Some algebraic theory for many-valued relation algebras

Abstract

We study MV-relation-algebras, appearing by abstracting away from the concrete many-valued relations and the operations on them, such as composition and converse. MV-relation-algebras are MV generalizations of the relation algebras developed by A. Tarski and his school starting from the late forties. Some facts about ideals, congruences, and various types of elements are proved. A characterization of the "natural" MV-relation-algebras (a parameterized analogue of the classical full proper relation algebras) is also provided, as well as a first-order elementary description of matrix MV-relation algebras.

Many-valued relation algebras

“…The desired equivalence follows at once from Proposition 2.21. (21) and the MV facts x = x and x ⊕ y = x y.…”

“…At the proof theoretical level, this is a consequence of the fact that for each x, we have x ≤ x ; x ; x, which in turn follows from the idempotency of the conjunction, so we depart here from the many-valued route. Still, interestingly enough, these properties are axiomatically considered is the paper [21], which is a further development of the present work.…”

Łukasiewicz-Moisil Relation Algebras

“…[11] and to the MV approach from [32] and [33]. Unlike in these latter cases, the connection with classical logic, being explicitly stated by means of the Chrysippian nuances, allows one to point out certain interesting similarities between LM and classical relational structures.…”

“…The only studies of many-valued or fuzzy relation algebras that we are aware of are [11], [32], and [33]. While [32] and [33] discuss relational structures based on BL [14] and MV algebras [5] (structures with little technical similarity to LM), in [11] there is a notion of "relation algebra", which is based on a complete Heyting algebra and satisfies an axiom similar to (A2), the socalled "Dedekind formula": (x; y) ∧ z = x; [y ∧ (x ; z)].…”

“…Unlike in these latter cases, the connection with classical logic, being explicitly stated by means of the Chrysippian nuances, allows one to point out certain interesting similarities between LM and classical relational structures. Therefore, we consider that an abstract study of LM-based relation algebras is useful, not only for enriching the many-valued relational algebraic theory developed in [11,32,33], but also for raising the issue of communication with classical relational structures. In this paper, we intend to study these structures, using the following route.…”