LEVEL SET OF INTUITIONTISTIC FUZZY SUBHEMIRINGS OF A HEMIRING

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1 LEVEL SET OF INTUITIONTISTIC FUZZY SUBHEMIRINGS OF HEMIRING N. NITH ssstant Proessor n Mathematcs, Peryar Unversty PG Extn Centre, Dharmapur Emal : anthaarenu@gmal.com BSTRCT: In ths paper, we made an attempt to study the algebrac nature o an ntutonstc uzzy subhemrngs o a hemrng MS Subject classcaton: 03F55, 06D72, KEY WORDS: Fuzzy set, ntutonstc uzzy set, ntutonstc uzzy subhemrng, homomorphsm, anthomomorphsm, level set. INTRODUCTION: There are many concepts o unversal algebras generalzng an assocatve rng R ; + ;. ). Some o them n partcular, nearrngs and several knds o semrngs have been proven very useul. Semrngs are algebras R ; + ;. ) share the same propertes as a rng except that R ; + ) s assumed to be a semgroup rather than a commutatve group. Semrngs appear n a natural manner n some applcatons to the theory o automata and ormal languages. n algebra R ; +,.) s sad to be a semrng R ; +) and R ;.) are semgroups satsyng a. b+c ) = a. b+a. c and b+c).a = b. a+c. a or all a, b and c n R. semrng R s sad to be addtvely commutatve a+b = b+a or all a, b and c n R. semrng R may have an dentty 1, dened by 1. a = a = a. 1 and a zero 0, dened by 0+a = a = a+0 and a.0 = 0 = 0.a or all a n R. semrng R s sad to be a hemrng t s an addtvely commutatve wth zero. ter the ntroducton o uzzy sets by L..Zadeh[13], several researchers explored on the generalzaton o the concept o uzzy sets. The concept o ntutonstc uzzy subsets IFS) was ntroduced by K.T.tanassov[4], as a generalzaton o the noton o uzzy set. The noton o Fuzzy let h-deals n hemrngs wth respect to a s-norm was ntroduced n [2].The noton o homomorphsm and anthomomorphsm o uzzy and ant-uzzy deal o a rng was ntroduced by N.Palanappan & K.rjunan. In ths paper, we ntroduce the some Theorems n ntutonstc uzzy subhemrng o a hemrng. 1.PRELIMINRIES: 1.1Denton: Let X be a non-empty set. uzzy subset o X s a uncton : X [0, 1]. 1.2 Denton: n ntutonstc uzzy subset IFS) n X s dened as an object o the orm = { < x, x), x) > / xx }, where : X [0,1] and : X [0,1] dene the degree o membershp and the degree o nonmembershp o the element xx respectvely and or every xx satsyng 0 x) + x) Denton: Let R be a hemrng. n ntutonstc uzzy subset o R s sad to be an ntutonstc uzzy subhemrngifshr) o R t satses the ollowng condtons: ) x + y) mn{ x), y) }, ) xy) mn{ x), y) }, ) x + y) max{ x), y) }, v) xy) max{ x), y) }, or all x and y n R. 1.4 Denton: I R, +,. ) and R, +,. ) are any two hemrngs, then the uncton : R R s called a homomorphsm x+y) = x)+y) and xy)=x)y), or all x and y n R. 1.5 Denton: I R, +,. ) and R, +,. ) are any two hemrngs, then the uncton : R R s called an ant-homomorphsm x+y) = y)+x) and xy)=y)x), or all x and y n R. 1.6 Denton: Let R and R be any two hemrngs. Let : R R be any uncton and let be an ntutonstc uzzy subhemrng n R, V be an ntutonstc uzzy subhemrng n R ) = R, dened by V y) = sup = n x 1 y) x 1 y) x) and V y) x), or all x n R and y n R. Then s called a premage o V under and s denoted by -1 V). 1.7 Denton: Let be an ntutonstc uzzy subset o a set X. For and n [0,1], the level subset o s the set,, ) ={xx : x) and x) }. Ths s called an ntutonstc uzzy level subset o. 2. LEVEL SET OF INTUITIONISTIC FUZZY SUBHEMIRING OF HEMIRING. 2.1 Theorem: Let be an ntutonstc uzzy subhemrng o a hemrng R. Then or and n [0,1] such that e) and e),, ) s a level subhemrng o R. Proo: For all x and y n, ), we have, x) and x) and y) and y). Now, x + y) 124

2 mn { x), y) } mn {, } =. Whch mples that x + y ). nd, xy) mn{ x), y) } mn{, }=.Whch mples that xy).nd also, x+y) max{ x), y)} max{, }=. Whch mples that x + y). nd, xy) max{ x), y)} max{, }=.Whch mples that xy).thereore, x+y) and x+y) and xy) and xy). nd thereore x + y and xy n, ). Hence, ) s a level subhemrng o a hemrng R. 2.1Denton: Let be an ntutonstc uzzy subhemrng o a hemrng R. The level subhemrng, ), or, [0,1] and e) and e) are called ntutonstc uzzy level subhemrng o. 2.2 Theorem: Let be an ntutonstc uzzy subhemrng o a hemrng R. Then two ntutonstc uzzy level subhemrng 1, 1), 2, 2) and 1, 2, 1, 2 are n [0,1] and 1, 2 e) and 1, 2 e) wth 2 < 1 and 1 < 2 o are equal there s no x n R such that 1 > x) > 2 and 1 < x) < 2. Proo: ssume that 1, 1) = 2, 2). Suppose there exsts x n R such that 1 > x)> 2 and 1 < x) < 2. Then 1, 1) 2, 2) mples x belongs to 2, 2), but not n 1, 1). Ths s contradcton to 1, 1) = 2, 2).Thereore there s no x R such that 1 > x) > 2 and 1 < x) < 2.Conversely there s no x R such that 1 > x) > 2 and 1 < x) < 2.Then 1, 1) = 2, 2). by the denton o level set ). 2.3 Theorem: Let R be a hemrng and be an ntutonstc uzzy subset o R such that, ) be a subhemrng R. I and n [0,1] satsyng e) and e), then s an ntutonstc uzzy subhemrng o R. Proo: Let R be a hemrng and x and y n R. Let x) = 1 and y) = 2, x) = 1 and y) = 2. Case ): I 1 < 2 and 1 > 2, then x, y 1, 1). s 1, 1) s a deal o R, x + y and xy n 1, 1). Now, x+y) 1 = mn{ 1, 2 } = mn{ x), y)} whch mples that x+y) mn{ x), y) }, or all x and y n R. lso, xy) 1 =mn{ 1, 2 }=mn{ x), y)}, whch mples that xy) mn{ x), y) }, or all x and y n R. nd, x+y) 1 = max{ 1, 2 }= max{ x), y)},whch mples that x+y) max{ x), y)}, or all x and y n R. lso xy) 1 = max{ 1, 2 } = max{ x), y)}, whch mples that xy) max{ x), y)}, or all x and y n R. Case ): I 1 < 2 and 1 < 2, then x and y n 1, 2). s 1, 2) s a subhemrng o R, x+y and xy 1, 2). Now, x+y) 1 =mn{ 1, 2 } =mn{ x), y)}whch mples that x+y) mn{ x), y)}, or all x and y n R. lso, xy) 1 = mn{ 1, 2 } = mn { x), y)}mples that xy) mn { x), y) }, or all x and y n R. nd, x+y) 2 = max{ 2, 1 }=max{ y), x)} mples that x+y) max{ x), y)}, or all x and y n R. lso, xy) 2 = max{ 2, 1 }= max{ y), x) } whch mples that xy) max{ x), y)}, or all x and y n R. Case ): I 1 > 2 and 1 > 2, then x and y n 2, 1). s 2, 1) s a subhemrng o R, x + y and xy n 2, 1). Now, x +y) 2 = mn{ 2, 1 } = mn{ y), x)} whch mples that x + y) mn{ x), y)}, or all x and y n R. lso, xy) 2 = mn { 2, 1 }= mn{ y), x)} whch mples that xy) mn{ x), y)}, or all x and y n R. nd, x+y) 1 = max{ 1, 2 } = max{ x), y)} whch mples that x+y) max{ x), y) }, or all x and y n R. lso, xy) 1 = max{ 1, 2 }= max{ x), y)} whch mples that xy) max{ x), y)}, or all x and y n R. Case v): I 1 > 2 and 1 < 2, then x and y n 2, 2). s 2, 2) s a subrng o R, x+y and xy n 2, 2). Now, x+y) 2 = mn{ 2, 1 }= mn{ y), x)} whch mples that x+y) mn{ x), y)}, or all x and y n R. lso, xy) 2 = mn{ 2, 1 }= mn { y), x) }mples that xy) mn{ x), y) }, or all x and y n R. nd, x+ y) 2 = max{ 2, 1 }=max{ y), x) } mples that x+y) max { x), y)}, or all x and y n R. lso, xy) 2 = max{ 2, 1 }= max{ y), x)} mples that xy) max{ x), y)}, or all x and y n R. Case v): I 1 = 2 and 1 = 2. It s trval. In all the cases, s an ntutonstc uzzy subhemrng o a hemrng R. 2.4 Theorem: Let be an ntutonstc uzzy subhemrng o a hemrng R. I any two level subhemrngs o belongs to R, then ther ntersecton s also level subhemrng o n R. Proo : Let 1, 2, 1, 2 [0,1] and 1, 2 e) and 1, 2 e). Case ) I 1 < x) < 2 and 1 > x)> 2, then 2, 2) 1, 1). Thereore, 1, 1) 2, 2) = 2, 2), but 2, 2) s a level subhemrng o. Case ) I 1 > x) > 2 and 1 < x) < 2, then 1, 1) 2, 2). Thereore, 1, 1) 2, 2) = 1, 1), but 1, 1) s a level subhemrng o. Case ) I 1 < x) < 2 and 1 < x)< 2, then 2, 1) 1, 2).Thereore, 2, 1) 1, 2) = 2, 1), but 2, 1) s a level subhemrng o. Case v) I 1 > x)> 2 and 125

3 1 > x)> 2, then 1, 2) 2, 1). Thereore, 1, 2) 2, 1) = 1, 2), but 1, 2) s a level subhemrng o. Case v) I 1 = 2 and 1 = 2, then 1, 1) = 2, 2). In all cases, ntersecton o any two level subhemrngs s a level subhemrng o. 2.5 Theorem: Let be an ntutonstc uzzy subhemrng o a hemrng R. I, j [0,1], e) and j e) and, j),, j I s a collecton o level subhemrngs o, then ther ntersecton s also a level subhemrng o. Proo: It s trval. 2.6 Theorem: Let be an ntutonstc uzzy subhemrng o a hemrng R. I any two level subhemrngs o belongs to R, then ther unon s also a level subhemrng o n R. Proo: Let 1, 2, 1, 2 [0,1] and 1, 2 e) and 1, 2 e). Case ) I 1 < x) < 2 and 1 > x)> 2, then 2, 2) 1, 1). Thereore, 1, 1) 2, 2) = 1, 1), but 1, 1) s a level subhemrng o. Case) I 1 > x) > 2 and 1 < x) < 2, then 1, 1) 2, 2). Thereore, 1, 1) 2, 2) = 2, 2), but 2, 2) s a level subhemrng o. Case) I 1 < x) < 2 and 1 < x) < 2, then 2, 1) 1, 2). Thereore, 2, 1) 1, 2) = 1, 2), but 1, 2) s a level subhemrng o. Casev) I 1 > x) > 2 and 1 > x) > 2, then 1, 2) 2, 1).Thereore, 1, 2) 2, 1) = 2, 1), but 2, 1) s a level subhemrng o. Casev) I 1 = 2 and 1 = 2, then 1,1) = 2, 2). In all cases, unon o any two level subhemrng s also a level subhemrng o. 2.7 Theorem: Let be an ntutonstc uzzy subhemrng o a hemrng R. I, j [0,1], e) and j e) and, j),, ji s a collecton o level subhemrngs o, then ther unon s also a level subhemrng o. Proo: It s trval. 2.8 Theorem: Let be an ntutonstc uzzy subhemrng o a hemrng R. I s an ntutonstc uzzy characterstc subhemrng o R, then each level subhemrng o s a characterstc subhemrng o R. Proo: Let be an ntutonstc uzzy characterstc subhemrng o a hemrng R. Let x and y n R and Im, Im ; utr) and x, ). Now, x)) = x). Thereore, x)). nd, x)) = x). Thereore, x)). Thereore, x), ). Hence,, ) ), ) ). For the reverse ncluson, let x, ) ) and let y n R be such that y) = x. Then, y) = y)) = x). nd, y)= y)) = x).thereore, y) and y). Hence, y, ), whe n x, ) ). Hence, ) ), ) --- 2). From 1) and 2), we get, ) s a characterstc subhemrng o a hemrng R. 2.9 Theorem: ny subhemrng H o a hemrng R can be realzed as a level subhemrng o some ntutonstc uzzy subhemrng o R. Proo: Let be the ntutonstc uzzy subset o a hemrng R dened by x) = x H, 0 < < 1 0 x H and x) = x H, 0 < < 1 0 x H and + 1, where H s subhemrng o a hemrng R. We clam that s an ntutonstc uzzy subhemrng o a hemrng R. Let x and y n R. I x and y n H, then x + y and xy n H. Snce H s a subhemrng o R, x + y) =, x) =, y) =, xy) =. So, x + y) mn { x), y)}.nd, xy) mn { x), y) }. lso, x+y) =, x) =, y) =, xy) =. So, x+y) max{ x), y)}. nd, xy) max{ x), y) }. I xh, yh, then x+y, xy H. Then, x+y) = 0, x) =, y) = 0, xy) = 0. Thereore, x+y) mn{ x), y) } and xy) mn{ x), y)}. nd x+y) = 0, x) =, y) = 0, xy) = 0. Thereore, x+y) max { x), y) } and xy) max { x), y) }. I x, y H, then x - y may or may not belong to H. Clearly x+y) mn{ x), y)} and xy) mn{ x), y)}. lso x+y) max{ x), y)}and xy) max{ x), y)}. In any case, x+y) mn{ x), y) }, xy) mn{ x), y)} and x+y) max{ x), y)}, xy) max{ x), y)}. Thus n all the cases, s an ntutonstc uzzy subhemrng o R Theorem: The homomorphc mage o a level subhemrng o an ntutonstc uzzy subhemrng o a hemrng R s a level subhemrng o an ntutonstc uzzy subhemrng o a hemrng R. Proo: Let R, +,. ) and R, +,. ) be any two hemrngs and : R R be a homomorphsm. That s, x + y) = x) + y) and xy) = x)y), or all x and y n R. 126

4 Let V = ), where s an ntutonstc uzzy subhemrng o a hemrng R. Clearly V s an ntutonstc uzzy subhemrng o a hemrng R. Let x and y n R, mples x) and y) n R. Let, ) s a level subhemrng o. That s, x) and x) ; y) and y) ; x+y), xy) and x+y), xy). We have to prove that, ) ) s a level subhemrng o V. Now, V x) ) x), whch mples that V x) ) ; and V y) ) y), whch mples that V y) ) and V x) + y) ) = V x + y) ), as s a homomorphsm x + y ), whch mples that V x) + y) ). lso, V x) y) ) = V x y) ) xy ), whch mples that V x) y) ). nd, V x)) x), whch mples that V x)) ; V y)) y), whch mples that V y)) and V x)+y) ) = V x + y) ) x + y ), whch mples that V x) + y) ). lso, V x)y) ) = V x y) ), as s a homomorphsm xy), whch mples that V x)y) ). Thereore, V x)+y) ), V x)+y) ), V x) y) ) and V x)y) ). Hence, ) ) s a level subhemrng o an ntutonstc uzzy subhemrng V o a hemrng R Theorem: The homomorphc pre-mage o a level subhemrng o an ntutonstc uzzy subhemrng o a hemrng R s a level subhemrng o an ntutonstc uzzy subhemrng o a hemrng R. Proo: Let R, +,. ) and R, +,. ) be any two hemrngs and : R R be a homomorphsm.that s, x+y) = x)+y) and xy) = x)y) or all x and y n R. Let V = ), where V s an ntutonstc uzzy subhemrng o a hemrng R. Clearly s an ntutonstc uzzy subhemrng o a hemrng R. Let x) and y) n R, mples x and y n R. Let, ) ) s a level subhemrng o V. That s, V x) ) and V x) ) ; V y) ) and V y) ) ; V x)+y)), V x)y)) and V x)+y) ), V x) y) ).We have to prove that, ) s a level subhemrng o. Now, x) = V x) ), mples that x) ; y) = V y) ), mples that y) and x+y ) = V x+y) )= V x)+y) ),whch mples that x+y). lso, xy ) = V xy)) = V x)y) ), whch mples that xy). nd, x)= V x) ), mples that x) ; y) = V y)), mples that y) and x+y ) = V x+y) ) = V x)+y) ), whch mples that x+y). xy ) = V xy) )= V x)y) ),whch mples that xy). Thereore, V x)+y)), V x)+y) ), V x)y) ) and V x)y) ). Hence,, ) s a level subhemrng o an ntutonstc uzzy subhemrng o R Theorem: The ant-homomorphc mage o a level subhemrng o an ntutonstc uzzy subhemrng o a hemrng R s a level subhemrng o an ntutonstc uzzy subhemrng o a hemrng R. Proo: Let R, +,. ) and R, +,. ) be any two hemrngs and : R R be an ant-homomorphsm. That s, x+y) = y)+x) and xy) = y)x), or all x and y n R. Let V = ), where s an ntutonstc uzzy subhemrng o R. Clearly V s an ntutonstc uzzy subhemrng o R. Let x and y n R, mples x) and y) n R. Let, ) s a level subhemrng o. That s, x) and x) ; y) and y). y+x), yx) and y+x), yx). We have to prove that, ) ) s a level subhemrng o V. Now, V x)) x), whch mples that V x) ) ; V y) ) y), whch mples that V y)). Now, V x)+y) ) = V x)+y) )= V y+x) ) y+x ), whch mples that, V x)+y) ). lso, V x)y) ) = V yx)) yx),whch mples that V x)y) ). nd, V x) ) x), whch mples that V x)) and V y)) y), whch mples that V y) ). Now, V x) + y) ) = V x) + y) )= V y+x) ), y + x),whch mples that V x) + y) ). lso, V x)y) ) = V yx )), y x),whch mples that V x) y) ). Thereore, V x)+y) ), V x)+ y) ) and V x)y) ), V x)y) ). Hence, ) ) s a level subhemrng o an ntutonstc uzzy subhemrng V o R Theorem: The ant-homomorphc pre-mage o a level subhemrng o an ntutonstc uzzy subhemrng o a hemrng R s a level subhemrng o an ntutonstc uzzy subhemrng o a hemrng R. Proo: Let R, +,. ) and R, +,. ) be any two hemrngs and : R R be an ant-homomorphsm. That s, x+y) = y) + x) and xy) = y)x), or all x and y n R. Let V = ), where V s an ntutonstc uzzy subhemrng o a hemrng R. Clearly s an ntutonstc uzzy subhemrng o a hemrng R. Let x) and y) n R, mples x and y n R. Let, ) ) s a level subhemrng o V. That s, V x) ) and V x) ) ; V y) ) and V y) ) ; V y) + x) ), V y) x) ) and V y) + x) ), V y) x) ).We have to prove that, ) s a level subhemrng o. Now, x) = V x) ), whch mples that x) ; y) = V y) ), whch mples that y). Now, x+y ) = V x+y) )= V y)+x) ) = V y)+x) ) 127

5 , whch mples that x+y). lso, xy ) = V xy) )= V y)x) ), whch mples that xy). nd, x) = V x) ), whch mples that x) and y) = V y) ), whch mples that y) and x+y ) = V x+y) ) = V y)+x) ), whch mples that x+y). xy ) = V xy) )= V y)x) ), whch mples that xy).thereore, V x)+y) ), V x)+y) ) and V x)y) ), V x)y) ). Hence, ) s a level subhemrng o an ntutonstc uzzy subhemrng o R Theorem: Let be any mappng rom a hemrng R 1 to R 2 and let be an ntutonstc uzzy subhemrng o R 1. Then or, [0,1], we have, ) ) = 10, 2 0 1, 2 Proo: Suppose that and n [0,1] and y = x)r 2. I y, ) ), then )y) = Sup x) and )y) = In 1 x y) x) x 1 y). Thereore, or every real number 1, 2 > 0, there exst x 0-1 y) such that x 0 ) >- 1 and x 0 ) < + 2. So, or every 1, 2 > 0, y = x 0 ) ) and hence 1, 2 ) REFERENCES 1. kram.m and Dar.K.H, On uzzy d-algebras, Punjab unversty journal o mathematcs, ), kram.m and Dar.K.H, Fuzzy let h-deals n hemrngs wth respect to a s-norm, Internatonal Journal o Computatonal and ppled Mathematcs, Volume 2 Number ), pp sok Kumer Ray, On product o uzzy subgroups, uzzy sets and systems, 105, ). 4. tanassov.k., Intutonstc uzzy sets, uzzy sets and systems, 201), ). 5. tanassov.k., Intutonstc uzzy sets theory and applcatons, Physca-Verlag, Sprnger-Verlag company, prl 1999, Bulgara. 6. Davvaz.B and Weslaw..Dudek, Fuzzy n-ary groups as a generalzaton o roseneld uzzy groups, RXIV VIMTH.R) 20 OCT 2007, Dxt.V.N., Rajesh Kumar, Naseem jmal., Level subgroups and unon o uzzy subgroups, Fuzzy sets and systems, 37, ). y 10, 20 10, 2 0 y 1, 2 1, 2 Therore,, ) ).1). Conversely, 10, 20 1, 2 )), then or each 1, 2 >0 we have y ) and there exst x 0, ) 1 2, ) 1 2 such that y=x 0 ). Thereore or each 1, 2 > 0, there exst x 0-1 y) and x 0 ) - 1 and x 0 ) + 2. Hence, )y) = x ) and )y) = Sup Sup 1 1 x y) 0 In 1 x ) In 2 1 x y) 0 2 Thereore, 10, 0 2 1, 2 )). So, y,, ) )..2). From 1) and 2) we get,, ) ) = 10, 2 0 1, 2 8. Palanappan. N & rjunan. K, The homomorphsm, ant homomorphsm o a uzzy and an ant uzzy deals, Varahmhr Journal o Mathematcal Scences, Vo.6 No.1, ). 9. Palanappan. N & rjunan. K, Operaton on uzzy and ant uzzy deals, ntartca J. Math., 41), ). 10. Palanappan. N & rjunan. K, Some propertes o ntutonstc uzzy subgroups, cta Cenca Indca, Vol.XXXIII M. No.2, ). 11. Rajesh Kumar, Fuzzy lgebra, Volume 1, Unversty o Delh Publcaton Dvson, July Svaramakrshna das.p, Fuzzy groups and level subgroups, Journal o mathematcal analyss and applcatons, 84, ). 13. Zadeh.L.., Fuzzy sets, Inormaton and control,vol.8, ) 128