On the hyperreal state space

The aim of this work is to study a class of non-archimedean valued measures on MV-algebras. We call them hyperreal states and their definition naturally arise from (the uniform version of) Di Nola representation theorem for MV-algebras (cf [5, 6]): for any MV-algebra A = (A,⊕,¬,>,⊥) there exists a ultrafilter U on the cardinality of A such that A embeds into (∗[0, 1]U) Spec(A) (where as usual Spec(A) denotes the space of prime ideals of A). Therefore, if A is any MV-algebra, there exists a non-archimedean extension ∗[0, 1]U of the real unit interval [0, 1] such that every element a of A can be regarded as a function fa : Spec(A)→ ∗[0, 1]U. Since MV-algebras are the equivalent algebraic semantics for Lukasiewicz logic, Di Nola’s theorem states that formulas of Lukasiewicz calculus are visualized as black and white pictures printed by a palette of infinitesimals grey levels ([9, §2]). As it is well known [3], the proper subclass of semisimple MV-algebras can be characterized, by Chang and Belluce theorem (cf. [2, 1]), as algebras of real-valued functions: up to isomorphisms every semisimple MValgebra A is an algebra of [0, 1]-valued functions defined over the space of maximal idealsM(A) of A. Therefore, following the above metaphor, if we interpret formulas of Lukasiewicz calculus into a semisimple MV-algebra, then by Chang and Belluce theorem there are no infinitesimals grey levels, and hence the palette used to print (i.e. interpret) the propositions, only has real grey levels (cf. [9, §2]). States on MV-algebras have been introduced by Mundici in [9]: for every MV-algebra A, state on A is a map s : A → [0, 1] that is, normalized and additive. Therefore states are real valued maps defined on (possibly) hyperreal-valued functions, and hence they do no preserve, the non-archimedean structure of the MV-algebra they are defined over. 1We refer the reader to [9, Definition 2.1].