On generalized interval valued fuzzy quasi-ideals of semigroups

In this paper, we give the concepts of generalized interval valued fuzzy subsemigroups, which are generalizations of the notion of interval valued fuzzy subsemigroups and of (ᾱ, β)-interval valued fuzzy subsemigroups, where ᾱ ≺ β. Also we prove some of related properties of generalized interval valued fuzzy quasi-ideals on semigroups. In the main results, we characterize regular and intra-regular semigroups in terms of generalized interval valued fuzzy ideal and quasi-ideals.Publisher's Versio

Bi-ideals of ordered semigroups based on the interval-valued fuzzy point

Interval-valued fuzzy set theory (advanced generalization of Zadeh’s fuzzy sets) is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, we introduce the notion of generalized quasi-coincident with (q(Formula Presented)) relation of an interval-valued fuzzy point with an interval-valued fuzzy set. In fact, this new concept is a more generalized form of quasi-coincident with relation of an interval-valued fuzzy point with an interval-valued fuzzy set. Applying this newly defined idea, the notion of an interval-valued (∈,∈vq(Formula Presented)) -fuzzy bi-ideal is introduced. Moreover, some characterizations of interval-valued (∈,∈vq(Formula Presented)) -fuzzy bi-ideals are described. It is shown that an interval-valued (∈,∈vq(Formula Presented)) -fuzzy bi-ideal is an interval-valued fuzzy bi-ideal by imposing a condition on interval-valued fuzzy subset. Finally, the concept of implication-based interval-valued fuzzy bi-ideals, characterizations of an interval-valued fuzzy bi-ideal and an interval-valued (∈,∈vq(Formula Presented)) - fuzzy bi-ideal are considered

A study on spherical fuzzy ideals of semigroup

In this paper, we introduce the notion of spherical fuzzy ideals of semigroup and establish the properties of it with suitable examples. Also, we introduce the concept of spherical fuzzy sub-semigroup, spherical fuzzy left(resp.right) ideal, spherical fuzzy bi-ideal, spherical fuzzy interior ideal, and homomorphism of a spherical fuzzy ideal in semigroups with suitable illustration. We show that every spherical fuzzy left(right) ideal is a spherical fuzzy bi-ideal

Cubic (1, 2)-ideals on semigroups

In this paper we introduce the concept of cubic (1, 2)-ideals on semigroups and we study basic properties of cubic (1, 2)-ideals. In particular, we find condition cubic bi-ideal is cubic (1, 2)-ideal coincide. Finally we can show that the images or inverse images of a cubic (1, 2)-ideal of a semigroup become a cubic (1, 2)-ideal.Emerging Sources Citation Index (ESCI)MathScinetScopu

Theory of Abel Grassmann\u27s Groupoids

It is common knowledge that common models with their limited boundaries of truth and falsehood are not su¢ cient to detect the reality so there is a need to discover other systems which are able to address the daily life problems. In every branch of science problems arise which abound with uncertainties and impaction. Some of these problems are related to human life, some others are subjective while others are objective and classical methods are not su¢ cient to solve such problems because they can not handle various ambiguities involved. To overcome this problem, Zadeh [67] introduced the concept of a fuzzy set which provides a useful mathematical toolfordescribingthebehaviorofsystemsthatareeithertoocomplexorare ill-dened to admit precise mathematical analysis by classical methods. The literature in fuzzy set and neutrosophic set theories is rapidly expanding and application of this concept can be seen in a variety of disciplines such as articialintelligence,computerscience,controlengineering,expertsystems, operating research, management science, and robotics. Zadeh introduced the degree of membership of an element with respect to a set in 1965, Atanassov introduced the degree of non-membership in 1986, and Smarandache introduced the degree of indeterminacy (i.e. neither membership, nor non-membership) as independent component in 1995 and defined the neutrosophic set. In 2003 W. B. Vasantha Kan- dasamy and Florentin Smarandache introduced for the rst time the I- neutrosophic algebraic structures (such as neutrosophic semigroup, neutro- sophic ring, neutrosophic vector space, etc.) based on neutrosophic num- bers of the form a + bI, wher

Characterizations of ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals

In this paper, we give characterizations of ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals. We characterize different classes regular (resp. intra-regular, simple and semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals (resp. (∈, ∈ ∨q)-fuzzy ideals). In this regard, we prove that in regular (resp. intra-regular and semisimple) ordered semigroups the concept of (∈, ∈ ∨q)-fuzzy ideals and (∈, ∈ ∨q)-fuzzy interior ideals coincide. We prove that an ordered semigroup S is simple if and only if it is (∈, ∈ ∨q)-fuzzy simple. We characterize intra-regular (resp. semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy ideals (resp. (∈, ∈ ∨q)-fuzzy interior ideals). Finally, we consider the concept of implication-based fuzzy interior ideals in an ordered semigroup, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed

Innovative types of fuzzy gamma ideals in ordered gamma semigroups

The fuzzification of algebraic structures plays an important role in handling many areas of multi-disciplinary research, such as computer science, control theory, information science, topological spaces and fuzzy automata to handle many real world problems. For instance, algebraic structures are particularly useful in detecting permanent faults on sequential machine behaviour. However, the idea of ordered T-semigroup as a generalization of ordered semigroup in algebraic structures has rarely been studied. In this research, a new form of fuzzy subsystem in ordered T-semigroup is defined. Specifically, a developmental platform of further characterizations on ordered T-semigroups using fuzzy subsystems properties and new fuzzified ideal structures of ordered semigroups is developed based on a detailed study of ordered T-semigroups in terms of the idea of belongs to (E) and quasicoincidence with (q) relation. This idea of quasi-coincidence of a fuzzy point with a fuzzy set played a remarkable role in obtaining several types of fuzzy subgroups and subsystems based on three contributions. One, a new form of generalization of fuzzy generalized bi T-ideal is developed, and the notion of fuzzy bi T-ideal of the form (E,E Vqk) in an ordered T-semigroup is also introduced. In addition, a necessary and sufficient condition for an ordered T-semigroup to be simple T-ideals in terms of this new form is stated. Two, the concept of (E,E Vqk)-fuzzy quasi T-ideals, fuzzy semiprime T-ideals, and other characterization in terms of regular (left, right, completely, intra) in ordered T-semigroup are developed. Three, a new fuzzified T-ideal in terms of interior T-ideal of ordered T-semigroups in many classes are determined. Thus, this thesis provides the characterizations of innovative types of fuzzy T-ideals in ordered T-semigroups with classifications in terms of completely regular, intra-regular, for fuzzy generalized bi T-ideals, fuzzy bi T-ideals, fuzzy quasi and fuzzy semiprime T-ideals, and fuzzy interior T-ideals. These findings constitute a platform for further advancement of ordered T-semigroups and their applications to other concepts and branches of algebra

On Relations between Some Types of (α,β)-Intuitionistic Fuzzy Ideals of Ternary Semigroups

In this article, the notion of (α,β)-intuitionistic fuzzy ideals (briefly, (α,β)-IF ideals) of ternary semigroups is described using ”belong to” relation (ϵ) and “quasi-coincidence with” relation (q) connecting two objects, i.e., an intuitionistic fuzzy point (IFP, for short) and an intuitionistic fuzzy set (briefly, IFS). Throughout this paper, α∈{ϵ,q,ϵ∨q} and β∈{ϵ,q,ϵ∨q,ϵ∧q}.  The main purposes of this research are to construct the definition of (α,β)-intuitionistic fuzzy ideals of ternary semigroups and to investigate the relations between some types of these ideals. To achieve these goals, we use literature review method to study previous researches regarding (α,β)-fuzzy ideals of ternary semigroups and (α,β)-IF ideals of semigroups. As a result, we find the conditions for an IFS and an ideal of a ternary semigroup to be classified as an (α,ϵ∨q)-IF ideal of ternary semigroup. Relations between some types of (α,β)-IF ideals of a ternary semigroup are also discussed here

Regular ag-groupoids characterized by (∈, ∈ ∨ q k)-fuzzy ideals

In this paper, we introduce a considerable machinery which permits us to characterize a number of special (fuzzy) subsets in AG -groupoids. Generalizing the concepts of (∈, ∈ ∨q) -fuzzy bi-ideals (interior ideal), we define (∈, ∈ ∨ q k) -fuzzy bi-ideals, (∈, ∈ ∨ q k )-fuzzy left (right)-ideals and ( , ) k ? ? ?q -fuzzy interior ideals in AG -groupoids and discuss some fundamental aspects of these ideals in AG -groupoids. We further define ( ∈, ∈ ∨ q k) -fuzzy bi-ideals and (∈, ∈ ∨ q k)-fuzzy interior ideals and give some of their basic properties in AG -groupoids. In the last section, we define lower/upper parts of (∈, ∈ ∨ q k ) -fuzzy left (resp. right) ideals and investigate some characterizations of regular and intera-regular AG -groupoids in terms of the lower parts of ( ∈, ∈ ∨ q k ) -fuzzy left (resp. right) ideals and ( ∈, ∈ ∨ q k )-fuzzy bi-ideal of AG -groupoids