Conuclei on varieties of hoops

Abstract

A conucleus $\delta$ on a partially ordered monoid $\mathbf{A}$ is an interior operator that satisfies $\delta(a) \cdot \delta(b) \leq \delta(a \cdot b)$ and $\delta(a) \cdot \delta(1) = \delta(a)$ for all $a,b \in A$. A conucleus is multiplicative if the equality $\delta(a \cdot b) = \delta(a) \cdot \delta(b)$ holds for all $a,b \in A$. In this article we focus on the study of conuclei on hoops, structures which generalize well-known classes of algebras, such as the class of MV-algebras and BL-algebras. Among several results, we provide a Glivenko-type theorem for conuclei. Special emphasis is given to term definable conuclei. The main result of this article is an explicit description of all terms that define a multiplicative conucleus on every structure of an arbitrary proper variety of Wajsberg hoops. We also show that the problem of finding terms that define (multiplicative) conuclei on a variety of basic hoops or BL-algebras is equivalent to finding such terms on some variety or some pair of varieties of Wajsberg hoops. We provide nontrivial interesting examples.