Projective MV-algebras and rational polyhedra
This paper deals with the geometry of (always finitely generated) projective MV-algebras. The free \(n\)-generator MV-algebra \(\mathcal{M}([0,1]^{n})\) of McNaughton functions on the \(n\)-cube is a trivial example of a finitely generated projective MV-algebra. A rational polyhedron \(P\subseteq [0,1]^{n}\) is defined as the pointset union of finitely many simplexes in \([0,1]^{n}\) with rational vertices. The main result of this paper is that if \(M\) is an \(n\)-generator projective MV-algebra, then there is a rational polyhedron \(P\) in the \(n\)-cube \([0,1]^{n}\) such that \(M\) is isomorphic to the MV-algebra \(\mathcal{M}(P)\) of restrictions to \(P\) of the McNaughton functions of the free \(n\)-generator MV-algebra. \(P\) necessarily contains a vertex \(v_{P}\) of the \(n\)-cube. It is well known that we can obtain from this that every finitely generated projective MV-algebra is finitely presented. The converse does not hold: for instance, if the rational polyhedron \(P\) does not contain any point of denominator 1, or \(P\) is not connected, then the finitely presented MV-algebra \(\mathcal{M}(P)\) is not projective. In this paper the authors characterize those polyhedra contained in the \(n\)-cube such that \(\mathcal{M}(P)\) is projective. In particular, if the rational polyhedron \(P\) is a union of segments originating at some fixed vertex \(v_{P}\) of the \(n\)-cube, then \(\mathcal{M}(P)\) is projective. Using this result, in the final section of the paper, the authors introduce two geometric constructions of new projective MV-algebras from old ones. They prove that if \(A=\mathcal{M}(P)\) and \(B=\mathcal{M}(Q)\) are projective, then so is the subalgebra of \(A\times B\) given by \(\{(f,g)\mid f(v_{P})=g(v_{Q})\}\), and so is the free product \(A\sqcup B\). The authors end their paper with the remark that in view of the categorical equivalence between MV-algebras and unital lattice-ordered abelian groups, the results of this paper have an equivalent counterpart for the latter ones. Thus, up to isomorphism, every finitely presented unital lattice-ordered abelian group has the form \(\overline{\mathcal{M}}(P)\) for some rational polyhedron \(P\) contained in some \(n\)-cube and vice versa. Here \(\overline{\mathcal{M}}(P)\) denotes the lattice-ordered abelian group generated by \(\mathcal{M}(P)\) in the lattice-ordered abelian group of all real-valued function on \(P\), with the constant 1 as the distinguished order-unit. Further, if \(P\subseteq [0,1]^{n}\) is star-shaped with a pole that is a vertex of \([0,1]^{n}\), then \(\overline{\mathcal{M}}(P)\) is projective. Free products of unital lattice-ordered abelian groups are projective.
scientific article; zbMATH DE number 5721487