Closing reciprocal relations w.r.t. stochastic transitivity

Abstract

We equip the set of reciprocal relations with an appropriate order relation capturing the bipolar semantics, turning this set into a complete join-semilattice. We also introduce the notion of a compatible family of reciprocal relations and show that such a family has an infimum. Moreover, we discuss the one-to-one correspondence between the set of 3-valued reciprocal relations and the set of complete crisp relations. We refine the theorem of Bandler and Kohout concerning the existence of closures of elements of a poset w.r.t. some given property, rendering it applicable to the above join-semilattice. The paper is solely dedicated to the transitivity property. Although uniquely defined for 3-valued reciprocal relations, general reciprocal relations can exhibit various types of transitivity. We characterize the mappings g and h for which the corresponding g-stochastic and h-isostochastic transitive closure exists for any reciprocal relation. In particular, it follows that weak and strong stochastic transitive closures always exist, while this is not the case for moderate stochastic transitivity. Moreover, max-isostochastic transitivity turns out to be the only practically relevant type of isostochastic transitivity. Finally, we provide algorithms realizing each of these transitive closures.