Notion Of Ideals Research Articles

On pseudo BCH-algebras

As a generalization of BCH-algebras, the notion of pseudo BHalgebra is introduced, and some of their properties are investigated. The notions of pseudo subalgebra, pseudo ideals, and pseudo atoms in pseudo BCH-algebras are introduced. Characterizations of their properties are provided. Mathematics Subject Classication: 06F35, 03G25

Fuzzy filters in Γ−semirings

We introduce the notion of fuzzy prime ideals and fuzzy filters in gamma semirings and study some of their properties.

Rough semigroups on approximation spaces

 Ozcan. This article is dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: In this paper, we introduced the notion of rough semigroups, rough subsemigroups, rough ideals, rough bi-ideals and homomorphisms of rough semigroups in approximation spaces. Then we give some properties of these rough structures.

Doubt Atanassov's intuitionistic fuzzy Sub-implicative ideals in -algebras

AbstractThe notions of doubt Atanassov's intuitionistic fuzzy sub-implicative ideals and doubt Atanassov's intuitionistic fuzzy P-ideals of BCI-algebras are introduced. We show that an Atanassov's intuitionistic fuzzy subset of BCI-algebras is an Atanassov's intuitionistic fuzzy sub-implicative ideal if and only if the complement of this Atanassov's intuitionistic fuzzy subset is a doubt Atanassov's intuitionistic fuzzy sub-implicative-ideal. We prove that any doubt Atanassov's intuitionistic fuzzy P-ideal is always a doubt Atanassov's intuitionistic fuzzy sub-implicative ideal. Moreover, some other properties about doubt Atanassov's intuitionistic fuzzy sub-implicative ideals and doubt Atanassov's intuitionistic fuzzy P-ideals of BCI-algebras are given.

QUASI-PRIMARY SUBMODULES SATISFYING THE PRIMEFUL PROPERTY II

Let $R$ be a commutative ring with identity and $M$ a unital $R$-module. In this article we extend the notion of quasi-primary ideals to submodules. A proper submodule $N$ of $M$ is called quasi-primary if whenever $rx\in N$ for $r\in R$ and $x\in M$, then $r\in \sqrt{(N:M)}$ or $x\in radN$ where $radN$ is the intersection of all prime submodules of $M$ containing $N$. Also, we say that a submodule $N$ of $M$ satisfies the primeful property if $M/N$ is a primeful $R$-module. For a quasi-primary submodule $N$ of $M$ satisfying the primeful property, $\sqrt{(N:M)}$ is a prime ideal of $R$. For the existence of a module-reduced quasi-primary decomposition, the radical of each term appeared in decomposition must be prime. We provide sufficient conditions, involving the saturation and torsion arguments, to ensure that this property holds as is valid in the ideal case. It is proved that for a submodule $N$ of $M$ over a Dedekind domain $R$ which satisffies the primeful property, $N$ is quasi-primary if and only if $radN is prime.

Ideals and fuzzy ideals on residuated lattices

This paper mainly focus on building the ideals theory of non regular residuated lattices. Firstly, the notions of ideals and fuzzy ideals of a residuated lattice are introduced, their properties and equivalent characterizations are obtained; at the meantime, the relation between filter and ideal is discussed. Secondly, two types prime ideals of a residuated lattice are introduced, the relations between the two types ideals are studied, in some special residuated lattices (such as MTL-algebras, lattice implication algebras, BL-algebras), prime ideal and prime ideal of the second kind are coincide. At the meantime, the notions of fuzzy prime ideal and fuzzy prime ideal of the second kind on a residuated lattice are introduced, aiming at the relation between prime ideal and prime ideal of the second kind, we mainly investigate the fuzzy prime ideal of the second kind. Finally, we investigated the fuzzy congruence relations induced by fuzzy ideal, we construct a new residuated lattice induced by fuzzy congruences, the homomorphism theorem is given.

Smarandache n-Structure on CI-Algebras

In this paper, the notions of CI-algebras, Smarandache CI-algebra, Q-Smarandache filters and Q-Smarandache ideals are intro- duced. We show that a nonempty subset F of a CI-algebra X is a Q-Smarandache filter if and only if A(x,y) ⊆ F ,w hichA(x,y )i s a Q-Smarandache upper set. Finally, we introduced the concepts of Sma- randache BE-algebra, Smarandache dual BCK-algebra and Smarandache n-structure on CI-algebra. Mathematics Subject Classification (2010). Primary 06F35; Secondary 03G25.

ON SIMPLE LEFT, RIGHT AND TWO-SIDED IDEALS OF AN ORDERED SEMIGROUP HAVING A KERNEL

Abstract. The intersection of all two-sided ideals of an ordered semi-group, if it is non-empty, is called the kernel of the ordered semigroup. Aleft ideal Lof an ordered semigroup (S,·,≤) having a kernel Iis said to besimple if I is properly contained in Land for any left ideal L ′ of (S,·,≤),I is properly contained in L ′ and L ′ is contained in Limply L ′ = L. Thenotions of simple right and two-sided ideals are defined similarly. In thispaper, the author characterize when an ordered semigroup having a ker-nel is the class sum of its simple left, right and two-sided ideals. Further,the structure of simple two-sided ideals will be discussed. 1. IntroductionThe intersection of all two-sided ideals of a semigroup, if it is non-empty, iscalled the kernel [5] of the semigroup.Let S be a semigrouphavinga kernelI. A left ideal L of S is said to be simpleif I ⊂ L, (The symbol ⊂ stands for proper inclusion of sets.) and for any leftideal L ′ of S, if I ⊂ L ′ ⊆ L, then L ′ = L. The notions of simple right and two-sided ideals are defined similarly. These concepts were introduced and studiedby Schwarz [6]. Indeed, the author characterized when a semigroup having akernel is the class sum of its simple left, right or two-sided ideals. Further, thestructure of simple two-sided ideals of some semigroups was investigated.The purpose of this paper is to extend Schwarz’s results to ordered semi-groups. Section 2 recalls some certain definitions and results used throughoutthis paper. In Section 3, we introduce the concepts of simple left, right and two-sided ideals of an ordered semigroup having a kernel. Necessary and sufficientconditions for an ordered semigroup having a kernel to be the class sum of itssimple left, right or two-sided ideals are given (Theorems 3.9-3.11). Further,we obtain important corollaries using annihilators of an ordered semigroup.Sections 4-5 investigate the structure of simple two-sided ideals of an orderedsemigroup having a kernel.

Generalized Soft Intersectional Ideals in Ternary Semirings

In this paper we introduce the notions of soft intersectional ternary subsemirings and soft intersectional ideals in ternary semirings. We also discuss some basic results associated with these notions. In the last part of the paper we characterize regular and weakly regular ternary semirings by their soft intersectional ideals.

Pure Ideals in Ordered Semigroups

In this paper the concepts of pure ideals, weakly pure ideals and purely prime ideals in ordered semigroups are introduced. We obtain some characterizations of pure ideals and prove that the set of all pure prime ideals is topologized. 1. Introduction and Preliminaries In (1), Ahsan and Takahashi introduced the notions of pure ideals and purely prime ideals in a semigroup without order. Recently, Bashir and Shabir (3) dened the concepts of pure ideals, weakly pure ideals and purely prime ideals in a ternary semigroup. The authors gave some characterizations of pure ideals and showed that the set of all purely prime ideals of a ternary semigroup is topologized. In this paper, we do in the line of Bashir and Shabir. We introduce the concepts of pure ideals, weakly pure ideals and purely prime ideals on an ordered semigroup. We characterize pure ideals and prove that the set of all purely prime ideals of an ordered semigroup is topologized. Note that the results on semigroups without order become then special cases. For the rest of this section, we recall some denitions and results used through- out the paper. A semigroup S with an order relation ≤ is called an ordered semigroup ((2), (5)) if for x;y;z ∈ S, x ≤ y implies zx ≤ zy and xz ≤ yz. An element 0 of S is called a zero element of S if 0x = x0 = 0 for all x ∈ S and 0 ≤ x for all x ∈ S. A nonempty subset A of S is called a subsemigroup of S if xy ∈ A for all x;y ∈ A. Note that every subsemigroup of S is an ordered semigroup under the order relation on S. For nonempty subsets A and B of S, let AB = {xy | x ∈ A;y ∈ B}. For x ∈ S, let Ax = A{x} and xA = {x}A. A nonempty subset A of S is a subsemigroup of S if and only if AA ⊆ A.

Green's relations in Partial \(\Gamma\)-Semirings

A \(\Gamma\)-so-ring is a structure possessing a natural partial ordering, an infinitary partial addition and a ternary multiplication, subject to a set of axioms. The partial functions under disjoint-domain sums and functional composition is a \(\Gamma\)-so-ring. In this paper we introduce the notions of dense and annihilator ideals of \(\Gamma\)-so-rings and obtain the Green's relations in Partial \(\Gamma\) semirings.

On N-Fold Positive Implicative Artinian and Positive Implicative Notherian BCK-Algebras

The notion of BCK-algebras was initiated by Imai and Iseki in 1966 as a generalization of both classical and non-classical positional calculus. In 1999, Huang and Chen introduced the notion of n-fold positive implicative ideals in BCK-algebras. In 2011, Satyanarayana and Durga Prasad introduced foldness of intuitionistic fuzzy positive implicative ideals in BCK-algebras. In this paper, we introduce the notion of n-fold positive implicative ideals, n-fold positive implicative Artinian (shortly, n PI -Artinian) and n-fold positive implicative Noe-therian (shortly, n PI -Noetherian) BCK-algebras and study some of its properties.

Fuzzy Soft Gamma Semigroups

The aim of this paper is to apply the concept of fuzzy soft sets over a Γ -semigroup. Here the notion of fuzzy soft ideals over a Γ -semigroup has been introduced. The special union, intersection and product of fuzzy soft ideals over a Γ -semigroup have been defined and proved that these are also fuzzy soft Γ -ideals over the Γ -semigroup.

Fuzzy Ideals and Fuzzy Interior Ideals in Ordered Semirings

The notions of fuzzy ideals and fuzzy interior ideals in ordered semirings are introduced and some of their characterizations are obtained through regularity criterion and with some operations on them.

On fuzzy ideals of BL-algebras.

In this paper we investigate further properties of fuzzy ideals of a BL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of a BL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime ideal ω is a fuzzy irreducible ideal if and only if ω(0) = 1 and |Im⁡(ω)| = 2. We give the Krull-Stone representation theorem of fuzzy ideals in BL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of a BL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.

Doubt fuzzy ideals of BF-algebra

The aim of this paper is to introduce the notion of Doubt fuzzy ideals of BF -algebra and to investigate some of their basic properties.

Hesitant Fuzzy Soft Subalgebras and Ideals inBCK/BCI-Algebras

As a link between classical soft sets and hesitant fuzzy sets, the notion of hesitant fuzzy soft sets is introduced and applied to a decision making problem in the papers by Babitha and John (2013) and Wang et al. (2014). The aim of this paper is to apply hesitant fuzzy soft set for dealing with several kinds of theories in BCK/BCI-algebras. The notions of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals are introduced, and related properties are investigated. Relations between a hesitant fuzzy soft subalgebra and a (closed) hesitant fuzzy soft ideal are discussed. Conditions for a hesitant fuzzy soft set to be a hesitant fuzzy soft subalgebra are given, and conditions for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal are provided. Characterizations of a (closed) hesitant fuzzy soft ideal are considered.

ON $I$-STATISTICALLY PRE-CAUCHY SEQUENCES

In this paper we continue our investigation of the recent summability notion of $I-$statistical convergence introduced in [8, 26] (where a new extension of the notion of natural density and statistical convergence was proposed using the notion of ideals of the set of positive integers $\mathbb{N}$) and introduce the notion of $I$-statistically pre-Cauchy sequences in line of [3]. We mainly show that $I-$statistical convergence implies $I-$statistical pre-Cauchy condition and give certain sufficient conditions for the converse to be true.

ON T-FUZZY ESSENTIAL IDEALS OF RINGS

In this paper our attempt is to extend the notion of essential fuzzy ideals of rings to T-fuzzy essential ideals. Various properties on T-fuzzy essential ideals are established. Some results are being discussed which establish necessary and sufficient condition between T-fuzzy essential ideal in fuzzy set theory and essential ideal in crisp set theory. Moreover it is proved that if µ,� andare non-zero T-fuzzy left ideals of R such that µ ⊆ � ⊆ �, then µ ⊆e(T) � if and only if µ ⊆e(T) � ⊆e(T) �.

Prime Fuzzy Ideals, Weakly Prime Fuzzy Ideals of Γ-semigroups

Using fuzzy points the notions of prime fuzzy ideals and weakly prime fuzzy ideals of a Γ-semigroup have been introduced. Some important properties and characterizations of these ideals have been obtained. The concluding result shows that our work sharpens previous works on prime fuzzy ideals of Γ-semigroups.