The Experts below are selected from a list of 2010 Experts worldwide ranked by ideXlab platform
Asghar Khan - One of the best experts on this subject based on the ideXlab platform.
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Union soft set theory applied to Ordered Semigroups
International Journal of Analysis and Applications, 2020Co-Authors: Raees Khan, Asghar Khan, Muhammad Uzair Khan, Nasir KhanAbstract:The uni-soft type of bi-ideals in Ordered Semigroup is considered. The notion of a uni-soft bi-ideal is introduced and the related properties are investigated. The concept of δ−exclusive set is given and the relations between uni-soft bi-ideals and δ−exclusive set are discussed. The concepts of two types of prime uni-soft bi-ideals of an Ordered Semigroup S are given and it is proved that, a non-constant uni-soft bi-ideal of S is prime in the second sense if and only if each of its proper δ−exclusive set is a prime bi-ideal of S. The characterizations of left and right simple Ordered Semigroups are considered. Using the notion of uni-soft bi-ideals, some semilattices of left and right simple Semigroups are provided. By using the properties of uni-soft bi-ideals, the characterization of a regular Ordered Semigroup is provided. In the last section of this paper, the characterizations of both regular and intra-regular Ordered Semigroups are provided.
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A Study of Non-Associative Ordered Semigroups in Terms of Semilattices via Smallest (Double-Framed Soft) Ideals
Etamaths Publishing, 2018Co-Authors: Faisal Yousafzai, Asghar Khan, Tauseef Asif, Bijan DavvazAbstract:Soft set theory, introduced by Molodtsov has been considered as a successful mathematical tool for modeling uncertainties. A double-framed soft set is a generalization of a soft set, consisting of union soft sets and intersectional soft sets. An Ordered AG-groupoid can be referred to as a non-associative Ordered Semigroup, as the main difference between an Ordered Semigroup and an Ordered AG-groupoid is the switching of an associative law. In this paper, we define the smallest left (right) ideals in an Ordered AG-groupoid and use them to characterize a strongly regular class of a unitary Ordered AG-groupoid along with its semilattices and double-framed soft (briefly DFS) l-ideals (r-ideals). We also give the concept of an Ordered A* G**-groupoid and investigate its structural properties by using the generated ideals and DFS l-ideals (r-ideals). These concepts will verify the existing characterizations and will help in achieving more generalized results in future works
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Generalized Bi-ideal of Ordered Semigroup Related to Intuitionistic Fuzzy Point
Zibeline International, 2017Co-Authors: Hidayat Ullah Khan, Asghar Khan, Faiz Muhammad KhanAbstract:Intuitionistic fuzzy generalized bi-ideals play an important role in the study of Ordered Semigroups. In this paper, we try obtain more general form of intuitionistic fuzzy generalized bi-ideal of an Ordered Semigroup. The concept of (∈, ∈ ∨qk)-intuitionistic fuzzy generalized bi-ideal is introduced and several related properties are investigated. We show that in regular and left weakly regular Ordered Semigroups the concepts of (∈, ∈ ∨qk)-intuitionistic fuzzy generalized biideal and (∈, ∈ ∨qk)-intuitionistic fuzzy bi-ideal coincide
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cartesian product of interval valued fuzzy ideals in Ordered Semigroup
Journal of Prime Research in Mathematics, 2016Co-Authors: Ullah Khan Hidayat, Asghar Khan, Nor Haniza SarminAbstract:Interval-valued fuzzy set theory is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, the concepts of interval-valued fuzzy (prime, semiprime) ideal and the Cartesian product of interval-valued fuzzy subsets have been introduced. Some interesting results about Cartesian product of interval-valued fuzzy ideals, interval-valued fuzzy prime ideals, interval- valued fuzzy semiprime ideals, interval-valued fuzzy bi-ideals and interval- valued fuzzy interior ideals in Ordered Semigroups are obtained. The pur- port of this paper is to link ordinary ideals with interval-valued fuzzy ideals by means of level subset of Cartesian product of interval-valued fuzzy sub- sets.
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A study of fuzzy soft interior ideals of Ordered Semigroups
Iranian Journal of Science and Technology Transaction A-science, 2013Co-Authors: Asghar Khan, Faiz Muhammad Khan, Nor Haniza Sarmin, Bijan DavvazAbstract:In this paper, we present the concepts of a fuzzy soft left (right) ideal and fuzzy soft interior ideal over an Ordered Semigroup S . Some basic results of fuzzy soft left (right) ideals and fuzzy soft interior ideals are investigated and the supported examples are provided. Different classes, regular, intra-regular, and simple Ordered Semigroups are characterized by means of fuzzy soft left (right) ideals and fuzzy soft interior ideals. It is shown that an Ordered Semigroup is simple if and only if it is fuzzy soft simple. Furthermore, left (right) regular and intra-regular Ordered Semigroups are characterized by means of fuzzy soft left (right) ideals and fuzzy soft ideals.
Jian Tang - One of the best experts on this subject based on the ideXlab platform.
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On C -ideals and the basis of an Ordered Semigroup
AIMS Mathematics, 2020Co-Authors: Xiang-yun Xie, Jian TangAbstract:In this paper, we characterize Ordered Semigroups containing the greatest ideal and give the conditions of the greatest ideal being a C-ideal in an Ordered Semigroup. Moreover, we introduce the concept of a basis of an Ordered Semigroup and study the relationship between the greatest C-ideal and the basis in an Ordered Semigroup.
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A novel study on fuzzy ideals and fuzzy filters of Ordered *-Semigroups
Journal of Intelligent & Fuzzy Systems, 2017Co-Authors: Xinyang Feng, Jian Tang, Bijan Davvaz, Yanfeng LuoAbstract:In this paper, we study the Ordered *-Semigroups in terms of fuzzy subsets in detail and define a unary operation *on the set of all the fuzzy subsets of an Ordered *-Semigroup S, which is a key notion to introduce the concepts of prime, weakly prime and semiprime fuzzy ideals of S. Furthermore, we establish the relationships among these three types of fuzzy ideals and give some characterizations of intra-regular Ordered *-Semigroups in terms of fuzzy ideals. Finally, we define and study the fuzzy filters of an Ordered *-Semigroup. In particular, we discuss the relationships between the filters and the fuzzy filters in Ordered *-Semigroups.
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Characterizations of regular Ordered Semigroups by generalized fuzzy ideals
Journal of Intelligent & Fuzzy Systems, 2014Co-Authors: Jian Tang, Xiang-yun XieAbstract:Let S be an Ordered Semigroup. In this paper we first introduce the concepts of ∈, ∈ ∨ qk-fuzzy ideals, ∈, ∈ ∨ qk-fuzzy bi-ideals and ∈, ∈ ∨ qk-fuzzy generalized bi-ideals of an Ordered Semigroup S by the Ordered fuzzy points of S, and investigate their related properties. Furthermore, characterizations of regular Ordered Semigroups by the properties of ∈, ∈ ∨ qk-fuzzy left ideals, ∈, ∈ ∨ qk-fuzzy right ideals and ∈, ∈ ∨ qk-fuzzy generalized bi-ideals are given.
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Characterizations of Ordered Semigroups by (∈,∈ ∨q)-Fuzzy Ideals
2012Co-Authors: Jian TangAbstract:Let S be an Ordered Semigroup. In this paper we first introduce the concepts of (∈,∈ ∨q)-fuzzy ideals, (∈,∈ ∨q)-fuzzy bi-ideals and (∈,∈ ∨q)-fuzzy generalized bi-ideals of an Ordered Semigroup S, and investigate their related properties. Furthermore, we also define the upper and lower parts of fuzzy subsets of an Ordered Semigroup S, and investigate the properties of (∈,∈ ∨q)-fuzzy ideals of S. Finally, characterizations of regular Ordered Semigroups and intra-regular Ordered Semigroups by means of the lower part of (∈ ,∈ ∨q)-fuzzy left ideals, (∈,∈ ∨q)-fuzzy right ideals and (∈,∈ ∨q)fuzzy (generalized) bi-ideals are given. Keywords—Ordered Semigroup; regular Ordered Semigroup; intraregular Ordered Semigroup; (∈,∈ ∨q)-fuzzy left (right) ideal of an Ordered Semigroup; (∈,∈ ∨q)-fuzzy (generalized) bi-ideal of an Ordered Semigroup.
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On Completely Semiprime, Semiprime and Prime Fuzzy Ideals in Ordered Semigroups
2011Co-Authors: Jian TangAbstract:In this paper, we first introduce the new concept of completely semiprime fuzzy ideals of an Ordered Semigroup S, which is an extension of completely semiprime ideals of Ordered Semigroup S, and investigate some its related properties. Especially, we characterize an Ordered Semigroup that is a semilattice of simple Ordered Semigroups in terms of completely semiprime fuzzy ideals of Ordered Semigroups. Furthermore, we introduce the notion of semiprime fuzzy ideals of Ordered Semigroup S and establish the relations between completely semiprime fuzzy ideals and semiprime fuzzy ideals of S. Finally, we give a characterization of prime fuzzy ideals of an Ordered Semigroup S and show that a nonconstant fuzzy ideal f of an Ordered Semigroup S is prime if and only if f is twovalued, and max{f(a), f(b)} = inf f((aSb]), ∀a, b ∈ S. Keywords—Ordered fuzzy point, fuzzy left (right) ideal of an Ordered Semigroup, completely semiprime fuzzy ideal, semiprime fuzzy ideal, prime fuzzy ideal.
Niovi Kehayopulu - One of the best experts on this subject based on the ideXlab platform.
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Magnifying elements and factorization of Ordered Semigroups
Algebra universalis, 2019Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:This note defines the left magnifying and the strongly left magnifying elements in an Ordered groupoid and discusses their properties. It is shown that in an Ordered Semigroup, every left magnifying element is of infinite order. The concept of factorizable Ordered Semigroups has been also introduced and, using the infinite order property, it is shown that every Ordered Semigroup having a strongly left magnifying element is factorizable.
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ON KERNEL OF Ordered SemigroupS – A CORRIGENDUM
2016Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:Abstract. According to the paper in [3], the kernel of an Ordered semi-group S is a completely regular subSemigroup of S. In this note we show that the kernel of an Ordered Semigroup S is not a completely regular subSemigroup of S, in general. 1
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Left strongly archimedean Ordered Semigroups
Semigroup Forum, 2014Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:We prove that if an Ordered Semigroup is a nil extension of a left strongly simple Ordered Semigroup, then it is left strongly archimedean, but, in contrast to the unOrdered case, the converse does not hold in general. However, a left strongly archimedean Ordered Semigroup is a nil extension of a simple Ordered Semigroup.
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On left regular and intra-regular Ordered Semigroups
Mathematica Slovaca, 2014Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:We study the decomposition of left regular Ordered Semigroups into left regular components and the decomposition of intra-regular Ordered Semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular Ordered Semigroups , Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular Ordered Semigroups , Semigroup Forum 46 (1993), 271–278]. We prove that an Ordered Semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular Semigroups, equivalently, it is a union of left regular subSemigroups of S . Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subSemigroups of S . The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an Ordered Semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple Semigroups. On the other hand, if an Ordered Semigroup is a semilattice (or a complete semilattice) of left simple Semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.
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ON Ordered SemigroupS WHICH ARE SEMILATTICES OF SIMPLE AND REGULAR SemigroupS
Communications in Algebra, 2013Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:We characterize the Ordered Semigroups which are decomposable into simple and regular components. We prove that each Ordered Semigroup which is both regular and intra-regular is decomposable into simple and regular Semigroups, and the converse statement also holds. We also prove that an Ordered Semigroup S is both regular and intra-regular if and only if every bi-ideal of S is an intra-regular (resp. semisimple) subSemigroup of S. An Ordered Semigroup S is both regular and intra-regular if and only if the left (resp. right) ideals of S are right (resp. left) quasi-regular subSemigroups of S. We characterize the chains of simple and regular Semigroups, and we prove that S is a complete semilattice of simple and regular Semigroups if and only if S is a semilattice of simple and regular Semigroups. While a Semigroup which is both π-regular and intra-regular is a semilattice of simple and regular Semigroups, this does not hold in Ordered Semigroups, in general.
Michael Tsingelis - One of the best experts on this subject based on the ideXlab platform.
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Magnifying elements and factorization of Ordered Semigroups
Algebra universalis, 2019Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:This note defines the left magnifying and the strongly left magnifying elements in an Ordered groupoid and discusses their properties. It is shown that in an Ordered Semigroup, every left magnifying element is of infinite order. The concept of factorizable Ordered Semigroups has been also introduced and, using the infinite order property, it is shown that every Ordered Semigroup having a strongly left magnifying element is factorizable.
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ON KERNEL OF Ordered SemigroupS – A CORRIGENDUM
2016Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:Abstract. According to the paper in [3], the kernel of an Ordered semi-group S is a completely regular subSemigroup of S. In this note we show that the kernel of an Ordered Semigroup S is not a completely regular subSemigroup of S, in general. 1
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Left strongly archimedean Ordered Semigroups
Semigroup Forum, 2014Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:We prove that if an Ordered Semigroup is a nil extension of a left strongly simple Ordered Semigroup, then it is left strongly archimedean, but, in contrast to the unOrdered case, the converse does not hold in general. However, a left strongly archimedean Ordered Semigroup is a nil extension of a simple Ordered Semigroup.
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On left regular and intra-regular Ordered Semigroups
Mathematica Slovaca, 2014Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:We study the decomposition of left regular Ordered Semigroups into left regular components and the decomposition of intra-regular Ordered Semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular Ordered Semigroups , Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular Ordered Semigroups , Semigroup Forum 46 (1993), 271–278]. We prove that an Ordered Semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular Semigroups, equivalently, it is a union of left regular subSemigroups of S . Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subSemigroups of S . The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an Ordered Semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple Semigroups. On the other hand, if an Ordered Semigroup is a semilattice (or a complete semilattice) of left simple Semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.
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ON Ordered SemigroupS WHICH ARE SEMILATTICES OF SIMPLE AND REGULAR SemigroupS
Communications in Algebra, 2013Co-Authors: Niovi Kehayopulu, Michael TsingelisAbstract:We characterize the Ordered Semigroups which are decomposable into simple and regular components. We prove that each Ordered Semigroup which is both regular and intra-regular is decomposable into simple and regular Semigroups, and the converse statement also holds. We also prove that an Ordered Semigroup S is both regular and intra-regular if and only if every bi-ideal of S is an intra-regular (resp. semisimple) subSemigroup of S. An Ordered Semigroup S is both regular and intra-regular if and only if the left (resp. right) ideals of S are right (resp. left) quasi-regular subSemigroups of S. We characterize the chains of simple and regular Semigroups, and we prove that S is a complete semilattice of simple and regular Semigroups if and only if S is a semilattice of simple and regular Semigroups. While a Semigroup which is both π-regular and intra-regular is a semilattice of simple and regular Semigroups, this does not hold in Ordered Semigroups, in general.
Xiang-yun Xie - One of the best experts on this subject based on the ideXlab platform.
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On C -ideals and the basis of an Ordered Semigroup
AIMS Mathematics, 2020Co-Authors: Xiang-yun Xie, Jian TangAbstract:In this paper, we characterize Ordered Semigroups containing the greatest ideal and give the conditions of the greatest ideal being a C-ideal in an Ordered Semigroup. Moreover, we introduce the concept of a basis of an Ordered Semigroup and study the relationship between the greatest C-ideal and the basis in an Ordered Semigroup.
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Characterizations of regular Ordered Semigroups by generalized fuzzy ideals
Journal of Intelligent & Fuzzy Systems, 2014Co-Authors: Jian Tang, Xiang-yun XieAbstract:Let S be an Ordered Semigroup. In this paper we first introduce the concepts of ∈, ∈ ∨ qk-fuzzy ideals, ∈, ∈ ∨ qk-fuzzy bi-ideals and ∈, ∈ ∨ qk-fuzzy generalized bi-ideals of an Ordered Semigroup S by the Ordered fuzzy points of S, and investigate their related properties. Furthermore, characterizations of regular Ordered Semigroups by the properties of ∈, ∈ ∨ qk-fuzzy left ideals, ∈, ∈ ∨ qk-fuzzy right ideals and ∈, ∈ ∨ qk-fuzzy generalized bi-ideals are given.
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Regular Ordered Semigroups and intra-regular Ordered Semigroups in terms of fuzzy subsets
Iranian Journal of Fuzzy Systems, 2010Co-Authors: Xiang-yun Xie, Jian TangAbstract:Let $S$ be an Ordered Semigroup. A fuzzy subset of $S$ is anarbitrary mapping from $S$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. In this paper, the concept of fuzzygeneralized bi-ideals of an Ordered Semigroup $S$ is introduced.Regular Ordered Semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.Finally, two main theorems which characterize regular OrderedSemigroups and intra-regular Ordered Semigroups in terms of fuzzyleft ideals, fuzzy right ideals, fuzzy bi-ideals or fuzzyquasi-ideals are given. The paper shows that one can pass fromresults in terms of fuzzy subsets in Semigroups to OrderedSemigroups. The corresponding results of unOrdered Semigroups arealso obtained.
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Fuzzy radicals and prime fuzzy ideals of Ordered Semigroups
Information Sciences, 2008Co-Authors: Xiang-yun Xie, Jian TangAbstract:Let S be an Ordered Semigroup. A fuzzy subset of S is, by definition, an arbitrary mapping f: S->[0,1], where [0,1] is the usual interval of real numbers. Motivated by studying prime fuzzy ideals in rings, Semigroups and Ordered Semigroups, and as a continuation of Kehayopulu and Tsingelis's works in Ordered Semigroups in terms of fuzzy subsets, in this paper we introduce the notion of Ordered fuzzy points of an Ordered Semigroup S, and give a characterization of prime fuzzy ideals of an Ordered Semigroup S. We also introduce the concepts of weakly prime fuzzy ideals, completely prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of an Ordered Semigroup S, and establish the relationship between the five classes of ideals. Furthermore, we characterize weakly prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of S by their level ideals. Finally, we introduce and investigate the fuzzy radicals of Ordered Semigroups by means of Ordered fuzzy points, and prove that the fuzzy radical of every completely semiprime fuzzy ideal of an Ordered Semigroup S can be expressed as the intersection of all weakly completely prime fuzzy ideals containing it. As an application of the results of this paper, the corresponding results of Semigroups (without order) are also obtained.
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On Strongly Ordered Congruences and Decompositions of Ordered Semigroups
Algebra Colloquium, 2008Co-Authors: Xiang-yun XieAbstract:In this paper, we introduce the concept of a strongly Ordered congruence on a directed Ordered Semigroup S. We prove that any strongly Ordered congruence on S is a strongly regular congruence. We characterize the finite direct product, subdirect product and full subdirect product of Ordered Semigroups by using the concepts of strongly Ordered congruence and regular congruence on an Ordered Semigroup S.
