J. Comp. & Math. Sci. Vol. 1(5), 592-597 (2010)
On Fuzzy Representations of Fuzzy G-Modules THAMPY ABRAHAM1 and SOURIAR SEBASTIAN2 1 Department of Mathematics, St. Peter's College, Kolenchery, Kerala - 682 311, India E-mail: thampyabraham2003@yahoo.co.in 2
Rev. Dr. A.O. Konnully Memorial Research Centre, Department of Mathematics, St. Albert's College, Ernakulam, Kerala - 682 018, India.
ABSTRACTS In this paper we study fuzzy representations of a fuzzy G- module of a G-module M/N onto a fuzzy G-module of a general linear space GL(V). This transformation is done on the basis of G-module representation theory. We also prove a fundamental theorem of Gmodule fuzzy representations. Keywords: G-module, fuzzy G-module, fuzzy homomorphism, fuzzy isomorphism, fuzzy representation.
INTRODUCTION The notion of fuzzy subsets was introduced by L.A. Zadeh7 in 1965. In 1971, Rosenfeld4 defined the fuzzy subgroups and gave some of its properties. Mathematicians like Abu Osman, Katsaras and Liu and Gu Wenxiang have studied fuzzy versions of various algebraic structures. G. Frobenius developed the theory of group representations at the end
of the 19th century. The theory of Gmodules originated in the 20 th century. Representation theory was developed on the basis of embedding a group G in to a linear group GL (V). As a continuation of the authors' work in6 to 11, in this paper, we prove a theorem related to the fuzzy homomorphism and a fundamental theorem for G-module fuzzy representations.
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
Thampy Abraham et al., J. Comp. & Math. Sci. Vol. 1(5), 592-597 (2010)
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Definition [1] Let G be a finite group. A vector space M over a field K (a subfield of C) is called a G-module if for every g G and m M, a product (called the right action of G on M) m.g M which satisfies the following axioms. 1. m.1G = m m M (1G being the identify of G) 2. m. (g. h) = (m.g). h m M, g, hG 3. (k1 m1 + k2 m2). G = k1 (m1. g) + k2 (m2. g), k1, k2 K, m1, m2 M & g G. In a similar manner the left action of G on M can be defined. 1.2. Definition [1] Let M and M* be G-modules. A mapping :MM* is a G-module homomorphism if 1. (k1 m1 + k2m2) = k1 (m1) + k2 (m2) 2. (gm) = g (m), k1, k2 K, m, m1, m2 M & g G. 1.3. Definition [1] Let M be a G-module. A subspace N of M is a G-submodule if N is also a Gmodule under the action of G. 1.4. Definition [3] Let G be a finite group and M be a G-module over K, which is a subfield of C. Then a fuzzy G-module on M is a fuzzy subset µ of M such that. i) µ (ax + by) > µ (x) µ (y), a, bK & x, y M ii) µ (gm) > µ (m), g G, m M It may be recalled that by a fuzzy subset of M we mean a function µ:M [0,1].
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Definition [1] Let G be a group and M be a vector space over a field K. A linear representation of G with representation space M is a homomorphism of G into GL(M), where GL (M) is the group of units in Homk (M, M) called the general linear group GL (M). 1.6 Definition [3] Let G and G1 be groups. Let µ be a fuzzy group on G and be a fuzzy group on G1. Let f be a group homomorphism of G onto G1. Then f is called a weak fuzzy homomorphism of µ into if f (µ) . The homomorphism f is a fuzzy homomorphism of µ onto f (µ ) =. We say that µ is fuzzy homomorphic to and we write µ . Let f : GG1 be an isomorphism. Then f is called a weak fuzzy isomorphism f (µ) and f is a fuzzy isomorphism if f (µ) = . 1.7 Definition [5] Let G be a group and M be vector space over K and T: GG 1 be a representation of G in M. Let µ be a fuzzy group on G and be a fuzzy group on the range of T. Then the representation T is a fuzzy representation if T is a fuzzy homomorphism of µ onto . 1.8 Definition [2, 3] Let f be a function defined from X to Y. The image of a fuzzy subset µ on X under 'f' is the fuzzy subset f (µ ) of Y defined by f ( µ) (y) = {µ (x) | x f–1 (y)}, y Y. The pre-image of the fuzzy subset
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on Y under 'f' is the fuzzy subset on Y under ‘f‘ is the fuzzy subset f –1 () of X defined by f –1 () (x) = { f (x) }, x X. 1.9 Definition [3] Let µ be a fuzzy subgroup of G and N be a normal subgroup of G. on G/N is defined as [x] = {µ(z) | [x] }, x G, where [x] denotes the coset xN. Then is a fuzzy subgroup of G/N. 1.10. A fundamental theorem of fuzzy representations (see [11]) Let G be a group and M be a vector space over a field K. If T : G GL (M) is a fuzzy representation of G then : G/NGL (M) is a fuzzy representation of G/N where N is a normal subgroup of G 1.11. Definition Let M and M* be G-modules Let T be a Gmodule homomorphism of M into M*. Let µ be a fuzzy G-module of M and be a fuzzy G-module of M*. Then T is called a G-module fuzzy homomorphism of µ onto if T (µ) =, written as µ . If T (µ) , T is called a weak fuzzy G-module homomorphism. If T:MM* is a G-module isomorphism and T (µ) = then T is a fuzzy G-module isomorphism of µ onto . If T (µ) then T is called a weak fuzzy Gmodule isomorphism of µ into . 1.12 Example Let G = { 1, -1 }, M = R4 over R and M* = R. Let { e1, e2, e3, e4} be the standard basis for M. Define µ : M [0,1] by µ (k1 e1 + k2 e2 + k3 e3 + k4 e4)
= 1 if ki = 0, i = 1/2 if k1 0, k2 = k3 = k4 = 0 = 1/3 if k2 0, k3 = k4 = 0 = 1/4 if k3 0, k4 = 0 = 1/5 if k4 0 Then µ is a fuzzy G-module of M. Define on R by (0) = 1, (x) = ½ if x 0. It can be verified that is a fuzzy G-module on R. Define f : R4 R by f (x) = 4i=1 xi, x = (x1 x2, x3, x4) For a, b R and x, y M, f(ax + by) = f { a (x1 x2, x3, x4) + b ( y1, y2,y3, y4)} = f {(a x1 + by1, ax2 + by2, ax3 + by3, ax4 + by4)} = 4i=1 (axi + byi) = a4i xi + b4i yi = a f (x) + b f (y) For g G & x M, f(gx) = f {g (x1 x2, x3, x4) } = 4i = 1 g x i = g 4i xi = g { f (x) } f is a G -module homomorphism and f (µ) (0) = {µ(x) | x f–1 (0) } = 1 For w 0, f (µ) (w) = {µ (x) | x f –1 (w) } = 1/2 f (µ) = . f is a G-module fuzzy homomorphism of µ onto v. 1.13 Definition Let G be a finite group and M and V be G-modules over K.Then T: M GL (V) is called a G-module representation if T is a G-module homomorphism of M into GL (V). 1.14 Definition Let G be a finite group and let M
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
Thampy Abraham et al., J. Comp. & Math. Sci. Vol. 1(5), 592-597 (2010)
and V be G-modules over K. Let T be a Gmodule homomorphism of M into GL (V). Let µ be a fuzzy G-module of M and be a fuzzy G-module of T (M). Then T is called a G-module fuzzy representation if T is a G-module fuzzy homomorphism of µ onto . 1.15 Example Let G = { 1, -1}. Let M = C and V be any G-module over R. Define T : MGL (V) by T(m) = Tm where Tm : VV is defined by Tm () = mv for m M and V. We can show that T is a Gmodule homomorphism. For k1, k2, R and m1, m2M, T (k1 m1 + k2m2) = Tk1m1 + k2m2 Tk1m1 +k2m2 () = (k1m1 + k2 m2) () = k1 (m1 ) + k2 (m2 ) = k1 Tm1 () + k2 Tm2 () = (k1 Tm1 + k2 Tm2) () T k1m1+ k2 m2 = k1 Tm1 + k2 Tm2 T (k1m1+ k2 m2) = k1 T (m1) + k2 T (m2) For g G and m M, T (gm) = Tgm Tgm () = (gm) = g (m) = g Tm () = (gTm) () Tgm = g Tm . ie, T (gm) = g T (m). T is a G-module homomorphism Define µ on M by (x + iy) = 1 if x = y = 0 = 1/2 if x 0, y = 0 = 1/3 if y 0 It can be verified that µ is a fuzzy G-module of M Define on T (m) by (Tm) = (Tx + iy) = 1 if x = y = 0 = 1/2 if x 0, y = 0 = 1/3 if y 0
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For k1, k2 R and Tm1, Tm2 T (M), ( k1 Tm1 + k2 Tm2) = (Tk1m1 + k2m2) Let m1 = x1 + iy1 and m2 = x2 + iy2.Then k1 m1 + k2 m2 = (k1 x1 + k2 x2) + i (k1 y1 + k2 y 2) If x1 = x2 = y1 = y2 = 0, (Tk1m1 + k2m2) = 1 If x1 0 or x2 0, y1 = 0, y2 = 0, (Tk1m1 + k2m2) = 1/2 If y1 0 or y2 0, (Tk1m1 + k2m2) = 1/3 Then v is a fuzzy G-module on T(M) T (µ) (Tx+iy) = {µ (x + iy) | x + iy T–1 (Tx + iy) T (µ) (Tm) = { (z) z T-1 (Tm) } = { µ (z) | T (z) = Tm} = { µ (m) |T (m) = Tm} = { µ (x + iy) | T (x+iy) = Tx + iy} = 1 when x = 0, y = 0 = 1/2 when x 0, y = 0 = 1/3 when y 0 = (Tm) T (µ)=. T is a G-module fuzzy representation of µ onto 1.16 Proposition [1] If M is a G-module and N is a Gsubmodule of M, then M/N is a G-module 1.17 Definition Let G be a finite group. Let M be a G-module over K and N be a G-submodule of M. Let µ be a fuzzy G-module on M. Then the fuzzy subset on M/N, defined by [x] = {µ (z) | z [x], x M}, where [x] = x + N, is a fuzzy G-module on M/N. The fuzzy G-module is called the quotient fuzzy G-module or factor fuzzy G-module of the fuzzy G-module µ of M
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relative to the G-submodule N. Proof For a, b k and [x], [y] M/N, {a [x] + b [y] = { [ax + Gy]} > [x] ^ [y] > [x] {g [x]} = [gx] > [x] is a fuzzy G-module on M/N ? 1.18 Theorem Let G be a finite group and M & M* be G-modules. Let µ be a fuzzy G-module of M and be a fuzzy G-module of T (M). Let T be a G-module fuzzy homomorphism of µ onto . Then : M/NM* is a Gmodule fuzzy homomorphism of onto where is a fuzzy G-module on M/N and N is a G-submodule of M. Proof Given that T is a G-module fuzzy homomorphism of µ onto . T (µ) = . Now we have to prove that : M/N M* is a Gmodule fuzzy homomorphism of onto . i.e., to show that is a G-module homomorphism of M/N into M* and () = . Define : M/NM* by [x] = T (x), x M where [x] = x + N. Then is a Gmodule homomorphism if { a [x] + b [y] = a [x] + b [y] and { g [x]} = g [x], [x], [y] M/N , a, b K & g G. {a [x] + b [y] } = { [ax + by] } = T (ax + by) = a T (x) + b T (y) = a [x] + b [y] {g [x] } = [gx] = T (gx)=g T (x) = g [x] is a G-module homomorphism. Now to show that () =
() () = { [x] | [x] –1(y),y (M/N)} = { [x] | [x] = y, y T (M) } = {µ { (z) |z [x] T (x) = yT (M)} = {µ(z) z [x], T (x) = y T (M), x M} = T (µ ) (y) () = T (µ) = is a G-module fuzzy homomorphism of onto 1.19 Theorem Let G be a finite group, M and V be G-modules over K. Let T : M GL (V) be a G-module fuzzy representation of µ onto where µ is a fuzzy G-module of M and is a fuzzy G-module of T (M). Then : M/ NGL (V) is a G-module fuzzy representation of onto where is a fuzzy G-module of M/N. Proof Given that T is G-module fuzzy representation of µ onto where µ is a fuzzy G-module of M and is a fuzzy Gmodule of T (M). We have to show that : M/NGL (V) is a G-module fuzzy representation of onto where is a fuzzy G-module of M/N, N being a G-submodule of M.We know that GL(V) is a G-module. Hence by theorem 3.15, is a G-module homomorphism of µ onto . Now to show that () = . () (Tx) = { [x] | [x] –1 (Tx)} = { [x] | [x] = Tx} = { {µ (z)|z [x]}, [x] = T [x], x M} = {µ (z) | z [x], [x] = Tx, x M} = T (µ ) (Tx)
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
Thampy Abraham et al., J. Comp. & Math. Sci. Vol. 1(5), 592-597 (2010)
() = T (µ ) = is a G-module fuzzy representation of onto 1.20 Example Let G = {1, -1}, M = C and M* = R. Then M and M* are G-modules over R. Define f : MM* by f (x + iy) = x + y. Then f is a G-module homomorphism. Define µ on M by µ (x + iy) = 1 if x = y = 0 = 1/2 if x 0, y = 0 = 1/3 if y 0 Define on T (M) by (0) = 1 and (z) = ½ when z 0. Then µ and are fuzzy Gmodules on M and T (M) respectively. Consider any G-submodule R of M and let M/R = {u + R | u M}. Then M/R is a Gmodule. Define : M/RR by {u + R} = f(u),uM.Then is a G-module homomorphism of M/R into R. Let : M/R[0,1] be defined by [u] = {u + R }, u M = v { µ (z) | z [u], u M} () (0) = { [u] | [u] –1 (0) } = 1 For r 0, ( ) ( r) = { [u] | [u] –1 ( r) } = 1/2. () = . is a G-module fuzzy representation of onto . REFERENCES 1. C ha rl e s W. Cu rt i es an d Irv in g Reiner, Representation Theory of finite groups and Associative Algevras, INC, (1962). 2. George J. Klir and Bo Yuan, Fuzzy sets and Fuzzy logic, Prentice-Hall of India (2000).
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Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)