2026/07/18 by Sebastián Buss, Diego Castaño, José Patricio Díaz Varela
#math.LO
A conucleus δ on a partially ordered monoid A is an interior operator that satisfies δ(a) ⋅ δ(b) ≤ δ(a ⋅ b) and δ(a) ⋅ δ(1) = δ(a) for all a,b ∈ A. A conucleus is multiplicative if the equality δ(a ⋅ b) = δ(a) ⋅ δ(b) holds for all a,b ∈ 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.