A discrete representation of free MV-algebras

The variety of MV-algebras is the equivalent algebraic semantics of the infinitely-valued Lukasiewicz logic [1], and MVn-algebras are algebraic models of the Lukasiewicz logic with n truth values (2 ≤ n < ω). Recall that an algebra A ∈ V is said to be a free algebra in a variety V, if there exists a set A0 ⊂ A such that A0 generates A and every mapping f from A0 to any algebra B ∈ V can be extended to a homomorphism h from A to B. In this case A0 is said to be the set of free generators of A. If the set of free generators has cardinality m ∈ ω then A is said m-generated or, in general, finitely generated. The structure of non-equivalent formulas of Lukasiewicz propositional logic forms the free ω-generated MV-algebra, through the well-known Tarki-Lindenbaum construction. If we restrict to non-equivalent formulas with m propositional variables, then we obtain the m-generated free MV-algebra. R. McNaughton described a set of special functions f : [0, 1] −→ [0, 1], given by all the continuous piecewise linear functions with integer derivatives, called after him McNaughton functions. Such a set, when endowed with MV-operations defined component-wise, is isomorphic to the m-generated free MValgebra [6]. It is worth to stress that the characterisation due to McNaughton was improved in [7, 8]. Another characterisation of the free MV-algebrea is given in [9]. Let MVn indicate the subvariety of MV (the variety of all MV-algebras), given by all MVn-algebras. In [3, 4] is given the description of the m-generated free MV-algebra as a subalgebra of the inverse limit of a system consisting of m-generated free MV-algebras in the variety MVn. A closely related family of varieties is given by MV for n ∈ ω, where each of them sits between MV and MVn. An algebra is locally finite if all its finitely generated subalgebras are finite. Recall also that a variety is called locally finite if all its finitely generated members are finite. The link between the two concepts is given by the fact that a variety is locally finite if, and only if, its free algebra over ω generators is locally finite. It is known that the variety MV is not locally finite but, remarkably, it is generated by all simple finite MV-algebras. In addition we have that the subvarieties of MV which are generated by finite families of simple finite (finite and linearly ordered) MV-algebras are locally finite. So finitely generated free algebras in any variety MVn, as well as MV, are finite, while finite generated free algebras in MV are infinite. For any integer m ≥ 1, we give a representation of the free m-generated MV-algebra in MV, denoted by FMV(m), using in a suitable manner, all the free m-generated algebras from MV, for every n, denoted by FMV(n)(m). A similar approach was studied in [2], although the authors only deal with the case with one generator.