Projective MV-algebras and rational polyhedra |
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Authors: | Leonardo Manuel Cabrer Daniele Mundici |
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Institution: | (1) Department of Mathematics “Ulisse Dini”, University of Florence, Viale Morgagni 67 A, 50134 Florence, Italy; |
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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 M(P){{\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 M(P){{\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 M(P){{\mathcal {M}}(P)} is projective. Using this result, we prove that if A = M(P){A = {\mathcal {M}}(P)} and B = M(Q){B = {\mathcal {M}}(Q)} are projective, then so is the subalgebra of A × B given by {(f, g) | f(v
P
) = g(v
Q
), and so is the free product
A \coprod B{A \coprod B} . |
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Keywords: | |
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