We prove that the unification type of Łukasiewicz (infinite-valued propositional) logic and of its equivalent algebraic semantics, the variety of MV-algebras, is nullary. The proof rests upon Ghilardi's algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the background to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra-a fundamental result that, albeit known to specialists, seems to appear in print here for the first time.

Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras / V. Marra, L. Spada. - In: ANNALS OF PURE AND APPLIED LOGIC. - ISSN 0168-0072. - 164:3(2013 Mar), pp. 192-210. [10.1016/j.apal.2012.10.001]

Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras

V. Marra
Primo
;
2013

Abstract

We prove that the unification type of Łukasiewicz (infinite-valued propositional) logic and of its equivalent algebraic semantics, the variety of MV-algebras, is nullary. The proof rests upon Ghilardi's algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the background to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra-a fundamental result that, albeit known to specialists, seems to appear in print here for the first time.
Łukasiewicz logic; Covering space; Fundamental group; Projective MV-algebras; Rational polyhedra; Unification
Settore MAT/01 - Logica Matematica
Settore MAT/02 - Algebra
Settore INF/01 - Informatica
mar-2013
Article (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/212602
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