- D.A.V. College, Jalandhar, Punjab, India.
- 09872006365
Poonam K Sharma
DAV College Jalandhar, Mathematics, Faculty Member
- Fuzzy Topology, Intuitionistic Logic, Algebra with fuzzy functions, Fuzzy algebra , Fuzzy Mathematics, Discrete Mathematics, Mathematics, and 19 moreFuzzy Mathematics, Fuzzy subgroups, UPPER FUZZY SUBRINGS, Fuzzy Modules Over Fuzzy Rings, Intuitionistic Fuzzy Right Cosets, Fuzzy Modules, Applied Mathematics, Decision Making, Logic And Foundations Of Mathematics, Intuitionistic fuzzy sets, Computer Science, Fuzzy set theory, Abstract Algebra, FUZZY IDEALS, Classical Logic, Intermediate Logics, Intuitionistic Fuzzy Set, NORMAL FUZZY SUBGROUPS, and Information Technologyedit
- Dr. P.K. Sharma completed his master degree in Mathematics from D.A.V. College, Jalandhar in 1988. He further pursued... moreDr. P.K. Sharma completed his master degree in Mathematics from D.A.V. College, Jalandhar in 1988. He further pursued his Ph.D degree in Modern Algebra from Jamia Millia Islamia University, New Delhi which he completed in 1993. His first appointment as lecturer in Mathematics was at D.A.V. College, Chandigarh in 1993. After serving for one year he joined D.A.V. College, Jalandhar in 1994. Till date he is serving at D.A.V. College, Jalandhar.
Publications:
He has published 171 research papers in National and International Journals and has written 4 books for post-graduation namely Topology, Modern Algebra, Real Analysis and Discrete Mathematics and one research book, Algebra: Relative relation modules of finite groups, published from Lambert Academia Publication Germany. His field of interest is Topology, Algebra and Fuzzy Mathematics.
Others Achievements:
Dr. P.K. Sharma is a member of editorial boards of 5 International Journals namely (i) International Journal of Fuzzy Mathematics and Systems (ii) Journal of Intelligent and Fuzzy System (iii) Neural Computing and Application (iv) Journal of the Egyptian Mathematical Society (v) Annals of Fuzzy Mathematics and Informatics (AFMI). He is a life member of 4 different Mathematical Societies namely (i) Ramanujan Mathematical Society (ii) Indian Mathematical Society (iii) Jammu Mathematical Society and (iv) Indian Society of Information Theory and Applications.
He has participated in 25 National and 20 International conferences and has chaired many sessions in Nationals and International conferences.
He has also organized three U.G.C. sponsored National conferences and one Teacher’s Enrichment Workshop at D.A.V. College, Jalandhar and has delivered nearly 25 extension lectures on various topics of mathematics in different colleges of Punjab.
He has visited U.S.A. in 2013 and Bulgaria in 2016 under the travel grant scheme of U.G.C. for presenting his research papers in the International Conference on scientific computing and Intuitionistic fuzzy sets and its Applications.
Project:
Dr. Sharma had completed one Major Research Project “ A study on Intuitionistic Fuzzy G-Modules” sponsored by U.G.C., in 2017 and four students have completed their Ph.D degree and two students are doing Ph.D under his guidance . Dr. Sharma has guided 25 students for their M.Phil. Dissertations
Honors and awards
• Honorary D.Sc. (Doctrin de Science Award Honoris Causa) in Mathematics, Reference No.: H- 456306092023PKS by International Agency for Standards and Ratings on 6th Sept. 2023.
• Awarded Lifetime achievement Award by VDGOOD Professional Association, in the International Scientist Award on Engineering, Science and Medicine, held at Salem, India on 08 & 09 Jan. 2021.
• Nominated in who’s who in the world for 2015edit - Dr. P.K. Sharma supervised four Ph.D. scholars and two scholars are still working under his supervision.edit
In this paper, we introduce the notion of expansion of intuitionistic fuzzy ideals of a commutative Γ-ring and by using this concept, we develop the notion of intuitionistic fuzzy f-primary ideals (2-absorbing f-primary ideals) which... more
In this paper, we introduce the notion of expansion of intuitionistic fuzzy ideals of a commutative Γ-ring and by using this concept, we develop the notion of intuitionistic fuzzy f-primary ideals (2-absorbing f-primary ideals) which unify the notion of intuitionistic fuzzy prime ideals (2-absorbing ideals) and intuitionistic fuzzy primary ideals (2-absorbing primary ideals) of Γ-ring. A number of important results about intuitionistic fuzzy prime ideals (2-absorbing ideals) and intuitionistic fuzzy primary ideals (2-absorbing primary ideals) are extended into this general frame work.
Research Interests:
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... more
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.
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In this paper, we study the properties of intuitionistic fuzzy modules from the categorical point of view by proving that the category CR-IFM of intuitionistic fuzzy modules has products, coproducts, equalizers and coequalizers. Then,... more
In this paper, we study the properties of intuitionistic fuzzy modules
from the categorical point of view by proving that the category
CR-IFM of intuitionistic fuzzy modules has products, coproducts,
equalizers and coequalizers. Then, we show that every intuitionistic
fuzzy coretraction (retraction) is an intuitionistic fuzzy equalizer (coequalizer). Further, categorical goodness of intuitionistic fuzzy
modules is illustrated by proving that the category of intuitionistic
fuzzy modules CR IFM is complete and co-complete.
from the categorical point of view by proving that the category
CR-IFM of intuitionistic fuzzy modules has products, coproducts,
equalizers and coequalizers. Then, we show that every intuitionistic
fuzzy coretraction (retraction) is an intuitionistic fuzzy equalizer (coequalizer). Further, categorical goodness of intuitionistic fuzzy
modules is illustrated by proving that the category of intuitionistic
fuzzy modules CR IFM is complete and co-complete.
Research Interests:
In this paper, we establish the intuitionistic fuzzy version of the Lasker-Noether theorem for a commutative Γ-ring. We show that in a commutative Noetherian Γ-ring, every intuitionistic fuzzy ideal A can be decomposed as the intersection... more
In this paper, we establish the intuitionistic fuzzy version of the Lasker-Noether theorem for a commutative Γ-ring. We show that in a commutative Noetherian Γ-ring, every intuitionistic fuzzy ideal A can be decomposed as the intersection of a finite number of intuitionistic fuzzy irreducible ideals (primary ideals). This decomposition is called an intuitionistic fuzzy primary decomposition. Further, we show that in case of a minimal intuitionistic fuzzy primary decomposition of A, the set of all intuitionistic fuzzy associated prime ideals of A is independent of the particular decomposition. We also discuss some other fundamental results pertaining to this concept.
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The aim of this paper is to introduce two special type of morphisms, namely Retraction and Coretraction in the category (C R-IFM) of intuitionistic fuzzy modules. We obtain the condition under which a morphism in C R-IFM , that is an... more
The aim of this paper is to introduce two special type of morphisms, namely Retraction and Coretraction in the category (C R-IFM) of intuitionistic fuzzy modules. We obtain the condition under which a morphism in C R-IFM , that is an intuitionistic fuzzy R-homomorphism, to be a retraction or a coretraction. Then, we acquire some equivalent statements for these two morphisms. Further, we study free, projective and injective objects in C R-IFM and establish their relation with morphism in C R-IFM and retraction, coretraction.
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In this paper, we introduce the notion of F-closure of intuitionistic fuzzy submodules of a module M. Our attempt is to investigate various characteristics of such an F-closure. If F is a non-empty set of intuitionistic fuzzy ideals of a... more
In this paper, we introduce the notion of F-closure of intuitionistic fuzzy submodules of a module M. Our attempt is to investigate various characteristics of such an F-closure. If F is a non-empty set of intuitionistic fuzzy ideals of a commutative ring R and A is an intuitionistic fuzzy submodule of M , then the F-closure of A is denoted by Cl M F (A). If F is weak closed under intersection, then (1) F-closure of A exhibits the submodule character, and (2) the intersection of F-closure of two intuitionistic fuzzy submodules equals the F-closure of intersection of the intuitionistic fuzzy submodules. If F is weak closed under intersection, then the submodule property of F-closure implies that F is closed. Moreover, if F is inductive, then F is a topological filter if and only if Cl M F (A) is an intuitionistic fuzzy submodule for any intuitionistic fuzzy submodule A of M .
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The aim of this paper is to present some characterizations of almost prime ideals and almost prime submodules in the intuitionistic fuzzy environment. We investigate various properties of these concepts and achieve many results.
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Let E be a free product of a finite number of cyclic groups, and S a normal subgroup of E such that E/S ∼= G is finite. For a prime p, Sˆ = S/S′Sp may be regarded as an FpG-module via conjugation in E. The aim of this article is to... more
Let E be a free product of a finite number of cyclic groups,
and S a normal subgroup of E such that E/S ∼= G is finite. For a prime
p, Sˆ = S/S′Sp may be regarded as an FpG-module via conjugation in
E. The aim of this article is to prove that Sˆ is decomposable into two
indecomposable modules for finite elementary abelian p-groups G.
and S a normal subgroup of E such that E/S ∼= G is finite. For a prime
p, Sˆ = S/S′Sp may be regarded as an FpG-module via conjugation in
E. The aim of this article is to prove that Sˆ is decomposable into two
indecomposable modules for finite elementary abelian p-groups G.
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Let L be a complete lattice. We introduce and characterise intuitionistic L-fuzzy classical prime submodule and intuitionistic L-fuzzy 2-absorbing submodules of a unitary module M over a commutative ring R with identity. We compare both... more
Let L be a complete lattice. We introduce and characterise intuitionistic L-fuzzy classical prime submodule and intuitionistic L-fuzzy 2-absorbing submodules of a unitary module M over a commutative ring R with identity. We compare both of these submodules with intuitionistic L-fuzzy prime submodules. It is proven that in the case of the multiplication module M , the two notions of intuitionistic L-fuzzy classical prime submodules and intuitionistic L-fuzzy prime submodules coincide. Many other related results concerning these notions are obtained.
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In this paper we try to study the intuitionistic-fuzzy aspects of socle of modules over rings. We demonstrate some properties of a socle of intuitionistic-fuzzy submodules and their relations with intuitionistic-fuzzy essential submodules... more
In this paper we try to study the intuitionistic-fuzzy aspects of socle of modules over rings. We demonstrate some properties of a socle of intuitionistic-fuzzy submodules and their relations with intuitionistic-fuzzy essential submodules and a family of intuitionistic-fuzzy complemented submodules of a module. Some related results are also established.
Research Interests:
In this paper, we study relative ordered (m, n)-hyperideals in ordered semihypergroups. We also study relative (m, 0)-hyperideals and relative (0, n)-hyperideals as well as characterize regular ordered semihypergroups, and obtain some... more
In this paper, we study relative ordered (m, n)-hyperideals in ordered semihypergroups. We also study relative (m, 0)-hyperideals and relative (0, n)-hyperideals as well as characterize regular ordered semihypergroups, and obtain some results based on these relative hyperideals. We prove that the intersection of all relative ordered (m, n)-hyperideals of S containing s is a relative ordered (m, n)-hyperideal of S containing s. Suppose that (S,•,≤) is an ordered semihypergroup, A ⊆ S and m,n are positive integers. We prove that if R(m,0) and L(0,n) be the set of all relative ordered (m,0)-hyperideals and the set of all relative ordered (0,n)-hyperideals of S, respectively. Then the following assertions are true: i) S is relative (m,0)-regular if and only if for all R ∈ R(m,0), R = (Rm •A]A. ii) S is relative (0,n)-regular if and only if for all L ∈ R(0,n), L = (A•Ln]A. Furthermore, suppose that (S, •, ≤) is an ordered semihypergroup and m, n are non-negative integers. Let A ⊆ S. Suppose that A(m,n) is the set of all relative ordered (m, n)-hyperideals of S. Then, we have the following: S is (m,n)-regular ⇐⇒ ∀A ∈ A(m,n),A = (Am •A•An]A.
Research Interests: Algebra and Hyper Algebra
The main goal of this paper is to count subgroups which are isomorphic to cyclic p-group, internal direct product of two cyclic p-group or semi direct product of two cyclic p-group of the non-Abelian p-group Zpn o Zp, n ≥ 2 where p may be... more
The main goal of this paper is to count subgroups which are isomorphic to cyclic p-group, internal direct product of two cyclic p-group or semi direct product of two cyclic p-group of the non-Abelian p-group Zpn o Zp, n ≥ 2 where p may be even or odd prime, by using simple-theoretical approach. AMS Subject Classification: Primary 20D60; Secondary 20D15
In this paper, we introduced the notion of intuitionistic fuzzy prime radical of an intuitionistic fuzzy ideal in Γ-rings. We also characterise intuitionistic fuzzy primary ideal of Γ-rings. We also analyse homomorphic behaviour of... more
In this paper, we introduced the notion of intuitionistic fuzzy prime radical of an intuitionistic fuzzy ideal in Γ-rings. We also characterise intuitionistic fuzzy primary ideal of Γ-rings. We also analyse homomorphic behaviour of intuitionistic fuzzy primary ideal and intuitionistic fuzzy prime radical of Γ-rings.
Research Interests:
In this paper, we initiate the study of a generalization of intuitionistic fuzzy primary ideals in Γ-ring by introducing intuitionistic fuzzy 2-absorbing primary ideals. We investigate the structural characteristics of intuitionistic... more
In this paper, we initiate the study of a generalization of intuitionistic fuzzy primary ideals in Γ-ring by introducing intuitionistic fuzzy 2-absorbing primary ideals. We investigate the structural characteristics of intuitionistic fuzzy 2-absorbing primary ideals and study their properties.
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The purpose of this paper is to introduce and investigate primary ideal and P-primary ideal in the intuitionistic fuzzy environment and lay down the foundation for the primary decomposition theorem in the intuitionistic fuzzy setting.... more
The purpose of this paper is to introduce and investigate primary ideal and P-primary ideal in the intuitionistic fuzzy environment and lay down the foundation for the primary decomposition theorem in the intuitionistic fuzzy setting. Also a suitable characterization of intuitionistic fuzzy P-primary ideal will be discussed.
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In this paper, we introduce some mathematical objects, viz., relative ordered prime, relative ordered completely prime, relative ordered semiprime, and relative ordered completely semiprime right Γ-hyperideals in ordered... more
In this paper, we introduce some mathematical objects, viz., relative ordered prime, relative ordered completely prime, relative ordered semiprime, and relative ordered completely semiprime right Γ-hyperideals in ordered Γ-semihypergroups. We also introduce the concepts of relative ordered prime right associated Γ-hyperideals. Moreover, we characterize ordered Γ-semihypergroups by relative ordered prime, relative ordered completely prime, relative ordered semiprime, and relative ordered completely semiprime right Γ-hyperideals.
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In this research article, we investigate and study the intuitionistic fuzzy structure space of a Γ-ring M set up by the class of intuitionistic fuzzy prime ideals of M called the intuitionistic fuzzy prime spectrum of Γ-ring. Apart from... more
In this research article, we investigate and study the intuitionistic fuzzy structure space of a Γ-ring M set up by the class of intuitionistic fuzzy prime ideals of M called the intuitionistic fuzzy prime spectrum of Γ-ring. Apart from studying basic properties of this structure space, we explore separation axioms, compactness, irreducibility and connectedness in this structure space.
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The purpose of this paper is to extend the notion of ordinary semiprime submodules to intuitionistic fuzzy semiprime submodules. Also we introduce and study new properties of intuitionistic fuzzy semiprime submodules. Many related results... more
The purpose of this paper is to extend the notion of ordinary semiprime submodules
to intuitionistic fuzzy semiprime submodules. Also we introduce and study new properties of
intuitionistic fuzzy semiprime submodules. Many related results are obtained.
to intuitionistic fuzzy semiprime submodules. Also we introduce and study new properties of
intuitionistic fuzzy semiprime submodules. Many related results are obtained.
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In this paper, we introduce and investigate the intuitionistic fuzzy multiplication modules over a commutative ring with non-zero identity. The basic properties of the intuitionistic fuzzy prime submodules of an intuitionistic fuzzy... more
In this paper, we introduce and investigate the intuitionistic fuzzy multiplication modules over a commutative ring with non-zero identity. The basic properties of the intuitionistic fuzzy prime submodules of an intuitionistic fuzzy multiplication modules are characterized.
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In this paper, we define the notion of intuitionistic fuzzy characteristic ideal (IFCI) of a Γ-ring which is analogue of a characteristic ideal in the ordinary ring theory and derive various new results. The correlation between the set of... more
In this paper, we define the notion of intuitionistic fuzzy characteristic ideal (IFCI) of a Γ-ring which is analogue of a characteristic ideal in the ordinary ring theory and derive various new results. The correlation between the set of all automorphisms of Γ-ring and the corresponding automorphisms of its operator rings have been innovated. Then a one to one correlation between the set of all intuitionistic fuzzy characteristic ideals of Γ-ring and that of its operator ring has been constituted. This is used to obtain a similar bijection for characteristic ideals.
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In this paper, we introduce the notion of translational invariant intuitionistic fuzzy subset of a Γ-ring and generalize some notions of a ring to a Γ-ring. Also, we define ideals of a Γ-ring generated by an intuitionistic fuzzy subset... more
In this paper, we introduce the notion of translational invariant intuitionistic fuzzy
subset of a Γ-ring and generalize some notions of a ring to a Γ-ring. Also, we define ideals
of a Γ-ring generated by an intuitionistic fuzzy subset with an element of Γ-ring and study their
properties. The notion of units, associate, prime element, irreducible element are also generalized
with respect to the intuitionistic fuzzy subset of a Γ-ring. Further, we study the properties of
homomorphic image and pre-image of translational invariant intuitionistic fuzzy subset under the
Γ-ring homomorphism and we prove that every homomorphic image of a prime ideal of a Γ-ring
generated by an Aγ-prime element and translational invariant and f-invariant intuitionistic fuzzy
subset is also a prime ideal.
subset of a Γ-ring and generalize some notions of a ring to a Γ-ring. Also, we define ideals
of a Γ-ring generated by an intuitionistic fuzzy subset with an element of Γ-ring and study their
properties. The notion of units, associate, prime element, irreducible element are also generalized
with respect to the intuitionistic fuzzy subset of a Γ-ring. Further, we study the properties of
homomorphic image and pre-image of translational invariant intuitionistic fuzzy subset under the
Γ-ring homomorphism and we prove that every homomorphic image of a prime ideal of a Γ-ring
generated by an Aγ-prime element and translational invariant and f-invariant intuitionistic fuzzy
subset is also a prime ideal.
Research Interests:
In this paper, we introduce the concept of relative ordered Γ-ideals, relative ordered quasi-Γ-ideals, relative ordered bi-Γ-ideals in ordered LA-Γ-semigroups. We characterize ordered LA-Γ-semigroups, LA ⋆-Γ-semigroups, ordered... more
In this paper, we introduce the concept of relative ordered Γ-ideals, relative ordered quasi-Γ-ideals, relative ordered bi-Γ-ideals in ordered LA-Γ-semigroups. We characterize ordered LA-Γ-semigroups, LA ⋆-Γ-semigroups, ordered LA-Γ-groups and left (resp. right) relative simple ordered LA-Γ-semigroups by the relative ordered Γideals in ordered LA-Γ-semigroups.
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In [1] , it was studied Indian Influence in Mathematics of Abu Ja'far Muhammad ibn Musa Al-Khwarizmi, in 2022. Also, contributions of Indian mathematicians can be seen in [3] in 2021. This survey will serve as an impetus to stimulate the... more
In [1] , it was studied Indian Influence in Mathematics of Abu Ja'far Muhammad ibn Musa Al-Khwarizmi, in 2022. Also, contributions of Indian mathematicians can be seen in [3] in 2021. This survey will serve as an impetus to stimulate the interests in basic Indian Mathematicians to record and recognize them as pioneers not only in the fields of mathematics but also in science, in applied and natural science, in engineering and social science for further teaching as well as research in their respective subject matters with mainly the choicest blessings of the absolute Supreme Creator, Founder God of the Universe, like that in Islam too, Allah has 99 names emphasizing on education leading to both basic as well as higher level mathematics like ideals, quasi-ideals and bi-ideals in semihypergroups, ordered semihypergroups, Γ-semigroups, ordered Γ-semigroups, Γ-near ring with the spirited hope for future perspective in teaching and in research.
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In this paper, we introduce G-hyperideal, G-prime hyperideal, G-prime-bihyperideal, and G-weakly prime hyperideal in ordered semihypergroups and we also define G-regular ordered hypersemigroups as well as G-semisimple ordered... more
In this paper, we introduce G-hyperideal, G-prime hyperideal, G-prime-bihyperideal, and G-weakly prime hyperideal in ordered semihypergroups and we also define G-regular ordered hypersemigroups as well as G-semisimple ordered hypersemigroups. Thereby, we extend the results to the broader and abstract setting of ordered hypersemigroups, this also strengthens previous generalizations relating to rings, semigroups and other algebraic structures.
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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Research Interests:
Indian mathematics has its deep roots in the Vedas, different from what is known as Vedic Mathematics. Vedic age gave rise to a new era of progress in the field of Science, Technology and Development. The Hindu Scripture Vedas is... more
Indian mathematics has its deep roots in the Vedas, different from what is known as Vedic Mathematics. Vedic age gave rise to a new era of progress in the field of Science, Technology and Development. The Hindu Scripture Vedas is synonymous with all kinds of original source of knowledge and intellectual wisdom in the universe leading to modern knowlege in modern mathematics. Indian mathematicians made tremendous contributions to the entire world of Mathematics and Science. Decimal number system as well as the invention of zero (0) are among the greatest contributions of Indian mathematicians. The theory of trigonometry, Mathematical Modelling, Algebra, algorithm, modern arithmetic, sine and cosine functions leading to modern trigonometry, Diophantine equation, square root, cube root, negative numbers are also developed by Indian mathematicians. In this review article, the work of some of the renowned Indian mathematicians from Indus Valley civilization and the Vedas to modern times are covered in short with the hope that it may reveal hidden fundamental mathematical ideas as basic ideal tools which may usefully motivate for further research work in every domain of mathematical sciences, natural and applied sciences, engineering and social sciences. Moreover, there are many more remarkable Indian mathematicians who contributed to the origin of mathematical sciences. They have made several general contributions to mathematics that have significantly influenced scientists and mathematicians in the modern times.
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Al-Khwarizmi's works is influenced from India developed from the sacred Hindu text the unique Vedas. His Algebra is influenced by the great Indian mathe matician Brahmagupta who called it Kuttaka. Al Khwarizmi's work on algebra gives us... more
Al-Khwarizmi's works is influenced from India developed from the sacred Hindu text the unique Vedas. His Algebra is influenced by the great Indian mathe matician Brahmagupta who called it Kuttaka. Al Khwarizmi's work on algebra gives us what is now known as Modern Algebra also called Abstract Algebra. The word 'Algebra' is just a broad part in mathematical sciences leading to the study of modern abstract mathematical objects like groups, semigroups, semihypergroups, rings, near rings, Γnearrings, Abstract Affine Γ-nearrings and fields. Algebra developed into a distinct branch of modern mathematics from the time of Indian mathematician Aryab hata I. Recently, Basar, Satyanarayana, Kumar and Mohammad Yahya [International Journal of Statistics and Applied Mathematics, 2021; 6(6): 62-67] studied about var ious Indian mathematicians based on the Hindu sacred scripture the unique Vedas.
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The category theory deals with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and acting as a... more
The category theory deals with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and acting as a unifying notion. In this paper, we study the relationship between the category of groups and the category of intuitionistic fuzzy groups. We prove that the category of groups is a subcategory of category of intuitionistic fuzzy groups and that it is not an Abelian category. We establish a function β : Hom(A, B) → [0, 1] × [0, 1] on the set of all intuitionistic fuzzy homomorphisms between intuitionistic fuzzy groups A and B of groups G and H, respectively. We prove that β is a covariant functor from the category of groups to the category of intuitionistic fuzzy groups. Further, we show that the category of intuitionistic fuzzy groups is a top category by establishing a contravariant functor from the category of intuitionistic fuzzy groups to the lattices of all intuitionistic fuzzy groups.
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Let R be a commutative ring with identity and M be an R-module. An intuitionistic L-fuzzy submodule (ILFSM) C of an intuitionistic L-fuzzy module A of R-module M , is called an intuitionistic L-fuzzy essential submodule in A, if C ∩ B = χ... more
Let R be a commutative ring with identity and M be an R-module. An intuitionistic L-fuzzy submodule (ILFSM) C of an intuitionistic L-fuzzy module A of R-module M , is called an intuitionistic L-fuzzy essential submodule in A, if C ∩ B = χ {θ} for any non-trivial ILFSM B of A. In this case we say that A is an essential extension of C. Also, if C has no proper essential extension in A, then C is called an intuitionistic L-fuzzy closed submodule in A. Further, for ILFSMs B, C of A, C is called complement of B in A if C is maximal with the property that B ∩ C = χ {θ}. We study these mentioned notations which are generalization of the notions of essential submodule, closed submodule and complement of a submodule in the intuitionistic L-fuzzy module theory. We prove many basic properties of both these concepts.
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The main aim of this paper is to investigate the relationship between the characteristic of a ring and the order of an element in the intuitionistic fuzzy environment. We also characterise the term divisor of zero in a ring and study... more
The main aim of this paper is to investigate the relationship between the characteristic of a ring and the order of an element in the intuitionistic fuzzy environment. We also characterise the term divisor of zero in a ring and study their connection with the ideal of the ring in the intuitionistic fuzzy setting. Some related results are also analyse.
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In this paper, we introduce the notion of -complement of intuitionistic fuzzy graph structure = (A, B1, B2,..., Bk) where is a permutation on {B1, B2,...,Bk} and obtain some results. We also define some elementary definitions like self... more
In this paper, we introduce the notion of -complement of intuitionistic fuzzy graph structure = (A, B1, B2,..., Bk) where is a permutation on {B1, B2,...,Bk} and obtain some results. We also define some elementary definitions like self complementary, totally self complementary, strong self complementary intuitionistic fuzzy graph structure and study their properties.
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Let M be an R-module, A and B are intuitionistic fuzzy submodules of M with A B. Then A is called an intuitionistic fuzzy cosmall submodule of B in M if B / A << IF M /A (= (M) / A *). In this paper an attempt has been to study... more
Let M be an R-module, A and B are intuitionistic fuzzy submodules of M with A B. Then A is called an intuitionistic fuzzy cosmall submodule of B in M if B / A << IF M /A (= (M) / A *). In this paper an attempt has been to study intuitionistic fuzzy cosmall submodules and investigate various properties of such intuitionistic fuzzy submodules. The notion of an intuitionistic fuzzy hollow module is also introduce and a relationship of this with the intuitionistic fuzzy indecomposable module and the factor module are established.
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The concept of intuitionistic fuzzy G-modules and their properties are defined and discussed by the author et al. in [16]. In this paper we develop the notion of exact sequence of intuitionistic fuzzy G-modules and study their properties.
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The main goal of this paper is to count subgroups which are isomorphic to cyclic p-group, internal direct product of two cyclic p-group or semi direct product of two cyclic p-group of the non-Abelian p-group Z p n Z p , n ≥ 2 where p may... more
The main goal of this paper is to count subgroups which are isomorphic to cyclic p-group, internal direct product of two cyclic p-group or semi direct product of two cyclic p-group of the non-Abelian p-group Z p n Z p , n ≥ 2 where p may be even or odd prime, by using simple-theoretical approach.
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In this paper,, the concept of bridge and cut vertices in an intuitionistic fuzzy graph structures (IFGS) are defined and their properties are studied. We describe the existence of bridge in an IFGS and obtain some equivalent conditions.... more
In this paper,, the concept of bridge and cut vertices in an intuitionistic fuzzy graph structures (IFGS) are defined and their properties are studied. We describe the existence of bridge in an IFGS and obtain some equivalent conditions. Also intuitionistic fuzzy bridges and intuitionistic fuzzy cut vertices are characterized using partial intuitionistic fuzzy spanning subgraph structures.. .
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In this article, we have given an explicit recursive formula for the number of intuitionistic fuzzy subgroups of a finite cyclic group
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In this paper, we have introduced the topological structure on the set of all intuitionistic fuzzy prime ideals of a ring. This topology is called the Zariski topology or the intuitionistic fuzzy prime spectrum of a ring. We have shown... more
In this paper, we have introduced the topological structure on the set of all intuitionistic fuzzy prime ideals of a ring. This topology is called the Zariski topology or the intuitionistic fuzzy prime spectrum of a ring. We have shown that this topology is always T0-space and is T1-space when R is a ring in which every prime ideal is maximal, but even in this case it is not T2-space. We have also studied a special subspace Y which is always compact and is connected if and only if 0 and 1 are the only idempotent in R. We have also shown that, when the ring R is Boolean ring, then the subspace Y is also T2space. An embedding of space X onto a subspace X* = {AX | A is f-invariant} has been established.
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In this paper we investigate the radical structure of an intuitionistic fuzzy polynomial ideal A x induced by an intuitionistic fuzzy ideal A of a ring and study its properties. Given an intuitionistic fuzzy ideal B of a ring R and a... more
In this paper we investigate the radical structure of an intuitionistic fuzzy polynomial ideal A x induced by an intuitionistic fuzzy ideal A of a ring and study its properties. Given an intuitionistic fuzzy ideal B of a ring R and a homomorphism f : R → R , we show that if f x : R[x] → R [x] is the induced homomorphism of f , that is, f x (n i=0 a i x i) = n i=0 (f (a i))x i , then f −1 x [(√ B) x ] = (f −1 (B)) x .
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In this paper, we further study the theory of intuitionistic fuzzy rings and give some new concepts such as intuitionistic fuzzy ring with operators, intuitionistic fuzzy ideal with operators , intuitionistic fuzzy quotient ring with... more
In this paper, we further study the theory of intuitionistic fuzzy rings and give some new concepts such as intuitionistic fuzzy ring with operators, intuitionistic fuzzy ideal with operators , intuitionistic fuzzy quotient ring with operators, etc. while their some elementary properties are discussed.
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In this paper, we introduce the concept of intuitionistic fuzzy small submodule with respect to an arbitrary intuitionistic fuzzy submodule of an R-module M. We derive the condition when an intuitionistic fuzzy submodule to be a small... more
In this paper, we introduce the concept of intuitionistic fuzzy small submodule with respect to an arbitrary intuitionistic fuzzy submodule of an R-module M. We derive the condition when an intuitionistic fuzzy submodule to be a small submodule with respect to another intuitionistic fuzzy submodule with the crisp small submodule of the R-module M. It is also shown that the sum of two intuitionistic fuzzy small submodules with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy submodule with respect to the same fixed intuitionistic fuzzy submodule. This result can be extended to an arbitrary sum of intuitionistic fuzzy submodules. Further, we prove that the homomorphic image of an intuitionistic fuzzy small submodule with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy small submodule with respect the homomorphic image of the fixed intuitionistic fuzzy submodule.
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In this paper, we study intuitionistic fuzzy prime submodules with the help of residual quotient subsets of ring and modules. Also a complete characterisation of an intuitionistic fuzzy prime submodule is given. A relationship between... more
In this paper, we study intuitionistic fuzzy prime submodules with the help of residual quotient subsets of ring and modules. Also a complete characterisation of an intuitionistic fuzzy prime submodule is given. A relationship between intuitionistic fuzzy prime submodule and intuitionistic fuzzy maximal submodule is established. Homeomorphic image and pre-image of intuitionistic fuzzy prime submodules are obtained.
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In this paper the concept of an intuitionistic L-fuzzy prime submodule of M is given, and some fundamental lemmas are proved. Also a characterization of an intuitionistic L-fuzzy prime submodule is given. Finally, we show that an... more
In this paper the concept of an intuitionistic L-fuzzy prime submodule of M is given, and some fundamental lemmas are proved. Also a characterization of an intuitionistic L-fuzzy prime submodule is given. Finally, we show that an intuitionistic L-fuzzy prime submodule is inherited by an R-module epimorphism.
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In this paper we introduce the notion of intuitionistic fuzzy polynomial ideal A x of a polynomial ring R[x] induced by an intuitionistic fuzzy ideal A of a ring R, and obtain an isomorphism theorem of a ring of intuitionistic fuzzy... more
In this paper we introduce the notion of intuitionistic fuzzy polynomial ideal A x of a polynomial ring R[x] induced by an intuitionistic fuzzy ideal A of a ring R, and obtain an isomorphism theorem of a ring of intuitionistic fuzzy cosets of A x. It is shown that an intuitionistic fuzzy ideal A of a ring is an intuitionistic fuzzy prime if and only if A x is an intuitionistic fuzzy prime ideal of R[x]. Moreover, we show that if A x is an intuitionistic fuzzy maximal ideal of R[x], then A is an intuitionistic fuzzy maximal ideal of R but converse is not true.
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In this article, we introduce chained-semigroups, cancellative-semigroups and obtain some equivalent conditions. Also, we prove that if is a chained-semigroup, then is an Archimedian-semigroup with no-idempotents if and only if satisfies... more
In this article, we introduce chained-semigroups, cancellative-semigroups and obtain some equivalent conditions. Also, we prove that if is a chained-semigroup, then is an Archimedian-semigroup with no-idempotents if and only if satisfies the concentric condition for every ∈. Furthermore, we prove that a cancellative Archimedian chained-semigroup is a-group if does not satisfy the concentric condition for some ∈. Finally, we prove that if is a chained-semigroup containing cancellable elements. Then, is a cancellative-semigroup provided satisfies the concentric condition for every ∈. The converse is true if is a Noetherian-semigroup without-idempotents.
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The main purpose of this paper is to investigate ordered-semihypergroups in the general terms of ordered-hyperideals. We introduce ordered (generalized) (m; n)-hyperideals in orderedsemihypergroups. Then, we characterize... more
The main purpose of this paper is to investigate ordered-semihypergroups in the general terms of ordered-hyperideals. We introduce ordered (generalized) (m; n)-hyperideals in orderedsemihypergroups. Then, we characterize ordered-semihypergroup by ordered (generalized) (0; 2)-hyperideals, ordered (generalized) (1; 2)-hyperideals and ordered (generalized) 0-minimal (0; 2)-hyperideals. Furthermore, we investigate the notion of ordered (generalized) (0; 2)-bi-hyperideals, ordered 0-(0; 2) bisimple ordered-semihypergroups and ordered 0-minimal (generalized) (0; 2)-bi-hyperideals in ordered-semihyperoups. It is proved that an ordered-semihypergroup S with a zero 0 is 0-(0; 2)-bisimple if and only if it is left 0-simple.
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In this paper, (i) we provide an explicit formula for the number of series of subgroups of group n p Z which contains a subgroup of order p and (ii) we provide an explicit formula for the the maximal chains of subgroups which contains a... more
In this paper, (i) we provide an explicit formula for the number of series of subgroups of group n p Z which contains a subgroup of order p and (ii) we provide an explicit formula for the the maximal chains of subgroups which contains a subgroup of order p and the recurrence relation techniques.
Research Interests:
Counting subgroup series of abelian p-group which contains a subgroup of order p
Research Interests:
Research Interests:
The emerging approach to computing is refer to Soft computing which is placed parallel to the remarkable ability of the human mind to reason and learn in a environment of uncertainty and imprecision. Some of it’s principle... more
The emerging approach to computing is refer to Soft computing which is placed parallel to the remarkable ability of the human mind to reason and learn in a environment of uncertainty and imprecision.
Some of it’s principle components includes:
Neural Network(NN)
Genetic Algorithm(GA)
Machine Learning (ML)
Probabilistic Reasoning(PR)
Fuzzy Logic(FL)
These methodologies (techniques) form the core of soft computing.
The main goal of soft computing is to develop intelligent machines to provide solutions to real world problems, which are not modeled, or too difficult to model mathematically.
It’s aim is to exploit the tolerance for Approximation, Uncertainty, Imprecision, and Partial Truth in order to achieve close resemblance with human like decision making.
Some of it’s principle components includes:
Neural Network(NN)
Genetic Algorithm(GA)
Machine Learning (ML)
Probabilistic Reasoning(PR)
Fuzzy Logic(FL)
These methodologies (techniques) form the core of soft computing.
The main goal of soft computing is to develop intelligent machines to provide solutions to real world problems, which are not modeled, or too difficult to model mathematically.
It’s aim is to exploit the tolerance for Approximation, Uncertainty, Imprecision, and Partial Truth in order to achieve close resemblance with human like decision making.
In this talk we introduce the notion of intuitionistic fuzzy polynomial ideal Ax of a polynomial ring R[x] induced by an intuitionistic fuzzy ideal A of a ring R. Then many properties of Ax will be discussed. We shall also establish an... more
In this talk we introduce the notion of intuitionistic fuzzy
polynomial ideal Ax of a polynomial ring R[x] induced by an
intuitionistic fuzzy ideal A of a ring R. Then many properties of Ax
will be discussed. We shall also establish an isomorphism theorem
of a ring of intuitionistic fuzzy cosets of Ax . It will be shown that an
intuitionistic fuzzy ideal A of a ring R is an intuitionistic fuzzy prime
if and only if Ax is an intuitionistic fuzzy prime ideal of R[x].
However, if Ax is an intuitionistic fuzzy maximal ideal of R[x], then
A is an intuitionistic fuzzy maximal ideal of R, but converse is not
true. We will also investigate the nil radical structure of Ax . The
homomorphic image and inverse image of an intuitionistic fuzzy
polynomial ideal Ax and nil radical of Ax when A is an intuitionistic
fuzzy prime ideal of a ring R will also be discussed.
polynomial ideal Ax of a polynomial ring R[x] induced by an
intuitionistic fuzzy ideal A of a ring R. Then many properties of Ax
will be discussed. We shall also establish an isomorphism theorem
of a ring of intuitionistic fuzzy cosets of Ax . It will be shown that an
intuitionistic fuzzy ideal A of a ring R is an intuitionistic fuzzy prime
if and only if Ax is an intuitionistic fuzzy prime ideal of R[x].
However, if Ax is an intuitionistic fuzzy maximal ideal of R[x], then
A is an intuitionistic fuzzy maximal ideal of R, but converse is not
true. We will also investigate the nil radical structure of Ax . The
homomorphic image and inverse image of an intuitionistic fuzzy
polynomial ideal Ax and nil radical of Ax when A is an intuitionistic
fuzzy prime ideal of a ring R will also be discussed.
Research Interests:
In this talk, we study the concept of residual quotient of IF subsets of ring and module and develop many properties out of these. Using the concept of residual quotient, we investigate some important characterization of annihilator of IF... more
In this talk, we study the concept of residual quotient of IF subsets of ring
and module and develop many properties out of these. Using the concept
of residual quotient, we investigate some important characterization of
annihilator of IF submodules, IF prime(primary) submodules and IF
prime(primary) decomposition.
and module and develop many properties out of these. Using the concept
of residual quotient, we investigate some important characterization of
annihilator of IF submodules, IF prime(primary) submodules and IF
prime(primary) decomposition.
Research Interests:
In this power point presentation, I try to explain the contribution of mathematics in our life and how it make our life easy and beautiful.
Research Interests:
The notion of fuzzy set was introduced by L.A. Zadeh as a generalization of the notion of classical set or crisp set. Fuzzy topological spaces were introduced by C.L. Chang and studied by many eminent authors like R. Lowen and C.K. Wong.... more
The notion of fuzzy set was introduced by L.A. Zadeh as a generalization of the notion of classical set or crisp set. Fuzzy topological spaces were introduced by C.L. Chang and studied by many eminent authors like R. Lowen and C.K. Wong. A. Rosenfeld applied the notion of fuzzy set to algebra and introduced fuzzy subgroup of a group. Shaoquan Sun introduced the notion of fuzzy Boolean subalgebra in a Boolean algebra. In this talk, we will discuss fuzzy topology by involving the Boolean algebraic structure on it and introduce the notion of Boolean algebraic fuzzy topological spaces. We will examine many properties of these spaces and obtain many results.
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Result concerning various ideals in near rings (as defined by Yakabe, Chelvam and Ganesan, Kim and Zhan and Xueling as well as some generalization thereof) that have been established over the past three decays, will be discussed. These... more
Result concerning various ideals in near rings (as defined by Yakabe, Chelvam and
Ganesan, Kim and Zhan and Xueling as well as some generalization thereof) that have been
established over the past three decays, will be discussed. These include the fuzzy ideals, fuzzy
quasi ideals and fuzzy bi-ideals in near rings. In this talk, we extend these notion to intuitionistic
fuzzy ideals, intuitionistic fuzzy quasi-ideals, intuitionistic fuzzy bi-ideals in near-rings. A relationship between various types of ideals has been established.
Ganesan, Kim and Zhan and Xueling as well as some generalization thereof) that have been
established over the past three decays, will be discussed. These include the fuzzy ideals, fuzzy
quasi ideals and fuzzy bi-ideals in near rings. In this talk, we extend these notion to intuitionistic
fuzzy ideals, intuitionistic fuzzy quasi-ideals, intuitionistic fuzzy bi-ideals in near-rings. A relationship between various types of ideals has been established.
Research Interests:
The Concept of a module over a ring is a generalization of the notion of vector space, where in the corresponding scalars are allowed to lie in an arbitrary ring. Modules also generalize the notion of abelian groups, which are modules... more
The Concept of a module over a ring is a generalization of the notion of vector space, where in the corresponding scalars are allowed to lie in an arbitrary ring. Modules also generalize the notion of abelian groups, which are modules over the ring of integers. Thus a module like a vector space, is an additive abelian group, a product is defined between the elements of a ring and the elements of the module, and this multiplication is associative (when used with multiplication in the ring) and distributive. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are widely used in algebraic geometry and algebraic topology.
In this talk , author differentiate between the ordinary or classical set theory given by George Cantor in 1876 and the Fuzzy Set Theory given by L.A.Zadeh in 1965 with the Intuitionistic fuzzy set theory given by K,T. Atanssov in 1983.... more
In this talk , author differentiate between the ordinary or classical set theory given by George Cantor in 1876 and the Fuzzy Set Theory given by L.A.Zadeh in 1965 with the Intuitionistic fuzzy set theory given by K,T. Atanssov in 1983. Author has given a detail work done in the intuitionistic fuzzy subgroups and discuss some of their importance.
Here in this talk, author give a detail about the fuzzy set theory. He also show by citing many examples the usefulness of this theory in the development of technology . He outline the various step of this fuzzy logic used in washing... more
Here in this talk, author give a detail about the fuzzy set theory. He also show by citing many examples the usefulness of this theory in the development of technology . He outline the various step of this fuzzy logic used in washing machine .
In this article, we have given an explicit recursive formula for the number of intuitionistic fuzzy subgroups of a finite cyclic group are distinct prime numbers. A method for constructing an intuitionistic fuzzy subgroup of a given group... more
In this article, we have given an explicit recursive formula for the number of intuitionistic fuzzy subgroups of a finite cyclic group are distinct prime numbers. A method for constructing an intuitionistic fuzzy subgroup of a given group in terms of double pinned flags is also proposed.
Research Interests:
Research Interests:
: Fuzzy Sets were introduced by L.A. Zadeh in 1965. One generalization of the notion of fuzzy sets was proposed by K. Atanassov in the beginning of 1983 and presented before the Seventh Scientific Session of ITKR, Sofia, June 1983. In... more
: Fuzzy Sets were introduced by L.A. Zadeh in 1965. One generalization of the notion of fuzzy sets was proposed by K. Atanassov in the beginning of 1983 and presented before the Seventh Scientific Session of ITKR, Sofia, June 1983. In addition to the degree of membership known from the fuzzy sets, here a new degree is introduced, called the degree of non-membership, with the requirement that their sum be less than or equal to 1. The complement of the sum of two degrees to 1 is regarded as a degree of uncertainty. This new extension of the fuzzy sets was named intuitionistic fuzzy set (IFS). The name of intuitionistic fuzzy sets is due George Gargov, with the motivation that their fuzzification denies the law of the excluded middle - one of the main ideas of intuitionism. Later in 1983 it turned out that the new sets allow for the definition of operators which are generalizations of the modal operators of necessity and possibility, which was the first serious result connecting IFS with classical logic and set theory.
Some of the area’s where the IFS Theory has found its important place:
Operations and relations over IFS. Algebraic research in the frame of the IFS theory ( IF groups, IF rings and Ideals , IF modules, IF fields , IF vector spaces etc. )
IF geometry, IF Analysis and IF Topology (IF numbers, IF Topology)
IF Logics (IF propositional and predicate calculus, IF modal and temporal logics, Connection between IF logics and other logical systems)
IF Approach to Artificial Intelligence(Decision making and machine learning, Neutral networks and pattern recognition, Expert system, logical programming )
IF Generalized Nets
IF Graphs
Applications of IFSs (Medicine, optimization, chemical engineering, economics, computer hardware, astronomy, sociology, biology)
Here, in this talk, I will discuss the algebraic research in the frame of the IFS theory and in particular, will introduce intuitionistic fuzzy group and explore some of their properties. Some latest development in this area will also be highlighted.
Some of the area’s where the IFS Theory has found its important place:
Operations and relations over IFS. Algebraic research in the frame of the IFS theory ( IF groups, IF rings and Ideals , IF modules, IF fields , IF vector spaces etc. )
IF geometry, IF Analysis and IF Topology (IF numbers, IF Topology)
IF Logics (IF propositional and predicate calculus, IF modal and temporal logics, Connection between IF logics and other logical systems)
IF Approach to Artificial Intelligence(Decision making and machine learning, Neutral networks and pattern recognition, Expert system, logical programming )
IF Generalized Nets
IF Graphs
Applications of IFSs (Medicine, optimization, chemical engineering, economics, computer hardware, astronomy, sociology, biology)
Here, in this talk, I will discuss the algebraic research in the frame of the IFS theory and in particular, will introduce intuitionistic fuzzy group and explore some of their properties. Some latest development in this area will also be highlighted.
