Classes of Int-Soft Filters in Residuated Lattices

The Scientific World JOURNAL ◽

10.1155/2014/595160 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Young Bae Jun ◽

Sun Shin Ahn ◽

Kyoung Ja Lee

Keyword(s):

Extension Property ◽

Residuated Lattices

The notions of int-soft filters, int-softG-filters, regular int-soft filters, andMV-int-soft filters in residuated lattices are introduced, and their relations, properties, and characterizations are investigated. Conditions for an int-soft filter to be an int-softG-filter, a regular int-soft filter, or anMV-int-soft filter are provided. The extension property for an int-softG-filter is discussed. Finally, it is shown that the notion of anMV-int-soft filter coincides with the notion of a regular int-soft filter inBL-algebras.

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The “Generator” of Int-Soft Filters on Residuated Lattices

Mathematics ◽

10.3390/math7030236 ◽

2019 ◽

Vol 7 (3) ◽

pp. 236 ◽

Cited By ~ 1

Author(s):

Huarong Zhang ◽

Minxia Luo

Keyword(s):

Extension Property ◽

Residuated Lattices ◽

Simple Method ◽

Basic Properties

In this paper, we give the “generator” of int-soft filters and propose the notion of t-int-soft filters on residuated lattices. We study the properties of t-int-soft filters and obtain some commonalities (e.g., the extension property, quotient characteristics, and a triple of equivalent characteristics). We also use involution-int-soft filters as an example and show some basic properties of involution-int-soft filters. Finally, we investigate the relations among t-int-soft filters and give a simple method for judging their relations.

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Ideals of Residuated Lattices

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2021.58.2.1493 ◽

2021 ◽

Vol 58 (2) ◽

pp. 182-205

Author(s):

Liviu-Constantin Holdon ◽

Arsham Borumand Saeid

Keyword(s):

Prime Ideal ◽

Residuated Lattice ◽

Extension Property ◽

Characterization Theorem ◽

Residuated Lattices ◽

Quotient Algebra ◽

Prime Ideals ◽

Characterization Theorems

In this article, we study ideals in residuated lattice and present a characterization theorem for them. We investigate some related results between the obstinate ideals and other types of ideals of a residuated lattice, likeness Boolean, primary, prime, implicative, maximal and ʘ-prime ideals. Characterization theorems and extension property for obstinate ideal are stated and proved. For the class of ʘ-residuated lattices, by using the ʘ-prime ideals we propose a characterization, and prove that an ideal is an ʘ-prime ideal iff its quotient algebra is an ʘ-residuated lattice. Finally, by using ideals, the class of Noetherian (Artinian) residuated lattices is introduced and Cohen’s theorem is proved.

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Residuated Lattices: An Algebraic Glimpse at Substructural Logics

10.1016/s0049-237x(07)x8002-9 ◽

2007 ◽

Keyword(s):

Substructural Logics ◽

Residuated Lattices

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Residuated Structures and Orthomodular Lattices

Studia Logica ◽

10.1007/s11225-021-09946-1 ◽

2021 ◽

Author(s):

D. Fazio ◽

A. Ledda ◽

F. Paoli

Keyword(s):

Algebraic Logic ◽

Residuated Lattices ◽

Quantum Structures ◽

Heyting Algebras ◽

Quantum Algebras ◽

Orthomodular Lattices ◽

Lattice Of Subvarieties ◽

Residuated Structures ◽

Common Framework ◽

Mv Algebras

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

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