some C. It is not hard to check that ∼ is an equivalence relation.If A ∼ C B and the congruence lattice of A × C happens to be modular, then we write A ∼ mod C B, in which case we say that A and B are modular isotopic over C. We call A and B modular isotopic in one step, denoted A ∼ mod 1 B, if they are modular isotopic over C for some C. Finally, A and B are modular isotopic, denoted A ∼ mod B if the pair (A, B) belongs to the transitive closure of ∼ mod 1 . Let Con A denote the congruence lattice of A. It is well known that A ∼ mod B implies Con A ∼ = Con B. The proof of this result appearing in [2] is a straightforward application of Dedekind's Transposition Principle. Since a version of this principle has been shown to hold even in the nonmodular case ([1]), we might hope that the proof technique used in [2] could be used to show that A ∼ C B implies Con A ∼ = Con B. But this strategy quickly breaks down, and the application of the perspectivity map, which works finePresented by E. Kiss.