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![Invention Journal of Research Technology in Engineering & Management (IJRTEM)
ISSN: 2455-3689
www.ijrtem.com Volume 1 Issue 12-Version-2 ǁ October. 2017 ǁ PP 09-17
| Volume 1 | Issue 12 | www.ijrtem.com | 9 |
Correspondence and Isomorphism Theorems for Intuitionistic
fuzzy subgroups
1,
B Venkat Rao , 2,
K Lakshmi
1 ,2
Basic Science and Humanities Department, Aditya Institute of Technology And management, India
ABSTRACT:The aim of this paper is basically to study the First Isomorphism Theorem, Second Isomorphism
Theorem, Third Isomorphism theorem, Correspondence Theorem etc. of intuitionistic fuzzy/vague sub groups of
a crisp group.
KEY WORDS: Intutionistic fuzzy or Vague Subset, Intutionistic fuzzy Image, Intutionistic fuzzy Inverse Image,
Intutionistic fuzzy/vague sub (normal) group, Correspondence Theorem, First (Second, Third) Isomorphism
Theorem.
I. INTRODUCTION
Zadeh introduced the concept of Fuzzy Subset of a set X , as a function µ from X to the closed interval [0,1] of
real numbers. He defined µ as the membership function which assigns to each member x of X its membership
value, µx in [0, 1]. In 1983, Atanassov[1] introduced intutionistic fuzzy subset extending the concept of fuzzy
subset, defined as A = (µA , νA ), where µA , νA are functions from the set X to the closed interval [0, 1] of real
numbers such that for each x ∈ X , µ(x) + ν (x) ≤ 1, where µA is called the membership function of A and νA is
called the non-membership function of A. Gau and Buehrer[6] in 1993, introduced the same concept with a
different name called Vague subset. Thus intutionistic fuzzy subset of a set or vague subset of a set are same. In
1989, Biswas[7] introduced the notion of intutionistic fuzzy subgroup or if/v-subgroup of a group and studied
some of its properties. In 2004, Hur-Jang-Kang[15] developed the concepts of if/v-normal subgroup of a group
and Huretal.[10,11,16] continued their studies of the same. In Huretal.[16], they established a one-one
correspondence between, if/v-normal subgroups and if/v-congruence’s. In 2003, Banergee-Basnet[6] explored
the notions of if/v-subrings and if/v-ideals of a ring. The same year Hur-Jang-Kang[10] introduced and studied
the notion if/v- subring of a ring. In Huretal.[17,18] continued their studies of if/v-ideals. In Huretal.[18], they
introduced and studied the notions of if/v-prime ideals, if/v-completely prime ideals and if/v-weakly completely
prime ideals. With regard to generalizations of topological structures on to the intuitionitic fuzzy/vague sets, one
can refer to the bibliography at the end of this paper for several interesting articles; we are not particularly
mentioning any one of them as our emphasis in this paper now is mainly the study of generalizations of
algebraic structures on to the intuitionitic fuzzy/vague sets. Coming back to the studies of intuitionistic
fuzzy/vague subgroups of a group, Feng[8] and Palaniappan etal.[22] initiated the study intuitionistic L-fuzzy/L-
vague subgroups of a group. Looking at all these and other papers, one thing which becomes evident is that
various (lattice) algebraic properties of if/v-images and if/v-inverse images which, incidentally, not only play a
crucial role in the study of both Intuitionistic L-Fuzzy Algebra and Intuitionistic L-Fuzzy Topology but also are
necessary for the individual/exclusive development of iv/f-Set Theory, are not yet studied, although these
concepts were existing since long. In one paper, we made an exclusive and somewhat detailed study of these
(lattice) algebraic properties of if/v-images and if/v-inverse images under a crisp map for if/v- subsets that take
their truth values in the closed interval of real numbers, [0,1]. In this paper, we apply parts of the above Theory
to generalize such results as First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism
theorem, Correspondence Theorem etc.. For any set X, the set of all if-subsets of X be denoted by A(X ). By
defining, for any pair of if-subsets A = (µA ,νA ) and B = (µB , νB ) of X , A ≤ B iff µA ≤ µB and νB ≤ νA , A(X )
becomes a complete infinitely distributive lattice. In this case for any family (Ai )i∈I of if-subsets of X , (∨i∈I Ai
)x = ∨i∈I Ai x and (∧i∈I Ai )x = ∧i∈I Ai x. For any set X , one can naturally associate, with X , the if-subset (µ(X) ,
ν(X) ) = (1, 0), where 1 is the constant map assuming the value 1 for each x ∈ X and 0 is the constant map
assuming the value 0 for each x ∈ X , which turns out to be the largest element in A(X ). Observe that then, the
if-empty subset φ of X gets naturally associated with the if-subset (µφ ,νφ ) = (0, 1), which turns out to be the
least element in A(X ). Let A = (µA ,νA ) be an if-subset of X . Then (νA , µA ) is also an if-subset of X , thus for
any if-subset A = (µA , νA ) the if-complement of A, denoted by Ac is defined by (νA , µA ). Observe that Ac
= X
− A = X ∧Ac
. Throughout this paper the capital letters X, Y, Z stand for arbitrary but fixed (crisp) sets, the small
letters f, g stand for arbitrary but fixed (crisp) maps f : X → Y and g : Y → Z , the capital letters A, B, C, D, E,
F together with their suffixes stand for if-subsets and the capital letters I and J stand for the index sets. Ingeneral](https://image.slidesharecdn.com/b0112209017-180131073146/75/Correspondence-and-Isomorphism-Theorems-for-Intuitionistic-fuzzy-subgroups-1-2048.jpg)
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Correspondence and Isomorphism Theorems for Intuitionistic fuzzy subgroups
- 1. Invention Journal ofResearch Technology in Engineering & Management (IJRTEM) ISSN: 2455-3689 www.ijrtem.com Volume 1 Issue 12-Version-2 ǁ October. 2017 ǁ PP 09-17 | Volume 1 | Issue 12 | www.ijrtem.com | 9 | Correspondence and Isomorphism Theorems for Intuitionistic fuzzy subgroups 1, B Venkat Rao , 2, K Lakshmi 1 ,2 Basic Science and Humanities Department, Aditya Institute of Technology And management, India ABSTRACT:The aim of this paper is basically to study the First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism theorem, Correspondence Theorem etc. of intuitionistic fuzzy/vague sub groups of a crisp group. KEY WORDS: Intutionistic fuzzy or Vague Subset, Intutionistic fuzzy Image, Intutionistic fuzzy Inverse Image, Intutionistic fuzzy/vague sub (normal) group, Correspondence Theorem, First (Second, Third) Isomorphism Theorem. I. INTRODUCTION Zadeh introduced the concept of Fuzzy Subset of a set X , as a function µ from X to the closed interval [0,1] of real numbers. He defined µ as the membership function which assigns to each member x of X its membership value, µx in [0, 1]. In 1983, Atanassov[1] introduced intutionistic fuzzy subset extending the concept of fuzzy subset, defined as A = (µA , νA ), where µA , νA are functions from the set X to the closed interval [0, 1] of real numbers such that for each x ∈ X , µ(x) + ν (x) ≤ 1, where µA is called the membership function of A and νA is called the non-membership function of A. Gau and Buehrer[6] in 1993, introduced the same concept with a different name called Vague subset. Thus intutionistic fuzzy subset of a set or vague subset of a set are same. In 1989, Biswas[7] introduced the notion of intutionistic fuzzy subgroup or if/v-subgroup of a group and studied some of its properties. In 2004, Hur-Jang-Kang[15] developed the concepts of if/v-normal subgroup of a group and Huretal.[10,11,16] continued their studies of the same. In Huretal.[16], they established a one-one correspondence between, if/v-normal subgroups and if/v-congruence’s. In 2003, Banergee-Basnet[6] explored the notions of if/v-subrings and if/v-ideals of a ring. The same year Hur-Jang-Kang[10] introduced and studied the notion if/v- subring of a ring. In Huretal.[17,18] continued their studies of if/v-ideals. In Huretal.[18], they introduced and studied the notions of if/v-prime ideals, if/v-completely prime ideals and if/v-weakly completely prime ideals. With regard to generalizations of topological structures on to the intuitionitic fuzzy/vague sets, one can refer to the bibliography at the end of this paper for several interesting articles; we are not particularly mentioning any one of them as our emphasis in this paper now is mainly the study of generalizations of algebraic structures on to the intuitionitic fuzzy/vague sets. Coming back to the studies of intuitionistic fuzzy/vague subgroups of a group, Feng[8] and Palaniappan etal.[22] initiated the study intuitionistic L-fuzzy/L- vague subgroups of a group. Looking at all these and other papers, one thing which becomes evident is that various (lattice) algebraic properties of if/v-images and if/v-inverse images which, incidentally, not only play a crucial role in the study of both Intuitionistic L-Fuzzy Algebra and Intuitionistic L-Fuzzy Topology but also are necessary for the individual/exclusive development of iv/f-Set Theory, are not yet studied, although these concepts were existing since long. In one paper, we made an exclusive and somewhat detailed study of these (lattice) algebraic properties of if/v-images and if/v-inverse images under a crisp map for if/v- subsets that take their truth values in the closed interval of real numbers, [0,1]. In this paper, we apply parts of the above Theory to generalize such results as First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism theorem, Correspondence Theorem etc.. For any set X, the set of all if-subsets of X be denoted by A(X ). By defining, for any pair of if-subsets A = (µA ,νA ) and B = (µB , νB ) of X , A ≤ B iff µA ≤ µB and νB ≤ νA , A(X ) becomes a complete infinitely distributive lattice. In this case for any family (Ai )i∈I of if-subsets of X , (∨i∈I Ai )x = ∨i∈I Ai x and (∧i∈I Ai )x = ∧i∈I Ai x. For any set X , one can naturally associate, with X , the if-subset (µ(X) , ν(X) ) = (1, 0), where 1 is the constant map assuming the value 1 for each x ∈ X and 0 is the constant map assuming the value 0 for each x ∈ X , which turns out to be the largest element in A(X ). Observe that then, the if-empty subset φ of X gets naturally associated with the if-subset (µφ ,νφ ) = (0, 1), which turns out to be the least element in A(X ). Let A = (µA ,νA ) be an if-subset of X . Then (νA , µA ) is also an if-subset of X , thus for any if-subset A = (µA , νA ) the if-complement of A, denoted by Ac is defined by (νA , µA ). Observe that Ac = X − A = X ∧Ac . Throughout this paper the capital letters X, Y, Z stand for arbitrary but fixed (crisp) sets, the small letters f, g stand for arbitrary but fixed (crisp) maps f : X → Y and g : Y → Z , the capital letters A, B, C, D, E, F together with their suffixes stand for if-subsets and the capital letters I and J stand for the index sets. Ingeneral
- 2. Correspondence and IsomorphismTheorems… | Volume 1 | Issue 12 | www.ijrtem.com | 10 | whenever P is an if-subset of a set X , always µP and νP denote the membership and non-membership function of the if-subset P respectively. Also we frequently use the standard convention that ∨φ = 0L and ∧φ = 1L . II. Homomorphisms And Isomorphisms Of Intuitionistic Fuzzy/ Vague-Subgroups In this section, we begin with a result on if/v-(inverse) image of an if/v-normal sub- group. Then we go on to introduce various notions of if/v-homo(iso)morphisms. Finally, in the end we show that ηA = B implies ηA∗ ≤ B∗ , under certain conditions. Theorem 2.1: For any pair of groups G and H and for any pairs of if/v- subgroups A, B of G and C , D of H , for any group homomorphism η: G → H such that A is an if/v- normal subgroup of B and C is a if/v-normal subgroup of D, 1. ηA is an if/v-normal subgroup of ηB. 2. η−1 C is an if/v-normal subgroup of η−1 D. Proof: (1): Since A, B are if/v-subgroups of G, ηA, ηB are if/v-subgroups of H . Further, A ≤ B implies ηA ≤ ηB. So ηA is an if/v-subgroup of ηB. Since (a) ηu = x, ηv = y impliy uvu−1 ∈ η−1 (xyx−1 ) and (b) A is an if/v-normal subgroup of B, we get that Hence ηA is an if/v-normal subgroup of ηB.
- 3. Correspondence and IsomorphismTheorems… | Volume 1 | Issue 12 | www.ijrtem.com | 11 | ⊆ η(A∗).Therefore η(A∗) = B∗.NA III. Correspondence and Isomorphism Theorems for Intuitionistic Fuzzy/Vague-Subgroups In this section, first we generalize the First Isomorphism Theorem. Then we move onto the Correspondence Theorem, the Second Isomorphism Theorem and the Third Isomorphism Theorem in that order. First isomorphism theorem :
- 4. Correspondence and IsomorphismTheorems… | Volume 1 | Issue 12 | www.ijrtem.com | 12 |
- 5. Correspondence and IsomorphismTheorems… | Volume 1 | Issue 12 | www.ijrtem.com | 13 | Correspondence Theorem Proof: Since the set of all if/v-(normal) subgroups of G is a complete lattice, the proof follows from the previous theorem. Second Isomorphism Theorem
- 6. Correspondence and IsomorphismTheorems… | Volume 1 | Issue 12 | www.ijrtem.com | 14 | Now we show that the above η, in fact, defines a if/v-weak isomorphism be
- 7. Correspondence and IsomorphismTheorems… | Volume 1 | Issue 12 | www.ijrtem.com | 15 | Third Isomorphism Theorem
- 8. Correspondence and IsomorphismTheorems… | Volume 1 | Issue 12 | www.ijrtem.com | 16 | REFERENCES 1. Atanassov, K., Intuitionistic fuzzy sets, in: V. Sgurev, Ed., VII ITKR’s Session, Sofia, June 1983 (Central Sci. and Techn. Library, Bulg. Academy of Sciences, 1984). 2. Atanassov, K. and Stoeva, S., Intuitionistic fuzzy sets, in: Polish Symp. On Interval and Fuzzy Mathematics, Poznan (Aug. 1983) 23 - 26. 3. Atanassov, K. and Stoeva, S., Intuitionistic L-fuzzy sets, in: R. Trappl, Ed., Cybernetics and Systems Research 2 (Elsevier Sci. Publ., Amsterdam, 1984 ) 539 - 540.
- 9. Correspondence and IsomorphismTheorems… | Volume 1 | Issue 12 | www.ijrtem.com | 17 | 4. Atanassov, K., Intuitionistic fuzzy relations, in: L. Antonov, Ed., III International School ”Automation and Scientific Instrumentation”, Varna (Oct, 1984) 56 - 57. 5. Atanassov, K., New Operations defined over the Intuitionistic fuzzy sets, in: Fuzzy Sets and Systems 61 (1994) 137 - 142, North - Holland. 6. Banerjee B. and Basnet D.,Kr., Intuitionistic fuzzy subrings and ideals, Journal of Fuzzy Mathematics Vol.11(1)(2003), 139-155. 7. Biswas R., Intuitionistic fuzzy subgroups, Mathematical Forum, Vol.10(1989), 37-46. 8. Feng Y., Intuitionistic L-fuzzy groups, Preprint, univ-savoie.fr 8. Gau, W.L. and Buehrer D.J., Vague Sets, IEEE Transactions on Systems, Man and Cybernetics, Vol. 23, 1993, , 610-614. 9. Gurcay H., Oker D.C. and Haydar A. Es, On fuzzy continuity in intuitionistic fuzzy topological spaces, Journal of Fuzzy Mathematics, Vol.5(1997), 365-378. 10. Hur K., Kang H.W. and Song H.K., Intuitionistic fuzzy subgroups and subrings, Honam Mathematical Journal Vol.25(1)(2003), 19-41. 11. Hur K., Jang S.Y. and Kang H.W., Intuitionistic fuzzy subgroups and cosets, Honam Mathematical Journal, Vol.26(1)(2004), 17-41. 12. Hur K., Jun Y.B. and Ryou J.H., Intuitionistic fuzzy topological groups, Honam Mathematical Journal, Vol.26(2)(2004), 163-192. 13. Hur K, Kim J.H. and Ryou J.H., Intuitionistic fuzzy topological spaces, Journal of Korea Society for Mathematical Education, Series B: Pure and Applied Mathematics, Vol.11(3)(2004), 243-265. 14. Hur K., Kim K.J. and Song H.K., Intuitionistic fuzzy ideals and bi-ideals, Honam Mathematical Journal, Vol.26(3)(2004), 309-330. 15. Hur K, Jang S.Y. and Kang H.W., Intuitionistic fuzzy normal subgroups and intuitionistic fuzzy cosets, Honam Mathematical Journal, Vol.26(4)(2004), 559-587. 16. Hur K., Kim S.R. and Lim P.K. Lim, Intuitionistic fuzzy normal subgroup 3 Correspondence And Isomorphism Theorems For Intuitionistic Fuzzy/Vague-Subgroups 11 and intuitionistic fuzzy-congruences, International Journal of Fuzzy Logic and Intelligent Systems, 17. Vol.9(1)(2009), 53-58.(we prove that every intuitionistic fuzzy congruence determines an intuitionistic fuzzy normal subgroup. Conversely, given an intuitionistic fuzzy normal subgroup, we can associate an intuitionistic fuzzy congruence.) 18. Hur K., Jang S.Y. and Kang H.W., intuitionistic fuzzy ideals of a ring, Journal of Korean Society of Mathematics Education, Series B-PAM, Vol12(3)(2005), 193-209. 19. Lee S.J. and Lee E.P., The category of intuitionistic fuzzy topological spaces, Bullettin of Korean Mathematical Society, Vol.37(1)(2000), 63-76. 20. Oker D.C., An introduction to intuitionistic fuzzy topological spaces, Fuzzy Set and Systems, Vol.88(1997), 81-89. 21. Oker D.C, and HaydarA.Es, On fuzzy compactness in intuitionistic fuzzy topological spaces, Journal of Fuzzy Mathematics, Vol.3(1995), 899-909. 22. Palaniappan N., Naganathan S. and Arjunan K., A study on Intutionistic L-fuzzy subgroups, Applied Mathematical Science, Vol.3(53)(2009), 2619-2624. 23. Ramakrishna N., Vague Normal Groups, International Journal of Computational Cognition, Vol.6(2)(2008), 10-13. 24. Vasanti.G, About the images and inverse images of Intuitionistic Or Vague fuzzy subsets, Bulletin of Society for Mathematical Services and Standards, vol. 1, No.3, March 2012, pp. 01-05, ISSN: 2277-8020. 25. Vasanti. G, More On L-Intuitionist Fuzzy Or L-Vague Quotient Rings, Advances In Fuzzy Sets and Systems, Vol.No. 16, No.2, Dec 2013,( Online Mar 2014), Pages 65-91. ISSN No: 0973-421X. 26. Laxmi.K, Vasanti.G, Some (Lattice) Algebraic Properties For Fuzzy (Inverse) Images Of Zadeh Fuzzy Subsets - Bulletin of Mathematical Science and Applications from Switzerland, Vol.3, No.3, Pages 135-147, August 2014, ISSN: 2278-9634. 27. Laxmi.K, Vasanti.G, Note on Intutionistic Fuzzy (Normal) Subgroups or vague (normal ) Subgroups, Global Journal of Science Frontier Research, Volume No 15 Issue 2, Feb 2015, pages 37-52, Online ISSN: 2249-4626 Print ISSN: 0975- 5896. 28. Vasanti.G, Correspondence And Isomorphism Theorems For L-Intuitionistic Or L-vague sub Rings, accepted in Global Journal of Pure and Applied Mathematics (GJPAM). 29. Wang J. and Lin X., Intuitionistic Fuzzy Ideals with thresholds (α,β) of Rings, International Mathematical Forum, Vol.4(23)(2009), 1119-1127.