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 ðB=$ G Þ; where ð $ G Þ; 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 ðB=$ G Þ; 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 ðB=$ G Þ. We also study some relations between states on MV-algebras and the corresponding R-generated Boolean algebras.Keywords MV-algebra Á Effect algebra Á R-generated Boolean algebra Á MV-pair Á State 1 MV-algebras and MV-effect algebras MV-algebras were introduced by Chang in 1958 (Chang 1959) as a model for many-valued logics. The original Chang's definition was simplified by Mangani in 1973. An MV-algebra ðM; È; à ; 0Þ is a (2, 1, 0) type of algebra, where the binary operation È is commutative and associative with the nullary operation 0 as neutral element;The last of the axioms is called Łukasiewicz axiom and it is the one which guarantees that MV-algebra is a lattice which is moreover distributive, according to the ordering given by a b iff a Ã È b ¼ 1: Moreover, in 1986, a famous representation theorem by Mundici furnishes the concept of MV-algebra by isomorphism between MValgebras and intervals [0, u] in Abelian lattice-ordered groups with strong unit u. We have a prototypical model of MV-algebra in the real unit interval [0, 1].It turns out that MV-algebras are a subclass of a more general algebraic structures called effect algebras. An effect algebra is a partial algebraic structure ðE; þ; 0; 1Þ with a partial binary operation ? satisfying (1) a þ b ¼ b þ a; (2) ða þ bÞ þ c ¼ a þ ðb þ cÞ (in the sense that if one side exists so does the other and equality holds); (3) to every a 2 E; there is a unique element a 0 such that a þ a 0 ¼ 1; (4) if a þ 1 is defined then a ¼ 0: We say that a and b are orthogonal (written a ? b) if a þ b is defined. Sometimes we write a þ b tacitly assuming that a ? b: Notice that the element a 1 þ a 2 þ Á Á Á þ a n ¼: P n i¼1 a i can be defined recurrently if all the partial sub-sums exist. An effect algebra is partially ordered by the relation a b if there is c 2 E with a þ c ¼ b: If such an element c exists, it is unique, and is denoted by c ¼ b À a: In the latter ordering, 0 is the smallest and 1 is the greatest element. Effect algebras were introduced in Foulis and Bennett (1994), in an alternative form called D-posets in Kôpka and Chovanec (1994).We may define a partial operation ? on an MV-algebra by the restriction of È to the pairs of elements for which a b à : Then the structure ðM; þ; 0; 1Þ is a lattice ordered effect algebra that satisfies some additional conditions. Also a reversed process can be done to obtain an