Retractions of free MV-algebras and unital $\ell$-groups

Abstract

A number of papers deal with the problem of counting the number of retractions of a structure $S$ onto a substructure $T.$ In the particular case when $S$ is a free algebra, this number is $\ge 1$ iff $T$ is projective. In this paper we consider the case when $T$ is a projective lattice-ordered abelian group with a distinguished strong order unit, or equivalently, a projective MV-algebra. Let $A$ be a retract of the free $n$-generator MV-algebra $\mathcal{M}([0,1{]}^n)$ of McNaughton functions on $[0,1{]}^n$. We prove that the number $\mathsf{r}(A)$ of retractions of $\mathcal{M}([0,1{]}^n)$ onto $A$ is finite if, and only if, the maximal spectral space ${\mu }_A$ is homeomorphic to a (Kuratowski) closed domain $M$ of $[0,1{]}^n$, in the sense that $M=\mathsf{cl}(\mathsf{int}(M))$. Further, the closed domain condition is decidable and $\mathsf{r}(A)$ is computable, once a retraction onto $A$ is explicitly given. Thus every finitely generated projective MV-algebra $B$ comes equipped with a new invariant $ι(B)=\sup\{\mathsf{r}(A) \mid \mbox{$A\cong B$for$A$aretractof$\mathcal{M}([0,1]^{k})$} \},$ where $k$ is the smallest number of generators of $B$. We compute $\iota (B)$ for many projective MV-algebras $B$ considered in the literature. Various problems concerning retractions of free MV-algebras are shown to be decidable. Via the $\Gamma$ functor, our results and computations automatically transfer to finitely generated projective abelian $\ell$-groups with a distinguished strong unit.

25 pages, 5 figures

References in corpus (2)

• A compact group which is not Valdivia compact
• Rational Simplicial geometry and projective unital lattice-ordered abelian groups