Projective MV-algebras and rational polyhedra

Abstract

Let M be 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 \({{\rm{\mathcal {M}}}(P)}\) of restrictions to P of the McNaughton functions of the free n-generator MV-algebra. P necessarily contains a vertex vP of the n-cube. We 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 vP of the n-cube, then \({{\mathcal {M}}(P)}\) is projective. Using this result, we prove that if \({A = {\mathcal {M}}(P)}\) and \({B = {\mathcal {M}}(Q)}\) are projective, then so is the subalgebra of A × B given by {(f, g) | f(vP) = g(vQ), and so is the free product \({A \coprod B}\) .