Idempotent endomorphisms of free MV-algebras and unital ℓ-groups

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• 作者：Leonardo Manuel Cabrer^a ; ¹ ; ^{lmcabrer@yahoo.com.ar} ; Daniele Mundici^b ; ^{mundici@math.unifi.it}
• 关键词：Primary ; 06D35 ; secondary ; 03B50 ; 08B30 ; 52B70 ; 55U10 ; 57Q05
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：221
• 期：4
• 页码：908-934
• 全文大小：1489 K
• 卷排序：221

文摘

Several papers deal with the problem of counting the number of idempotent endomorphisms of a structure S onto a substructure T. 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. Suppose A is the image (=range) of an idempotent endomorphism of the free n  -generator MV-algebra M([0,1]n)M([0,1]n) of McNaughton functions on [0,1]n[0,1]n. We prove that the number r(A)r(A) of idempotent endomorphisms of M([0,1]n)M([0,1]n) onto A   is finite if, and only if, the maximal spectral space μAμA is homeomorphic to a (Kuratowski) closed domain M   of [0,1]n[0,1]n, in the sense that M=cl(int(M))M=cl(int(M)). Further, the closed domain condition is decidable and r(A)r(A) is computable, once an idempotent endomorphism of M([0,1]n)M([0,1]n) onto A is explicitly given. Thus every finitely generated projective MV-algebra B   comes equipped with a new invariant ι(B)=sup⁡{r(A)|ι(B)=sup⁡{r(A)|A≅BA≅B, for A   the image of an idempotent endomorphism of M([0,1]k)}M([0,1]k)}, and k the smallest number of generators of B  . We compute ι(B)ι(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. Via the Γ functor, our results and computations automatically transfer to finitely generated projective abelian ℓ-groups with a distinguished strong unit.