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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.
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
Full Text: DOI
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