Representation of MV-algebras by regular ultrapowers of [0, 1]

Abstract

We present a uniform version of Di Nola Theorem, this enables to embed all MV-algebras of a bounded cardinality in an algebra of functions with values in a single non-standard ultrapower of the real interval [0,1]. This result also implies the existence, for any cardinal α, of a single MValgebra in which all infinite MV-algebras of cardinality at most α embed. Recasting the above construction with iterated ultrapowers, we show how to construct such an algebra of values in a definable way, thus providing a sort of "canonical" set of values for the functional representation.

Another proof of the completeness of the Łukasiewicz axioms and of the extensions of Di Nola’s Theorem

“…Second, by using the same arguments as found in [13,Section 4], for every infinite cardinal α, there is an iterated ultrapower (see [6, Section 6.5]) α of (Q ∩ [0, 1]) N , definable in α, in which every MV-algebra of cardinality at most α embeds.…”

Representation of perfect and local MV-algebras

“…We aim now at giving a uniform version of the previous theorem. The idea is basically the same of [14], we only have to check that the construction goes through also in the restricted case of perfect MV-algebras. For any set I of infinite cardinality α there exists an α-regular ultrafilter over I [9].…”

“…Theorem 2.6. [14] For any infinite cardinal α there exists an MV-algebra of functions A such that any MV-algebra of infinite cardinality at most α embeds in A.…”

“…Theorem 2.7. [14] For every infinite cardinal α there is an iterated ultrapower Π α of [0, 1], definable 1 in α, such that every MV-algebra of cardinality α embeds in an algebra of functions with values in Π α .…”

Representations of zero-cancellative pomonoids

“…Following Di Nola's work (see [7,8]), our (see [3]) recent work in the MV-algebra setting and the work of Cignoli and Mundici (see [6]) for totally ordered abelian groups reveals a simple method how to establish this (uniform) representation. This shapes the idea that our method could be used also for other structures than MV-algebras.…”