Retractions of free MV-algebras and unital $\ell$-groups
L.M. Cabrer, D. Mundici

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 onto a substructure In the particular case when is a free algebra, this number is iff is projective. In this paper we consider the case when is a projective lattice-ordered abelian group with a distinguished strong order unit, or equivalently, a projective MV-algebra. Let be a retract of the free -generator MV-algebra of McNaughton functions on . We prove that the number of retractions of onto is finite if, and only if, the maximal spectral space is homeomorphic to a (Kuratowski) closed domain of , in the sense that . Further, the closed domain condition is decidable and is computable, once a retraction onto is…
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Taxonomy
TopicsAdvanced Algebra and Logic · Logic, Reasoning, and Knowledge · Rough Sets and Fuzzy Logic
