Geometrical methods in Wajsberg hoops

Abstract

The aim of this paper is to show that certain geometric methods - originally developed for investigating Lukasiewicz many-valued logics - may be profitably applied to the study of classes of partially ordered residuated. monoids. We substantiate this claim by characterizing all free Wajsberg hoops in terms of McNaughton functions over the n-cube, and by giving complete axiomatizations for all varieties of Wajsberg hoops

Amalgamation and interpolation in ordered algebras

“…(2) ⇒ (1) Suppose now that A satisfies condition (2). Then A has the CEP: just let Θ be the identity congruence.…”

“…It suffices to show that (2) implies (1). Let D be the direct product of the algebras D xy for all two-element sets {x, y} as in the statement of (2). By the co-universality of D, the homomorphisms h xy : B → D xy and k xy :…”

“…Then A has the CEP: just let Θ be the identity congruence. Hence it suffices to show that also every homomorphic image of A satisfies (2). In other words, let A 1 be a homomorphic image of A, and let π : A → A 1 be the corresponding surjective homomorphism.…”

“…Consider an element (x 1 , y 1 ) ∈ (Θ 1 ∨Cg A 1 (R 1 ))∩B 2 1 . There exist x, y ∈ B such that x 1 = π(x) and y 1 = π(y).…”

“…Model-theoretic proofs (based on quantifier-elimination) of the deductive interpolation property and amalgamation property for semilinear varieties of commutative residuated lattices can be found in [13], [49], and [50]. 2 In the remainder of this section, we describe some general conditions for the amalgamation property in varieties of pointed residuated lattices. In particular, we show that a variety V of semilinear residuated lattices satisfying the congruence extension property has the amalgamation property iff the class V lin of totally ordered members of V has the amalgamation property (Theorem 49).…”