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

TL;DR

This paper investigates the conditions under which the number of retractions of free MV-algebras onto projective subalgebras is finite, providing a decidable criterion and computable invariant, with implications for related algebraic structures.

Contribution

It establishes a necessary and sufficient condition for finiteness of retractions, introduces a new invariant for projective MV-algebras, and proves the decidability of related problems.

Findings

Finite retraction count characterized by closed domain spectral spaces.

Decidability of the closed domain condition.

Computability of the number of retractions once a retraction is given.

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 $\geq 1$ 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 $\iota(B)=\sup\{\mathsf{r}(A) \mid \mbox{$A\cong B$ for $A$ a retract of $\mathcal{M}([0,1]^{k})$} \},$ 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.