A representation theorem for integral rigs and its applications to residuated lattices

Abstract

We prove that every integral rig in Set is (functorially) the rig of global sections of a sheaf of really local integral rigs. We also show that this representation result may be lifted to residuated integral rigs and then restricted to varieties of these. In particular, as a corollary, we obtain a representation theorem for pre-linear residuated join-semilattices in terms of totally ordered fibers. The restriction of this result to the level of MV-algebras coincides with the Dubuc-Poveda representation theorem.

Split exact sequences of finite MTL-chains

“…An ordered commutative monoid is a structure S = (S, •, ≤, 1) such that (S, •, 1) is a commutative monoid, (S, ≤) is a poset and • is monotone with respect to ≤. If 1 is the top element of (S, ≤), then S is called integral 3 . If S is convex, then we say that S is an integral commutative convex monoid.…”

“…Since its foundation, algebraic logic has been devoted to studying logics from the perspective of universal algebra. But it is important to recall that there are some works in the literature where-starting from algebraic structures which are closer to rings-results of interest in algebraic logic have been obtained (see [3]). These facts bring out the question of the potential of employing tools of classical algebra (rings, modules, etc.)…”

On finite MTL-algebras that are representable as poset products of archimedean chains

“…Observe that (5) guarantees that h is well defined. To check that h ∈ i∈T M i , let us suppose that i ∈ T and h(i) = 0 i .…”

“…A straightforward verification shows that ∼ is a congruence on M. For every a ∈ M, we write [a] for the equivalence class of a in M/F . Recall that (Section 3 of [5]) the canonical homomorphism h : A → A/M has the universal property of forcing all the elements of M to be 1; i.e, for every MTL-algebra B and every MTL-morphism f : A → B such that f (a) = 1 for every a ∈ M, there exists a unique MTL-morphism g : A/M → B making the diagram below…”

MVW-rigs

“…In the present work the class of MV-algebras with products is characterized in a wider context than that presented by Dinola. From the properties of the universal algebra found in the MV-algebras of closed continuous functions for products it will be shown that this general context is more convenient to work properties analogous to commutative algebra.As a result of this characterization a new algebraic structure is defined, which is an MV-algebra endowed with a product operation, which we will call MVW-rig (Weak-Rig Multivalued) because of its close relation with the rigs defined in [2]. This structure is defined with axioms of universal algebra, a good number of natural examples are presented in the MV-algebras environment and the first results concerning homomorphisms, ideals, quotients and subdirect products are established.…”