Compatible operations in some subvarieties of the variety of weak Heyting algebras

Martín, Hernán Javier San

doi:10.2991/eusflat.2013.72

Cited by 10 publications

(12 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…However, H is locally affine complete in the sense that any restriction of a compatible function to a finite subset is a polynomial. Moreover, the variety CRL is locally affine complete [5, Corollary 9], and also the variety RWH [16,Corollary 7]. This proves that f (x) is an upper bound of T x .…”

Section: We Say That F Is Compatible With a Congruencementioning

confidence: 93%

“…In [5], compatible functions were studied in commutative residuated lattices, following basically the characterization of compatible functions by means of the relationship between congruences and convex subalgebras ( [12]). In [16], compatible functions were studied in the weak Heyting algebra (A, ∧, ∨, →, 0, 1), which satisfy the inequality a ∧ (a → b) ≤ b, using essentially the description of compatible functions by means of the relationship between congruences and open filters [7]. In the present work, we study compatible functions in a new variety that includes the previous ones, providing a common framework to the results given in [5,16].…”

Section: Introductionmentioning

confidence: 99%

“…In [16], compatible functions were studied in the weak Heyting algebra (A, ∧, ∨, →, 0, 1), which satisfy the inequality a ∧ (a → b) ≤ b, using essentially the description of compatible functions by means of the relationship between congruences and open filters [7]. In the present work, we study compatible functions in a new variety that includes the previous ones, providing a common framework to the results given in [5,16].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Compatible operations on commutative weak residuated lattices

Martín

2015

Algebra Univers.

View full text Add to dashboard Cite

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.

show abstract

Section: We Say That F Is Compatible With a Congruencementioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Compatible operations on commutative weak residuated lattices

Martín

2015

Algebra Univers.

View full text Add to dashboard Cite

show abstract

“…Hence, in every subresiduated lattice the equation 1 → (a → b) = a → b is verified. The variety of Heyting algebras 2 is a proper subvariety of the variety of SRL and SRL is a proper subvariety of the variety of WH-algebras (see for instance [10]).…”

Section: Algebraic Preliminariesmentioning

confidence: 99%

Dualities for subresiduated lattices

Celani

Nagy²,

Martín

2021

Algebra Univers.

View full text Add to dashboard Cite

A subresiduated lattice is a pair (A, D), where A is a bounded distributive lattice, D is a bounded sublattice of A and for every a, b ∈ A there is c ∈ D such that for all d ∈ D, d ∧ a ≤ b if and only if d ≤ c. This c is denoted by a → b. This pair can be regarded as an algebra ⟨A, ∧, ∨, →, 0, 1⟩ of type (2, 2, 2, 0, 0) where D = {a ∈ A : 1 → a = a}. The class of subresiduated lattices is a variety which properly contains to the variety of Heyting algebras.In this paper we present dual equivalences for the algebraic category of subresiduated lattices. More precisely, we develop a spectral style duality and a bitopological style duality for this algebraic category. Finally we study the connections of these results with a known Priestley style duality for the algebraic category of subresiduated lattices.1 The bounded distributive lattice D endowed with the binary operation → is a Heyting algebra.

show abstract

“…We also find conditions on a binary function g : A × A → A that imply that the function a → min{b ∈ A : g(a, b) ≤ b} is compatible when defined. We will employ similar ideas to those used in [10,17,27,28,29]. Definition 5.…”

Section: Compatible Functionsmentioning

confidence: 99%