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Idempotent endomorphisms of free MV-algebras and unital $$\ell$$-groups. (English) Zbl 1362.06003
For counting the number of idempotent endomorphisms of a structure $$S$$ onto a substructure $$T$$, the authors consider the case when $$T$$ is a projective lattice-ordered abelian group with a distinguished strong order unit, or equivalently, a projective MV-algebra. Suppose $$A$$ is the image (= range) of an idempotent endomorphism of the free $$n$$-generator MV-algebra $$\mathcal M ([0,1]^n)$$ of McNaughton functions on $$[0,1]^n$$. They prove that the number $$\mathrm r(A)$$ of idempotent endomorphisms 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 $$M=\mathrm{cl}(\operatorname{int}(M))$$. Further, the closed domain condition is decidable and $$\mathrm r(A)$$ is computable, once an idempotent endomorphism of $$\mathcal M([0,1]^n)$$ onto $$A$$ is explicitly given. Thus, every finitely generated projective MV-algebra $$B$$ comes equipped with a new invariant $$\iota(B)=\sup\{\mathrm r(A)\mid A\cong B$$, for $$A$$ the image of an idempotent endomorphism of $$\mathcal M([0,1]^k)\}$$, and $$k$$ the smallest number of generators of $$B$$. We compute $$\iota(B)$$ for many projective MV-algebras $$B$$ existing in the literature. Various problems concerning idempotent endomorphisms of free MV-algebras are shown to be decidable.
##### MSC:
 06D35 MV-algebras 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 08B30 Injectives, projectives 52B70 Polyhedral manifolds 55U10 Simplicial sets and complexes in algebraic topology 57Q05 General topology of complexes
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