MV-pairs and states

Abstract

An MV-pair is a BG-pair (B; G) (where B is a Boolean algebra and G is a subgroup of the automorphism group of B) satisfying certain conditions. Recently, it was proved by Jenca that, given an MV-pair (B; G), the quotient $(B/\!\sim_G),$ where $(\sim_G),$ is an equivalence relation naturally associated with G, is an MV-algebra, and conversely, to every MV-algebra there corresponds an MV-pair. In this paper, we introduce a new definition of so called MV*-pair, and we show that $(B/\!\sim_G),$ is an effect algebra iff the first of the defining properties of the MV*-pair is satisfied, while the second property guarantees the MV-algebra structure of $(B/\!\sim_G)$. We also study some relations between states on MV-algebras and the corresponding R-generated Boolean algebras.