Miriam Kurtzhals (University of Barcelona) will give a talk on
The Beth companion: making implicit operations explicit
Abstract:
Complements in distributive lattices, when they exist, are unique and preserved by homomorphisms. Thus, complementation can be regarded as a partial operation of the variety of distributive lattices that, while not explicitly defined by a term in the language of lattices, can be implicitly defined by a formula describing complements. The ability to make implicit operations explicit resembles the Beth definability property in logic and is closely related to the surjectivity of epimorphisms. In fact, the implicit, but not explicit, definability of complements can be thought of as the cause of the existence of non-surjective epimorphisms between distributive lattices.
In this talk we will introduce the notion of a Beth companion as an optimal solution to the problem of making all implicit operations of a given class of algebras explicit. We will see that the Beth companion of a quasivariety, if it exists, is a quasivariety that is unique up to term equivalence. Notably, the variety of Boolean algebras is the Beth companion of the variety of distributive lattices. Other examples of Beth companions will be presented and some varieties not admitting a Beth companion will be discussed. Moreover, we will see that properties of the original class of algebras, such as congruence distributivity and the amalgamation property, are often preserved, and in some cases improved, by passing to the Beth companion.