MV-pairs and state operators

Flaminio and Montagna (2008) enlarged the language of MV-algebras by a unary operation σ, called internal state or a state operator, equationally defined so as to preserve the basic properties of a state in its usual meaning. The resulting class of MV-algebras is called state MV-algebras. Jenča (2007) and Vetterlein (2008), using different approaches, represented MV-algebras through the quotient of a Boolean algebra B by a suitable subgroup G of the group of all automorphisms of B. Such a couple (B,G) is called an MV-pair. We introduce the notion of a state MV-pair as a triple (B,G, σ), where (B,G) is an MV-pair and σ is a state operator on B, and show that there are relations between state MV-pairs and state MV-algebras similar to the relations between MV-pairs and MV-algebras. We also give a characterization of those MV-pairs, resp. state MV-pairs, that induce subdirectly irreducible MV-algebras, resp. state MV-algebras. MV-algebras were introduced as algebraic bases for many-valued logic [3]. That is, MV-algebras stand in relation to the Lukasziewicz infinite valued logic as Boolean algebras stand to classical two-valued logic. A key relationship between Boolean algebras and MV-algebras lies in the fact that the set of all idempotents of an MV-algebraM is a Boolean algebra, in fact the greatest Boolean subalgebra of M . The Boolean algebra of idempotents can be considered as a system of classical propositions, while the surrounding algebra M can be considered as an extension of the classical logic by fuzzy, resp. unsharp propositions. Another relation between MV-algebras and Boolean algebras was described in [10], where a representation theorem for MV-algebras is given in terms of Boolean algebras and their automorphism groups. Actually, it is shown in [10] that given a Boolean algebra B and a subgroup G of its automorphism group satisfying certain conditions, the pair (B,G) can be canonically associated with an MV-algebra. Such pairs (B,G) are called MV-pairs. Conversely, given an MValgebra M , if B(M) denotes its R-generated Boolean algebra [11] and G(M) is a special subgroup of the automorphism group of B(M), it turns out that (B(M), G(M)) forms an MV-pair. In [7], a categorical development of the results in [10] is presented. ? This contribution is based on a joint work with E. Vinceková. ?? This work was supported by grant VEGA 2/0059/12 and by Science and Technology Assistance Agency under the contract no. APVV-0178-11.