Many-valued relation algebras

Abstract

We introduce MV-relation algebras (MVRAs) and distributive MV-relation algebras (DMVRAs), many-valued generalizations of classical relation algebras and study some of their arithmetical properties. We provide corresponding notions of group relation algebra and complex algebra and generalize some results about them from the classical case. For this, we work with more general structures than MVRAs and DMVRAs, by replacing the MV part with a BL-algebra, obtaining what we call fuzzy relation algebras and distributive fuzzy relation algebras.

Some algebraic theory for many-valued relation algebras

“…MV-relation-algebras were defined in [14], using only the first three axioms, (A0)-(A2) below. The reader may wish to consult [14] for introductory definitions and results.…”

“…The reader may wish to consult [14] for introductory definitions and results. What we shall call here "MV-relation-algebra" is required to satisfy more axioms, (A3)-(A5) below.…”

“…This equality is equivalent, consecutively, to: Proof. That algebras satisfying (A0)-(A3) form a variety (implying that MVRAs do so) was proved in [14]. We recall from [14] the set of equations which can replace (A2) in the specification of MVRAs:…”

“…Some examples of algebras of MVRA-type satisfying (A0)-(A3) were already given in [14]. Many of those examples satisfy (A4) and (A5) as well.…”

“…(7) z ≤ x ⊕ y iff z x ≤ y, (8) x i∈I y i = i∈I (x y i ), (9) x ⊕ i∈I y i = i∈I (x ⊕ y i ), (10) d(x, y) = 0 iff x = y, (11) d(x, y) = d(y, x), (12) d(x, z) ≤ d(x, y) ⊕ d(y, z), (13) d(x, y) = d(x, y), (14) …”