MV-pairs and state operators

Flaminio and Montagna (2008) enlarged the language of MV-algebras by a unary operation σ, called internal state or 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)(B,G) is called an MV-pair. We introduce the notion of a state MV-pair as a triple (B,G,σ)(B,G,σ), where (B,G)(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.