Probabilities in varieties of MV-algebras

The MV (many-valued) algebras were introduced by Chang (1958) as algebraic models of the infinite-valued Lukasiewicz logic. MV-algebras constitute a variety. For every n⩽2, the class MV_n of all n-valued algebras is a subvariety of the variety of all MV-algebras. Each variety MV_n is generated by the finite-chain MV-algebra having n elements. The notion of probability (=state) on an MV-algebra was first studied by Mundici (1995). States on Abelian lattice-ordered groups with a strong unit [ALOG(su)] were introduced as a natural generalization of states on partially-ordered real vector spaces with an order unit. Recalling that the category of ALOG(su) is equivalent to the category of MV-algebras, then a natural definition of states can be given, paralleling the notion of states on an ALOG(su). If (G,u) is an ALOG(su) with strong unit u, and R is the additive group of real numbers, then a state on (G,u) is any normalized positive homomorphism from (G,u) to (R,1), i.e. any additive map s from G to n such that s(G ⁺)⊆R⁺ and s(u)=1, where G⁺ and R ⁺ are the positive cones of G and R respectively