Fuzzy ideals and congruences of lattices (original) (raw)
On fuzzy ideals of fuzzy lattice
2012 IEEE International Conference on Fuzzy Systems, 2012
We characterize a fuzzy lattice through a fuzzy partial order relation, define a fuzzy ideal and fuzzy filter of fuzzy lattice, characterize a fuzzy ideal of fuzzy lattice using its level set and its support and show that a subset of a fuzzy lattice is a fuzzy ideal if and only if its support is a crisp ideal. Similarly, we show the same for its level set.
2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), 2013
The main goal of this paper is to investigate the properties of fuzzy ideals of a ring R. It provides a proof that there exists an isomorphism of lattices of fuzzy ideals when ever the rings are isomorphic. Finite-valued fuzzy ideals are also described and a method is created to count the number of fuzzy ideals in finite and Artinian rings.
Types of fuzzy ideals in fuzzy lattices
In this paper we consider the notion of Fuzzy Lattices, which was introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374). We propose some new notions for Fuzzy Ideals and Filters and provide a characterization of Fuzzy Ideals via α-level Sets and Support. Some types of ideals and filters, such as: Fuzzy Principal Ideals (Filters), Proper Fuzzy Ideals (Filters), Prime Fuzzy Ideals (Filters) and Fuzzy Maximal Ideals (Filters) are also provided. Some properties (analogous to the classical theory) are also proved and the notion of Homomorphism from fuzzy lattices as well as the demonstration of some important propositions about it are also provided.
We consider the fuzzy lattice notion introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374), define an αideals and α-filters for fuzzy lattices and characterize α-ideals and α-filters of fuzzy lattices by using its support and its level set. Moreover, we prove some similar properties to the classical theory of α-ideals and α-filters, such as, the class of α-ideals and α-filters are closed under union and intersection.
Fuzzy Prime Ideal Theorem in Residuated Lattices
International Journal of Mathematics and Mathematical Sciences, 2021
is paper mainly focuses on building the fuzzy prime ideal theorem of residuated lattices. Firstly, we introduce the notion of fuzzy ideal generated by a fuzzy subset of a residuated lattice and we give a characterization. Also, we introduce different types of fuzzy prime ideals and establish existing relationships between them. We prove that any fuzzy maximal ideal is a fuzzy prime ideal in residuated lattice. Finally, we give and prove the fuzzy prime ideal theorem in residuated lattice.
Generalizations of prime intuitionistic fuzzy ideals of a lattice
Notes on Intuitionistic Fuzzy Sets, 2024
As a generalization of the concepts of an intuitionistic fuzzy prime ideal and a prime intuitionistic fuzzy ideal, the concepts of an intuitionistic fuzzy 2-absorbing ideal and a 2-absorbing intuitionistic fuzzy ideal of a lattice are introduced. Some results on such intuitionistic fuzzy ideals are proved. It is shown that the radical of an intuitionistic fuzzy ideal of L is a 2-absorbing intuitionistic fuzzy ideal if and only if it is a 2-absorbing primary intuitionistic fuzzy ideal of L. We also introduce and study these concepts in the product of lattices.
Ideals and filters of an almost distributive fuzzy lattice
In this paper, Almost Distributive Fuzzy Lattice (ADFL) is characterized by ADL's in terms of level set A α of a Fuzzy Relation A. The concepts of ideals, filters, the smallest ideal and the smallest filter containing a non empty subset of R of an ADFL are introduced, and some results are proved. In addition, the meet " " and join " " of two ideals(filters) of an ADFL are defined, and finally it is proved that for any two ideals I and J of an ADFL L, I J and I ∨ J are also ideals of L, and for any two filter F and G of an ADFL L, F G and F ∨ G are also filters of L.
Lattice structure on some fuzzy algebraic systems
Soft Computing, 2008
In this paper, we study the lattice structure of some fuzzy algebraic systems such as (G-)fuzzy groups, some fuzzy ordered algebras and fuzzy hyperstructures. We prove that under suitable conditions, these structures form a distributive or modular lattice.
Characterizations of intuitionistic fuzzy ideals and filters based on lattice operations
Journal of Fuzzy Set Valued Analysis, 2017
In a recent paper, Thomas and Nair have introduced the notions of intuitionistic fuzzy ideal and intuitionistic fuzzy filter on a lattice and some basic properties were proved. In this paper, we characterize these notions in terms of the lattice operations and in terms of their associated crisp sets. We introduce the notions of prime intuitionistic fuzzy ideal and filter as interesting kinds, and then we investigate their various characterizations and different properties.
L-Fuzzy Congruences and L-Fuzzy Kernel Ideals in Ockham Algebras
2021
In this paper, we study fuzzy congruence relations and kernel fuzzy ideals of an Ockham algebra (A, f), whose truth values are in a complete lattice satisfying the infinite meet distributive law. Some equivalent conditions are derived for a fuzzy ideal of an Ockham algebra A to become a fuzzy kernel ideal. We also obtain the smallest (respectively, the largest) fuzzy congruence on A having a given fuzzy ideal as its kernel.