On Some Categories of Involutive Centered Residuated Lattices

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFLew of the substructural logic FLew. In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFLew (namely, a certain variety of FLew-algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result.The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFLew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL ew .

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“…Some results related to Theorem 1.1 have subsequently been obtained independently by Castiglioni, Menni, and Segastume in [11].…”

Section: Added In Proofmentioning

confidence: 84%

Constructive Logic with Strong Negation is a Substructural Logic. I

Spinks

Veroff

2008

Stud Logica

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“…We define

in the following way:

{sans-serifK}^{@ t h •} @ t h •

where A 0 is the dual order of A (cf. , § 7]). For

A \in {IRL}_{0}

, define the following operations in

\begin{matrix} true \\ right (a, b) \lor (d, e) & : = & left (a \lor d, b \land e), \\ right (a, b) \land (d, e) & : = & left (a \land d, b \lor e), \\ right \sim (a, b) & : = & left (b, a), \\ right (a, b) * (d, e) & : = & left (a . d, (a \to e) \land (d \to b)), \\ right (a, b) \to (d, e) & : = & left ((a \to d) \land (e \to b), a . e) . \end{matrix}

…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…An involutive residuated lattice is said to be centered if it has a distinguished element, called a center, that is, a fixed point for the involution. A

c

‐differential residuated lattice is an integral involutive residuated lattice with bottom 0 and center

c

, satisfying the following condition [, Definition 7.2]:

For any x, y \in T, (x * y) \land c = ((x \land c) * y) \lor (x * (y \land c)) .

…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…For any

T \in DRL

, consider

C (T) : = {x \in T : x \geq c}

. This defines a functor

C : DRL \to {IRL}_{0}

which is left adjoint to

{sans-serifK}^{@ t h •} @ t h •

[, Theorem 7.6].…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…In , the authors proved that the equivalence between Heyting and Nelson algebras (in fact between Heyting algebras and Nelson residuated lattices) can be lifted to an equivalence between the category IRL 0 of integral residuated lattices with bottom, and a category of involutive residuated lattices called DRL′, whose objects are

c

‐differential residuated lattices satisfying the condition (

C {sans-serifK}^{@ t h •} @ t h •

) (cf. ). Later it was proved in that a

c

‐differential residuated lattice satisfies the characterizing condition of DRL′ if and only if we can define a unary operation κ that satisfies some “quantifier‐like” conditions.…”

Section: Introductionmentioning

confidence: 99%

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The logic Ł^•

Sagastume

Martín

2014

Math. Log. Quart.

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The algebraic category MV • is the image of MV, the category whose objects are the MV-algebras, by the equivalence K • (cf. [7,8]). In this paper we define the logic Ł • whose Lindenbaum algebra is an MV • -algebra (object of MV • ), and establish a link between Ł • and the infinite valued Łukasiewicz logic Ł. We define cU-operators, that have properties of universal quantifiers, and establish a bijection that maps an MV-algebra endowed with a U-operator (cf. [20][21][22]) into an MV • -algebra endowed with a cU-operator. This map extends to a functor that is a categorical equivalence.

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The subvariety of commutative residuated lattices represented by twist-products

Busaniche

Cignoli

2014

Algebra Univers.

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On Some Categories of Involutive Centered Residuated Lattices

Cited by 15 publications

References 9 publications

Constructive Logic with Strong Negation is a Substructural Logic. I

Constructive Logic with Strong Negation is a Substructural Logic. I

The logic Ł^•

The subvariety of commutative residuated lattices represented by twist-products

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On Some Categories of Involutive Centered Residuated Lattices

Cited by 15 publications

References 9 publications

Constructive Logic with Strong Negation is a Substructural Logic. I

Constructive Logic with Strong Negation is a Substructural Logic. I

The logic Ł•

The subvariety of commutative residuated lattices represented by twist-products

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Product

Resources

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The logic Ł^•