Search results for "MV-algebra"

showing 3 items of 3 documents

A complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite MV-chain

2010

We consider extensions of the unique additive measure on a finite MV-chain to uncertainty measures on its canonical Girard algebra extension. If the underlying MV-chain has more than two non-trivial elements, in a previous paper we have proved the non-existence of strongly additive measure extensions, where strong additivity is defined as additivity not for all disjoint unions but only restricted to the so-called divisible disjoint unions. This negative result motivates to look for weakly additive measure extensions which are defined to be additive only on all MV-subalgebras of the canonical Girard algebra extension. We obtain a characterization of all such MV-subalgebras which are in fact …

CombinatoricsArtificial IntelligenceLogicWeakly additiveAdditive functionState (functional analysis)MV-algebraDisjoint setsExtension (predicate logic)Characterization (mathematics)Measure (mathematics)MathematicsFuzzy Sets and Systems
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Uncertainty measures—Problems concerning additivity

2009

Additivity of an uncertainty measure on an MV-algebra has a clear meaning. If the divisibility is dropped, we come up to a so-called Girard algebra. There we discuss strong resp. weak additivity based on so-called divisible disjoint unions resp. on additivity for all sub-MV-algebras. We obtain a description of those extensions from additive measures on an MV-algebra to the canonical Girard algebra extension of pairs which are strongly additive and valuation measures. Finally, we prove the non-existence of strongly additive measure extensions, if the underlying MV-algebra is a finite chain with more than two non-trivial elements.

Discrete mathematicsArtificial IntelligenceLogicAdditive functionMV-algebraExtension (predicate logic)Divisibility ruleDisjoint setsSigma additivityMeasure (mathematics)Valuation (algebra)MathematicsFuzzy Sets and Systems
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Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras

2013

AbstractWe 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 Ghilardiʼs 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 …

Fundamental groupPure mathematicsUnificationŁukasiewicz logic; Unification; Projective MV-algebras; Rational polyhedra; Fundamental group; Covering spaceLogicCovering spaceDuality (mathematics)Projective MV-algebrasMV-algebraCovering spaceŁukasiewicz logicRational polyhedraAlgebraAlgebraic semanticsUnificationVariety (universal algebra)Algebraic numberŁukasiewicz logicMathematicsAnnals of Pure and Applied Logic
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