Generalized Bosbach and Riečan states based on relative negations in residuated lattices

Abstract

Bosbach and Riečan states on residuated lattices both are generalizations of probability measures on Boolean algebras. Recently, two types of generalized Bosbach states on residuated lattices were introduced by Georgescu and Mureşan through replacing the standard MV-algebra in the original definition with arbitrary residuated lattices as codomains. However, several interesting problems there remain still open. The first part of the present paper gives positive answers to these open problems. It is proved that every generalized Bosbach state of type II is of type I and the similarity Cauchy completion of a residuated lattice endowed with an order-preserving generalized Bosbach state of type I is unique up to homomorphisms preserving similarities, where the codomain of the type I state is assumed to be Cauchy-complete. Consequently, many existing results about generalized Bosbach states can be further strengthened. The second part of the paper introduces the notion of relative negation (with respect to a given element, called relative element) in residuated lattices, and then many issues with the canonical negation such as Glivenko property, semi-divisibility, generalized Riečan state of residuated lattices can be extended to the context of such relative negations. In particular, several necessary and sufficient conditions for the set of all relatively regular elements of a residuated lattice to be special residuated lattices are given, of which one extends the well-known Glivenko theorem, and it is also proved that relatively generalized Riečan states vanishing at the relative element are uniquely determined by their restrictions on the MV-algebra consisting of all relatively regular elements when the domain of the states is relatively semi-divisible and the codomain is involutive.