L-fuzzy Ideals Research Articles

Ideals in residuated lattices

"The notion of ideal in residuated lattices is introduced in [Kengne, P. C., Koguep, B. B., Akume, D. and Lele, C., L-fuzzy ideals of residuated lattices, Discuss. Math. Gen. Algebra Appl., 39 (2019), No. 2, 181–201] and [Liu, Y., Qin, Y., Qin, X. and Xu, Y., Ideals and fuzzy ideals in residuated lattices, Int. J. Math. Learn & Cyber., 8 (2017), 239–253] as a natural generalization of that of ideal in MV algebras (see [Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D., Algebraic Foundations of many-valued Reasoning, Trends in Logic-Studia Logica Library 7, Dordrecht: Kluwer Academic Publishers, 2000] and [Chang, C. C., Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467–490]). If A is an MV algebra and I is an ideal on A then the binary relation x ∼I y iff x^{*}Ꙩ y; x Ꙩy^{*} ∈ I , for x; y ∈ A; is a congruence relation on A. In this paper we find classes of residuated lattices for which the relation ∼ I (defined for MV algebras) is a congruence relation and we give new characterizations for i-ideals and prime i-ideals in residuated lattices. As a generalization of the case of BL algebras (see [Lele, C. and Nganou, J. B., MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems, 218 (2013), 103–113]), we investigate the relationship between i-ideals and deductive systems in residuated lattices."

English

English

Residuated lattice of L-fuzzy ideals of a ring

In 1988, given a complete Brouwerian lattice $${\mathbb {L}}:=(L;~\wedge ,~\vee ;~0,~1)$$ and a ring $${\mathcal {A}}:=(A;~+,~\cdot ;~-;~0)$$ with unity 1, Swamy and Swamy (J Math Anal Appl 134:94–103, 1988) built a lattice structure, on the set of L-fuzzy ideals of $${\mathcal {A}}$$, and investigated some of its arithmetic properties. Since the residuation theory is richer than the lattice theory [see, Ciungu (Non-commutative multiple-valued logic algebras, Springer monographs in mathematics, Springer, Berlin, 2014), Galatos et al. (An algebraic glimpse at substructural logics, volume 151 of studies in logic and the foundations of mathematics, Elsevier, Amsterdam, 2007), Jipsen and Tsinakis (in: Martinez (ed) Ordered algebraic structures, Kluwer Academic Publisher, Dordrecht, 2002), Piciu (Algebras of fuzzy logic, Editura Universitaria Craiova, Craiova, 2007)], in this paper, we consider the notion of fuzzy ideals rather under a complete Brouwerian residuated lattice $${\mathcal {L}}:=(L;~\wedge ,~\vee ,~\ominus ,~\twoheadrightarrow ,~\multimap ;~0,~1)$$. A residuated lattice $${\mathcal {F}}id({\mathcal {A}},L):=\big ( Fid({\mathcal {A}},L);~\wedge ,~+,~\otimes ,~\hookrightarrow ,~\looparrowright ; ~\chi _0,~{\underline{1}}\big )$$ is built on the set $$Fid({\mathcal {A}},L)$$ of L-fuzzy ideals of $${\mathcal {A}}$$ and it is shown that the latter is both an extension of $${\mathcal {L}}$$ and the residuated lattice $${\mathcal {I}}d({\mathcal {A}}):=\big (Id({\mathcal {A}});~\cap ,~+,~\odot ,~\rightarrow ,~\rightsquigarrow ;~\{0\},~A \big )$$ on the set $$Id({\mathcal {A}})$$ of ideals of $${\mathcal {A}}$$.

L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras

Abstract In this paper, we apply the concept of fuzzy sets to Novikov algebras, and introduce the concepts of L-fuzzy ideals and L-fuzzy subalgebras. We get a sufficient and neccessary condition such that an L-fuzzy subspace is an L-fuzzy ideal. Moreover, we show that the quotient algebra A/μ of the L-fuzzy ideal μ is isomorphic to the algebra A/Aμ of the non-fuzzy ideal Aμ. Finally, we discuss the algebraic properties of surjective homomorphic image and preimage of an L-fuzzy ideal.

L-Fuzzy Ideals in Couple Γ-Semirings

Let M be a Γ-semiring. In this paper we obtain some properties of L-fuzzy ideals in M x M. Our results take inspiration from [M. Murali Krishna Rao and B. Vekateswarlu, L-fuzzy ideals in Γ-semiring, Annals of Fuzzy Mathematics and Informatics 10(1) (2015), 1-16]. The readers are left with a conjecture.

Spectrum of prime L-fuzzy ideals of an ordered semigroup

Spectrum of prime L-fuzzy ideals of an ordered semigroup

Homomorphisms and L-Fuzzy Ideals in Universal Algebras

In this paper, we study the homomorphic images (respectively pre-images) of -fuzzy ideals, L −fuzzy prime ideals, maximal −fuzzy ideals and L-fuzzy semi-prime ideals of universal algebras

L-fuzzy ideals of residuated lattices

L-fuzzy ideals of residuated lattices

L-fuzzy ideals of a poset

L-fuzzy ideals of a poset

Fuzzy Prime Ideals of ADL's

In this paper the concept of prime L-fuzzy ideals and L-fuzzy prime ideals of an ADL A with truth values in a complete lattice L satisfying the infinite meet distributive law are introduced. All prime L-fuzzy ideals of a given ADL A are determined by establishing a one-to- one correspondence between prime L-fuzzy ideals of an ADL A and the pairs (P;a), where P is a prime ideal of A and a is a prime element in L. Also, here minimal prime L-fuzzy ideals and L-fuzzy minimal prime ideals of an ADL A are introduced and characterized.

Interval valued L-fuzzy prime ideals, triangular norms and partially ordered groups

We introduce interval valued equiprime, 3-prime and c-prime L-fuzzy ideals of a nearring N by using interval valued t-norms and interval valued t-conorms. We characterize interval valued prime L-fuzzy ideals in terms of their level subsets. We define interval valued equisemiprime, 3-semiprime and c-semiprime L-fuzzy ideals of nearrings and study their properties. We find interrelations among different interval valued prime L-fuzzy ideals. We study these concepts further in a partially ordered group and define implications based on interval valued L-fuzzy ideals.

When do L-fuzzy ideals of a ring generate a distributive lattice?

Abstract The notion of L-fuzzy extended ideals is introduced in a Boolean ring, and their essential properties are investigated. We also build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals. By defining an operator “⇝” between two arbitrary L-fuzzy ideals in terms of L-fuzzy extended ideals, the result that “the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra” is immediately obtained. Furthermore, the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal, L-fuzzy extended ideals relative to an L-fuzzy subset, L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are studied in this paper.

Interval valued L-fuzzy ideals based on t-norms and t-conorms

In this paper, we present an unified approach which defines interval valued L-fuzzy ideals of nearrings by incorporating interval valued t-norms and t-conorms. We characterize interval valued L-fuzzy ideals in terms of their level subsets. We study insertion of factors property and relate interval valued fuzzy points with interval valued fuzzy ideals

Interval valued L-fuzzy cosets of nearrings and isomorphism theorems

In this paper, we study homomorphic images of interval valued L-fuzzy ideals of a nearring. If $$f:N_1\rightarrow N_2$$ is an onto nearring homomorphism and $$\hat{\mu }$$ is an interval valued L-fuzzy ideal of $$N_2$$ then we prove that $$f^{-1}(\hat{\mu })$$ is an interval valued L-fuzzy ideal of $$N_1$$ . If $$\hat{\mu }$$ is an interval valued L-fuzzy ideal of $$N_1$$ then we show that $$f(\hat{\mu })$$ is an interval valued L-fuzzy ideal of $$N_2$$ whenever $$\hat{\mu }$$ is invariant under $$f$$ and interval valued t-norm is idempotent. Finally, we define interval valued L-fuzzy cosets and prove isomorphism theorems.

Application of L-fuzzy soft sets to semirings

L-fuzzy soft set is a generalization of fuzzy soft set. In the recent paper L-fuzzy soft sets have emerged as a new tool to characterize semirings. Concepts of L-fuzzy soft subsemirings and L-fuzzy ideals are introduced and some of their properties are studied. It is seen that L-fuzzy soft ideals have ability to characterize regular and weakly regular semirings. Finally fully idempotent semirings are characterized with the help of L-fuzzy soft ideals.

On L-fuzzy ideals in semirings II

We study some properties of L-fuzzy left (right) ideals of a semiring R related to level left (right) ideals

Prime and primary L-fuzzy ideals of L-fuzzy rings

The aim of this paper is to give a definition of prime L-fuzzy ideal and primary L-fuzzy ideal of a fuzzy ring, and to give their first properties as well as the relationship of these concepts with the concept of L-fuzzy radical.

On L-fuzzy Ideals in Semirings I

In this paper we extend the concept of an L-fuzzy (characteristic) left (resp. right) ideal of a ring to a semiring R, and we show that each level left (resp. right) ideal of an .L-fuzzy left (resp. right) ideal μ of R is characteristic iff μ is L-fuzzy characteristic.

Normal L-fuzzy ideals in semirings

In this paper we introduce the notion of a normal L-fuzzy left (resp. right) ideal in a semiring, and investigate some properties.

Some results on a Galois connection between lattices of L-fuzzy ideals

In this paper, first we present a cl-semigroup structure on L-fuzzy ideals, and conclude that the set of all L-fuzzy quasi-ideals is a cl-monoid with zero. Then we obtain a Galois connection between two cl-semigroups of L-fuzzy ideals of homomorphic rings. Finally, by using Galois connections, we give the concepts of contracted and extended L-fuzzy ideals, and some related results are given.