Algebraic Foundations of Many-Valued Reasoning

Cignoli, Roberto; D’Ottaviano, Ítala M. Loffredo; Mundici, Daniele

doi:10.1007/978-94-015-9480-6

Cited by 998 publications

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“…The set of all infinitesimals in M is denoted by Infinit(M). Then according to [CDM,Proposition 3.6.4], we have…”

Section: Introductionmentioning

confidence: 99%

Loomis-sikorski theorem for σ-complete MV-algebras and ℓ-groups

Dvurečenskij¹

2000

J Aust Math Soc A

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We show that every a -complete MV-algebra is an MV-cr-homomorphic image of some a -complete MV-algebra of fuzzy sets, called a tribe, which is a system of fuzzy sets of a crisp set £2 containing l n and closed under fuzzy complementation and formation of min{5^n/ n , 1}. Since a tribe is a direct generalization of a a -algebra of crisp subsets, the representation theorem is an analogue of the LoornisSikorski theorem for MV-algebras. In addition, this result will be extended also for Dedekind CT-complete £-groups with strong unit.1991 Mathematics subject classification (Amer. Math. Soc): primary 03B50,03G12.

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“…The set of all infinitesimals in M is denoted by Infinit(M). Then according to [CDM,Proposition 3.6.4], we have…”

Section: Introductionmentioning

confidence: 99%

Loomis-sikorski theorem for σ-complete MV-algebras and ℓ-groups

Dvurečenskij¹

2000

J Aust Math Soc A

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“…An MV-algebra (cf. [9,18,19]) is a system M = (M, ⊕, ¬, 0 M ) of type (2, 1, 0) such that the reduct (M, ⊕, 0 M ) is a commutative monoid, and the following equations hold:…”

Section: Lukasiewicz Logic and Mv-algebrasmentioning

confidence: 99%

“…The variety of MV-algebras MV is generated, as a variety and as a quasi-variety, by [0, 1] MV (cf. [8,9]). This means that, in order to show that a given equality, or quasi-equality, written in the algebraic language of MValgebras, holds in every MV-algebra, it is sufficient to check whether it holds in [0, 1] MV .…”

Section: Lukasiewicz Logic and Mv-algebrasmentioning

confidence: 99%

On the set of intermediate logics between the truth- and degree-preserving Łukasiewicz logics

2016

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The aim of this paper is to explore the class of intermediate logics between the truth-preserving Lukasiewicz logic L and its degree-preserving companion L ≤ . From a syntactical point of view, we introduce some families of inference rules (that generalize the explosion rule) that are admissible in L ≤ and derivable in L and we characterize the corresponding intermediate logics. From a semantical point of view, we first consider the family of logics characterized by matrices defined by lattice filters in [0, 1], but we show there are intermediate logics falling outside this family. Finally, we study the case of finite-valued Lukasiewicz logics where we axiomatize a large family of intermediate logics defined by families of matrices (A, F ) such that A is a finite MV-algebra and F is a lattice filter.

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“…The above classes of algebras constitute the three main subvarieties of BL algebras, and are the equivalent algebraic semantics of the Lukasiewicz infinitely-valued logic, the Gödel-Dummett logic, and the Product logic (see [3,5,8]). LΠ algebras were defined in [7] as a combination of both MV algebras and Product algebras.…”

Section: Preliminariesmentioning

confidence: 99%

“…The presentation of MV algebras in that signature is term-wise equivalent to the presentation as residuated lattices, by the following definitions (see [5]):…”

Section: Preliminariesmentioning

confidence: 99%

Advances in the theory of L algebras

Marchioni

Spada

2010

Logic Journal of IGPL

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Recently an expansion of LΠ 1 2 logic with fixed points has been considered [23]. In the present work we study the algebraic semantics of this logic, namely µ LΠ algebras, from algebraic, model theoretic and computational standpoints.We provide a characterisation of free µ LΠ algebras as a family of particular functions from [0, 1] n to [0,1]. We show that the first-order theory of linearly ordered µ LΠ algebras enjoys quantifier elimination, being, more precisely, the model completion of the theory of linearly ordered LΠ 1 2 algebras. Furthermore, we give a functional representation of any LΠ 1 2 algebra in the style of Di Nola Theorem for MV-algebras and finally we prove that the equational theory of µ LΠ algebras is in PSPACE.

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Algebraic Foundations of Many-Valued Reasoning

Cited by 998 publications

References 0 publications

Loomis-sikorski theorem for σ-complete MV-algebras and ℓ-groups

Loomis-sikorski theorem for σ-complete MV-algebras and ℓ-groups

On the set of intermediate logics between the truth- and degree-preserving Łukasiewicz logics

Advances in the theory of L algebras

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