Mal’tsev conditions, lack of absorption, and solvability

Abstract

Abstract. We provide a new characterization of several Mal'tsev conditions for locally finite varieties using hereditary term properties. We show a particular example how lack of absorption causes collapse in the Mal'tsev hierarchy, and point out a connection between solvability and lack of absorption. As a consequence, we provide a new and conceptually simple proof of a result of Hobby and McKenzie, saying that locally finite varieties with a Taylor term possess a term which is Mal'tsev on blocks of every solvable congruence in every finite algebra in the variety.

The collapse of the bounded width hierarchy

“…Such a weaker version would be sufficient for our purposes. However, to avoid some technicalities, we use this general version from [5].…”

Asking the Metaquestions in Constraint Tractability

“…[5], Theorem 1.3). Let t be a k-ary idempotent polymorphism of H satisfying:t[A] = t[B]where A and B are k × k matrices with entries in {x, y} such that a ii = x and b ii = y for all i and a ij = b ij for all i = j.…”

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“…Pointing operations were first used in [6]. More details, as well as a proof of the characterization theorem we need, can be found in [11]. Definition 4.5.…”

“…This property is characterized (for cores) by congruence meet-semidistributivity; the characterization was conjectured, and the "only if" part proved, in [35]. The proof of the Bounded Width Theorem uncovered a new characterization of SD(∧) algebras via so-called pointing operations as well as the concept of absorbing subuniverse, which turned out to be quite useful even outside of the realm of congruence meet-semidistributivity (see [5,11,8]).…”