Compatible operations on commutative residuated lattices

Castiglioni, José Luis; Menni, Matías; Sagastume, Marta

doi:10.3166/jancl.18.413-425

Cited by 8 publications

(25 citation statements)

References 9 publications

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“…

This work extend to residuated lattices the results of [7]. It also provides a possible generalization to this context of frontal operators in the sense of [9].

Let L be a residuated lattice, and f : L k → L a function.

…”

mentioning

confidence: 89%

“…However, H is locally affine complete in the sense that any restriction of a compatible function to a finite subset is a polynomial. Morever, the variety CRL is locally affine complete too (see [7]). We prove in this paper that this is also the case for the variety RL (section 3).…”

Section: Introductionmentioning

confidence: 98%

“…However, it is not clear what can be considered as a natural or good extension of a substructural logic by means of new connectives. In section 3 we give an algebraic approach to that problem, analogous to that given in [7] for the commutative case, which basically follows from the characterization of [5] of compatible functions by means of the relationship between congruences and some particular subalgebras.…”

Section: Introductionmentioning

confidence: 99%

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Compatible Operations on Residuated Lattices

Castiglioni

Martín

2011

Stud Logica

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This work extend to residuated lattices the results of [7]. It also provides a possible generalization to this context of frontal operators in the sense of [9].Let L be a residuated lattice, and f : L k → L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L. We use this characterization of compatible functions in order to prove that the variety of residuated lattices is locally affine complete.We study some compatible functions on residuated lattices which are a generalization of frontal operators. We also give conditions for two operations P (x, y) and Q(x, y) on a residuated lattice L which imply that the function x → min{y ∈ L : P (x, y) ≤ Q(x, y)} when defined, is equational and compatible. Finally we discuss the affine completeness of residuated lattices equipped with some additional operators.

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“…

This work extend to residuated lattices the results of [7]. It also provides a possible generalization to this context of frontal operators in the sense of [9].

Let L be a residuated lattice, and f : L k → L a function.

…”

mentioning

confidence: 89%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Compatible Operations on Residuated Lattices

Castiglioni

Martín

2011

Stud Logica

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“…It is shown in [9] that any bounded residuated lattice is an l-algebra satisfying condition (A). In commutative residuated lattices a family of compatible operations S n , n ≥ 1, is considered, see [5], such that S 1 corresponds to the operation S in Heyting algebras. If some conditions hold (see [5]) in the residuated lattice, then S n , defined by the formula: S n (x) = min{y : y n → x ≤ y}, is compatible (it may happen that S 1 is defined but S 2 is not, etc.).…”

Section: Now We Present Three Important Examples Of Frontal Operationsmentioning

confidence: 99%

“…In commutative residuated lattices a family of compatible operations S n , n ≥ 1, is considered, see [5], such that S 1 corresponds to the operation S in Heyting algebras. If some conditions hold (see [5]) in the residuated lattice, then S n , defined by the formula: S n (x) = min{y : y n → x ≤ y}, is compatible (it may happen that S 1 is defined but S 2 is not, etc.). Not much is known about unification types in commutative integral bounded residuated lattices with additional compatible operations, e.g.…”

Section: Now We Present Three Important Examples Of Frontal Operationsmentioning

confidence: 99%

Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

Dzik

Radeleczki

2016

B Sect Log

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We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.

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Compatible operations on commutative weak residuated lattices

Martín

2015

Algebra Univers.

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Compatibility of functions is a classical topic in Universal Algebra related to the notion of affine completeness. In algebraic logic, it is concerned with the possibility of implicitly defining new connectives.In this paper, we give characterizations of compatible operations in a variety of algebras that properly includes commutative residuated lattices and some generalizations of Heyting algebras. The wider variety considered is obtained by weakening the main characters of residuated lattices (A, ∧, ∨, •, →, e) but retaining most of their algebraic consequences, and their algebras have a commutative monoidal structure. The order-extension principle a ≤ b if and only if a → b ≥ e is replaced by the condition: if a ≤ b, then a → b ≥ e. The residuation property c ≤ a → b if and only if a • c ≤ b is replaced by the conditions: if c ≤ a → b, then a • c ≤ b, and if a • c ≤ b, then e → c ≤ a → b. Some further algebraic conditions of commutative residuated lattices are required.

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Compatible operations on commutative residuated lattices

Cited by 8 publications

References 9 publications

Compatible Operations on Residuated Lattices

Compatible Operations on Residuated Lattices

Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

Compatible operations on commutative weak residuated lattices

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