Roughness in ordered Semigroups. Muhammad Shabir and Shumaila Irshad

Transcription

1 World Applied Sciences Journal 22 (Special Issue of Applied Mah): , 2013 ISSN IDOSI Publicaions, 2013 DOI: /idosiwasj22am Roughness in ordered Semigroups Muhammad Shabir and Shumaila Irshad Deparmen of Mahemaics, Quaid-i-Azam Universiy, Islamabad, Pakisan Absrac In his paper we iniiaed he sudy of roughness in ordered semigroups based on pseudoorder We inroduced he noions of upper (lower) rough ideal, bi-ideal, prime ideal and also he noions of upper (lower) rough fuzzy ideal, rough prime fuzzy ideal in ordered semigroups We sudied some properies of such ideals Key words and phrases Upper (lower) rough ideal, upper (lower) rough bi-ideal, upper (lower) rough fuzzy ideal, upper (lower) rough prime fuzzy ideal 1 Inroducion The fundamenal concep of rough se, inroduced by Pawlak in his definiive paper [1] of 1982, provides a new mahemaical approach o deal wih inexac, uncerain or vague knowledge Rough se heory has many applicaions in knowledge discovery approximae classificaion, conflic analysis, machine learning, daa analysis and so on, see [2, 3, 4, 5, 6, 7, 8] In rough se heory one consrucs upper and lower approximaion operaors based on available informaion The Pawlak approximaion operaors based on equivalence relaion In Pawlak approximaion, he lower approximaion of a se is he union of all he equivalence classes which are conained in his se, and he upper approximaion is he union of all equivalence classes which have a nonempy inersecion wih his se Many researchers used his concep o generalize several basic noions of algebras Thus, for example, Biswas and Nanda [9] applied he noion o groups and inroduced he noion of rough subgroups Kuroki and Wang [10] and Kuroki and Mordeson [11] also sudied rough groups On he oher hand Kuroki [12] iniiaed he sudy of rough ideals in semigroups Rough prime ideals in semigroups are sudied in [13] Davvaz and his colleagues sudied roughness in rings and Modules (see for example [14, 15, 16]) Since he requiremen of an equivalence relaion in Pawlak rough se models seem o be a very resricive condiion, so many researchers generalized he Pawlak rough se approximaions They used general binary relaions, coverings, compleely disribuive laices, fuzzy laices and Boolean algebras o generalize he Pawlak rough se, see [5, 7, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] An ordered semigroup (po-semigroup) S is a parial ordering se ( S, ) a he same ime a semigroup such ha: for any su,, S, s implies us u and su u Ordered semigroups and fuzzy ordered semigroups are sudied in [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39] If ρ is a congruence relaion on an ordered semigroup S, hen S / ρ is no an ordered semigroup, in general (see [32]) In [31] Kehayopulu and Tsingelis, inroduced he concep of Pseudoorder of ordered semigroups and shown ha: If ρ is a pseudoorder on an ordered semigroup S, hen here exiss a congruence ρ on S such ha S / ρ is an ordered Corresponding Auhor: Muhammad Shabir, mshabirbhai@yahoocouk 84

2 semigroup In his paper we use pseudoorder o define lower and upper approximaions of a se in an orederd semigroup We inroduced he noions of upper (lower) rough ideal, bi-ideal, prime ideal and also upper (lower) rough fuzzy ideal, rough fuzzy prime ideal in ordered semigroups We sudy some properies of such ideals 1 Preliminaries An ordered semigroup ( S,, ), is a pose ( S, ), a he same ime a semigroup ( S,) such ha a b implies ax bx and xa xb for all abx,, S A non-empy subse A of an ordered 2 semigroup S is called a subsemigroup of S if A A A non-empy subse I of an ordered semigroup S is called a righ (resp lef) ideal of S if IS I (resp SI I ) and a I and b S such ha b a implies b I I is called an ideal of S if i is boh a righ and a lef ideal of S An ideal P of an ordered semigroup S is called a prime ideal if xy P implies x P or y P, for all x, y S [ 8 ] A subsemigroup B of an ordered semigroup S is called a bi-ideal of S if BSB B and a B and b S such ha b a implies b B A relaion on an ordered semigroup S is called a pseudoorder (see [ 12 ]) if 1 2 is ransiive, ha is ( ab, ),( bc, ) 3 is compaible, ha is if ( ab, ) implies ( ac, ) for all,, hen ( ax, bx) and ( xa, xb) abc S, for all abx,, S An equivalence relaion on an ordered semigroup S is called a congruence relaion if ab, ax, bx xa, xb, for all abx,, S implies an A congruence on S is called a complee congruence if [ a] [ b] = [ ab] for all ab, S Where [ a] is he congruence class conaining he elemen a S A fuzzy subse f of a non-empy se X is a funcion from X ino he uni closed inerval [ 0,1 ], ha is f : X [ 0,1] Le f and g be fuzzy subses of a se X Then he fuzzy subses f defined as follows { } { } ( f g)( x) = f ( x) g ( x) ( f g)( x) = max f ( x), g ( x) for all x X For a S, define min,, {(, ) : } A = yz S S a yz a g and f g of X are 85

3 Le f and g be wo fuzzy subses of ( S,, ) The produc f ( yz, f g x ) { } min f y, g z, if Ax φ A = x 0 if Ax = φ g is defined by for all x S S,, be an ordered semigroup A fuzzy subse f of S is called a fuzzy subsemigroup of Le S if f ( ab) f ( a) f ( b) for all a, b S A fuzzy subse f of S is called a fuzzy righ (lef) ideal of S if ( 1 ) f ( xy) f ( x) ( f ( xy) f ( y) ) for all x, y S and 2 x y f x f y for all x, y S A fuzzy subse f of S is called a fuzzy ideal of S if i is boh a fuzzy righ and a fuzzy lef ideal of S A fuzzy subsemigroup f of S is called a fuzzy bi-ideal of S if implies ( 1 ) f ( xyz) f ( x) f ( z) ( 2 ) x y, implies f ( x) f ( y) for all xyz,, S for all x, y S A fuzzy ideal f of an ordered semigroup S is called a fuzzy prime ideal if f ( xy) = f ( x) or f ( xy) f ( y) = for all x, y S 2 Rough ses in ordered semigroups Suppose is a binary relaion on a universe U A pair of approximaion operaions, : PU PU are defined by ( X) = { x : y, xy y X} = { x : N( x) X} ( X) = { x : y X such ha xy} = { x : N( x) X φ} Where, N( x) { y U : xy} respecively (see [ 35 ] ) = They are called lower and upper approximaion operaions, Theorem [ 35 ] Le and λ be relaions on U If A and B are non-empy subses of U, hen he following hold: 1 ( U) = U = ( U) (2) ( φ) = φ = φ (3) A A A 86

4 ( 4 ) ( A B) = ( A) ( B) ( 5 ) ( A B) = ( A) ( B) ( 6 ) A B implies ( A) ( B) (7) A B implies ( A) ( B) ( 8 ) ( A B) ( A) ( B) ( 9 ) ( A B) ( A) ( B) ( 10) λ implies ( A) λ( A) ( 11) λ implies ( A) ( A) 32 Definiion Le be a pseudoorder on an ordered semigroup S and A be a non-empy subse of S Then he ses ( A) = { x S : y, xy y A} = x S : N( x) A and { } ( A) = { x S : y A such ha xy} = x S : N( x) A φ { } are called he -lower and -upper approximaions of A For a non-empy subse A of S, ( A) ( A), ( A) ( A) ( A) Where, N( x) { y S : xy} = 33 Remark Le be a pseudoorder on an ordered semigroup S, hen for each, N( a) N( b) N( ab) If N( a) N( b) N( ab) = is called a rough se wih respec o if =, hen is called a complee pseudoorder 34 Example Consider he ordered semigroup S { abc,, } ab S we have = defined by he muliplicaion and he order below 87

5 a b c a a a c b a b c c a c c {( aa) ( ac) ( bb) ( bc) ( cc) } = :,,,,,,,,, Le and λ be he pseudoorders on S defined by = aa,, ab,, ac,, bb,, bc,, cc, and { } ( aa, ),( ac, ),( ba, ), ( bb, ), ( bc, ), ( ca, ), ( cc, ) Then N( a) = { abc,, } N( b) = { bc, } N( c) = { c} N( ca) = { abc,, } and N( c) N( a) = { ac, } Thus, N( ca) N( c) N( a) { } λ = This implies ha is no a complee pseudoorder λ N a = ac, Now, { } λ N( b) = { abc,, } λ N( c) = { ac, } λ N( ab) = { ac, } and λn( a) λ N( b) = { ac, } λ N( ba) = { ac, } and λn( b) λ N ( a) = { ac, } λ N( ac) = { ac, } and λn( a) λ N( c) = { ac, } λ N( ca) = { ac, } and λn( c) λ N( a) = { ac, } λ N( bc) = { ac, } and λn( b) λ N( c) = { ac, } λ N( cb) = { ac, } and λn( c) λ N( b) = { ac, } So λ is a complee pseudoorder 35 Theorem Le be a pseudoorder on an ordered semigroup S and AB, are non-empy subses of S Then ( A) ( B) ( AB) Proof Proof is sraighforward 88

6 36 Example Consider he ordered semigroup S { abc,, } a b c a a a c b a b c c a c c {( aa) ( ac) ( bb) ( bc) ( cc) } = :,,,,,,,,, Consider he pseudoorder defined by, = aa,, ab,, ac,, bb,, bc,, cc, { } Then, N( a) = { abc,, } N( b) = { bc, } N( c) = { c} Le A= { a} and B { c} ( A) = { a} and ( B) = { abc,, } AB= { c} and ( AB) = { abc,, } ( A) ( B) = { ac, } So, ( A) ( B) ( AB) Thus, in general ( A) ( B) ( AB) = defined by he muliplicaion and he order below = be non-empy subses of S Then 37 Theorem Le be a complee pseudoorder on an ordered semigroup S and AB, are non empy subses of S, hen ( A) ( B) ( AB) Proof Proof is sraighforward 38 Example Consider he ordered semigroup S { abc,, } a b c a a a c b a b c c a c c = defined by he muliplicaion and he order below 89

7 {( aa) ( ac) ( bb) ( bc) ( cc) } = :,,,,,,,,, Consider he complee pseudoorder λ on S, λ = aa,, ac,, ba,, bb,, bc,, ca,, cc, { } λ N( a) = { ac, } λ N( b) = { abc,, } λ N( c) = { ac, } Le A= { a} and B { abc,, } ( A) = φ and ( B) = { abc,, } ( A) ( B) = φ AB = { ac, } and ( AB) = { ac, } So, ( A) ( B) ( AB) Thus, in general ( A) ( B) ( AB) = be non-empy subses of S Then 39 Theorem Le ρ and σ be pseudoorders on an ordered semigroup S and A a non-empy subse of S, hen ( ρ σ)( A) ρ( A) σ ( A) Proof Proof is sraighforward 310 Example Consider he ordered semigroup S { abcde,,,, } below a b c d e a a e c d e b a b c d e c a e c d e d a e c d e e a e c d e = : {( aa, ),( bb, ),( cc, ),( dd, ), ( ee, )} Le ( aa, ),( ad, ), ( bb, ), ( cc, ),( ce, ),( d, d), ( ee, ) { } ρ = and = defined by he muliplicaion and he order 90

8 {( aa, ),( ae, ),( bb, ),( cc, ),( dd, ), ( ee, )} ( ρ σ) {( aa, ),( bb, ),( cc, ),( d, d), ( ee, )} Le A { de, } ρ ( A) = { acde,,, } σ ( A) = { ade,, } ( ρ σ) ( A) = { d, e} So, ( ρ σ) ( A) ρ( A) σ ( A) σ = be he pseudoorders on S, hen = = be a non-empy subse of S, hen Bu equaliy does no hold in general 311 Theorem Le ρ and σ be pseudoorders on an ordered semigroup S If A is a non-empy subse of S, hen ( A) ( A) ( )( A) ρ σ ρ σ Proof Proof is sraighforward 312 Example Consider he ordered semigroup S { abc,, } a b c a a a c b a b c c a c c {( aa) ( ac) ( bb) ( bc) ( cc) } = :,,,,,,,,, Le and λ be he pseudoorders on S defined by = aa,, ab,, ac,, bb,, bc,, cc, { } ( aa, ),( ac, ),( ba, ), ( bb, ), ( bc, ), ( ca, ), ( cc, ) λ {( aa, ),( ac, ), ( bb, ), ( bc, ),( cc, )} { abc,, } { bc, } { c} { ac, } { } λ = = N a = N b = N c = λ N a = = defined by he muliplicaion and he order below 91

9 λ N( b) = { abc,, } λ N( c) = { ac, } ( λ) N( a) = { ac, } ( λ) N( b) = { bc, } ( λ) N( c) = { c} Le A= { ac, } be a subse of S Then ( λ)( A) { ac, } And ( A) = { c}, λ ( A) = { ac, } ( A) λ( A) = { c} So, ( A) λ( A) ( λ)( A) Bu equaliy does no hold in general = 3 Rough ideals in ordered semigroups 41 Definiion Le be a pseudoorder on an ordered semigroup S Then a non-empy subse A of S is called an upper rough subsemigroup of S if ( A) is a subsemigroup of S A is called an upper rough lef (righ, wo sided) ideal of S if ( A) is a lef (righ, wo sided) ideal of S 42 Theorem Le be a pseudoorder on an ordered semigroup S 1 If A is a subsemigroup of S, hen A is an upper rough subsemigroup of S ( 2 ) If A is a lef (righ, wo sided) ideal of S, hen A is an upper rough lef (righ, wo sided) ideal of S Proof ( 1 ) Le A be a subsemigroup of S From Theorem 31, par ( 3 ), A ( A), so ( A) φ Then by Theorem 35, ( A) ( A) ( AA) ( A) Thus ( A) is a subsemigroup of S, ha is A is an upper rough subsemigroup of S ( 2 ) Le A be a lef ideal of S, ha is SA A Since ( S) = S, herefore i follows from Theorem 35, ( i ) S( A) = ( S) ( A) ( SA) ( A) ( ii ) Le a ( A), hen here exiss x A such ha a x If b S and by ransiiviy of, b x Thus, b ( A) This shows ha ( A) is a lef ideal of S, ha is A is an upper rough lef ideal of S 92 such ha b a, hen ba

10 The following example shows ha if is a pseudoorder on S hen ( A) is a subsemigroup (lef ideal) of S even if A is no a subsemigroup (lef ideal) of S 43 Example Consider he ordered semigroup S { abcde,,,, } below a b c d e a b b d d d b b b d d d c d d c d c d d d d d d e d d c d c {( aa) ( ab) ( bb) ( cc) ( db) ( dc) ( d d) ( ec) ( ee) } = :,,,,,,,,,,,,,,,,, = defined by he muliplicaion and he order Le be he pseudoorder on S defined as follows: = {( aa, ),( ab, ),( ac, ),( ad, ), ( ba, ), ( bb, ), ( bc, ),( bd, ), ( cc, ),( da, ), ( db, ), ( dc, ), ( dd,,) ( ec,,) ( ee, )} N a = abcd,,, Then { } N( b) = { abcd,,, } N( c) = { c} N( d) = { abcd,,, } N( e) = { ce, } ( 1 ) Le A= { abc,, }, hen A is no a subsemigroup of S bu ( A) { abcde,,,, } subsemigroup of S A= abc,,, hen A is no a lef ideal of S because e c bu e A ( 2 ) Le { } Bu ( A) { abcde,,,, } = is a lef ideal of S 44 Theorem 93 = is a Le be a complee pseudoorder on an ordered semigroup S If A is a subsemigroup of S, A, if i is non-empy, is a subsemigroup of S hen Proof Le A be a subsemigroup of S and ( A) ( A) ( A) ( AA) ( A) φ Then by Theorem 37,

11 Thus ( A) is a subsemigroup of S, ha is A is a lower rough subsemigroup of S 45 Theorem Le be a complee congruence on an ordered semigroup S such ha (righ, wo sided) ideal of S, hen ( A), if i is non-empy, is a lef (righ, wo sided) ideal of S Proof Le A be a lef ideal of S Then i follows from Theorem 37, ha S A = S A SA A ( i ) ( ii ) Le a ( A), hen [ a] A Le b S [ a] = [ b] Since [ a] A, so [ b] A Hence b ( A) This shows ha ( A) 94 If A is a lef be such ha b a Then ba This implies, if i is non-empy, is a lef ideal of S The following example shows ha if is a pseudoorder on S hen ( A) is no necessarily a lef ideal of S even if A is a lef ideal of S 46 Example Consider he ordered semigroup S { abcde,,,, } below a b c d e a b b d d d b b b d d d c d d c d c d d d d d d e d d c d c {( aa) ( ab) ( bb) ( cc) ( db) ( dc) ( dd) ( ec) ( ee) } = :,,,,,,,,,,,,,,,, Le be he pseudoorder on S defined by = defined by he muliplicaion and he order ( aa) ( ab) ( ac) ( ad) ( ba) ( bb) ( bc) ( bd) ( cc) ( da) ( db) ( dc) ( dd) ( ec) ( ee) = {,,,,,,,,,,,,,,,,,,,,,,,,,,,,, } { abcd,,, } { abcd,,, } { c} { abcd,,, } { ce, } N a = N b = N c = N d = N e =

12 Le A { cde,, } Bu ( A) { ce, } = be a lef ideal of S = is no a lef ideal of S, because ac= d ( A) 47 Definiion A subse A of an ordered semigroup S is called an upper rough bi-ideal of S if ( A) is a biideal of S 48 Theorem Le be a pseudoorder on an ordered semigroup S If A is a bi-ideal of S, hen i is an upper rough bi-ideal of S Proof Le A be a bi-ideal of S Then by Theorem 35, we have ( i ) ( A) S( A) = ( A) ( S) ( A) ( ASA) ( A) ( ii ) Le a ( A) and b S be such ha b a Then here exiss x A, such ha a x and ba Since is ransiive, so bx b ( A) Thus by Theorem 42, we obain ha ( A) is a bi-ideal of S, ha is, A is a upper rough biideal of S The following example shows ha if is a pseudoorder on S hen ( A) is a bi-ideal of S even if A is no a bi-ideal of S 49 Example Consider he ordered semigroup S { abc,, } a b c a a a c b a b c c a c c {( aa) ( ac) ( bb) ( bc) ( cc) } = :,,,,,,,,, Consider he pseudoorder on S defined below = aa,, ab,, ac,, bb,, bc,, cc, { } N( a) = { abc,, } N( b) = { bc, } N( c) = { c} Le B = { c}, hen ( B) { abc,, } = defined by he muliplicaion and he order below = is a bi-ideal of S, bu B is no a bi-ideal of S because b c 95

13 bu b B 410 Theorem Le be a complee congruence on an ordered semigroup S such ha of S, hen ( A), if i is non-empy, is a bi-ideal of S Proof Le A be a bi-ideal of S Then by Theorem 37, we have A S A = A S A ASA A ( i ) ( ii ) Le a ( A) and b S be such ha b a, hen [ a] Since b a, so [ b] = [ a] and hence [ b] A Thus b ( A) From Theorem 45, we obain ha ( A) S 96 A If A is a bi-ideal is a subsemigroup of S Hence ( A) is a bi-ideal of The following example shows ha if is a pseudoorder on S hen ( A) is no necessarily a biideal of S even if A is a bi-ideal of S 411 Example Consider he ordered semigroup S { abcde,,,, } below a b c d e a b b d d d b b b d d d c d d c d c d d d d d d e d d c d c {( aa) ( ab) ( bb) ( cc) ( db) ( dc) ( d d) ( ec) ( ee) } = :,,,,,,,,,,,,,,,,, Le be he pseudoorder on S defined as follows: = defined by he muliplicaion and he order ( aa) ( ab) ( ac) ( ad) ( ba) ( bb) ( bc) ( bd) ( cc) ( da) ( db) ( dc) ( dd) ( ec) ( ee) = {,,,,,,,,,,,,,,,,,,,,,,,,,,,,, } N a = abcd,,, { } { abcd,,, } { c} { abcd,,, } { ce, } N b = N c = N d = N e =

14 Le A= { cde,, }, hen A is a bi-ideal of S Bu ( A) { ce, } d c and d ( A) 97 = is no a bi-ideal of S, because 412 Theorem Le be a pseudoorder on an ordered semigroup S If A and B are righ and lef ideals of S, respecively Then, ( AB) ( A) ( B) Proof Proof is sraighforward 413 Example Consider he ordered semigroup S { abcde,,,, } below a b c d e a b b d d d b b b d d d c d d c d c d d d d d d e d d c d c {( aa) ( ab) ( bb) ( cc) ( db) ( d c) ( dd) ( ee) } = :,,,,,,,,,,,,,,, Le be he pseudoorder on S defined as follows: = = defined by he muliplicaion and he order ( aa) ( ab) ( ac) ( ad) ( ba) ( bb) ( bc) ( bd) ( cc) ( da) ( db) ( dc) ( dd) ( ee) {,,,,,,,,,,,,,,,,,,,,,,,,,,, } N( a) = { abcd,,, } N( b) = { abcd,,, } N( c) = { c} N( d) = { abcd,,, } N( e) = { e} Le R = { cde,, } and L= { cde,, } be righ and lef ideals of S, respecively Then RL= { cd, } and ( RL) = { abcd,,, } ( R) = { abcde,,,, } ( L) = { abcde,,,, }

15 ( RL) ( R) ( L) Bu equaliy does no hold in general 414 Theorem Le be a pseudoorder on an ordered semigroup S If A and B are righ and lef ideals of S, respecively, hen, ( AB) ( A) ( B) Proof Proof is sraighforward 415 Example Consider he ordered semigroup S { abcde,,,, } below a b c d e a b b d d d b b b d d d c d d c d c d d d d d d e d d c d c {( aa) ( ab) ( bb) ( cc) ( db) ( d c) ( dd) ( ee) } = :,,,,,,,,,,,,,,, Le be he pseudoorder on S defined as follows: = defined by he muliplicaion and he order ( aa) ( ab) ( ac) ( ad) ( ba) ( bb) ( bc) ( bd) ( cc) ( da) ( db) ( dc) ( dd) ( ee) = {,,,,,,,,,,,,,,,,,,,,,,,,,,, } N a = abcd,,, { } N( b) = { abcd,,, } N( c) = { c} N( d) = { abcd,,, } N( e) = { e} Le R = { cde,, } and L { cde,, } RL= { cd, } and ( RL) = {} c ( R) = {,} ce ( L) = {,} ce ( RL) ( R) ( L) Bu equaliy does no hold in general = be righ and lef ideals of S, respecively 98

16 416 Proposiion Le and λ be pseudoorders on an ordered semigroup S Then λ is a pseudoorder if λ= λ Proof Proof is sraighforward 417 Theorem Le and λ be pseudoorders on an ordered semigroup S such ha λ= λ If A is a subsemigroup of S, hen ( A) ( A) ( )( A) λ λ Proof Proof is sraighforward 4 Rough prime ideals in ordered semigroups Recall ha an ideal A of an ordered semigroup S is called a prime ideal if xy A implies x A or y A, for all x, y S 51 Theorem Le be a complee congruence on an ordered semigroup S such ha ideal of S Then ( A) is, if i is non empy, a prime ideal of S 99 and A be a prime Proof Le A be an ideal of S Then by Theorem 45, ( A) is an ideal of S Suppose on he conrary ha ( A) is no a prime ideal of S Then here exis x, y S such ha xy ( A), bu x ( A) and y ( A) This implies [ x] [ y] = [ xy] A bu [ x] A and [ y] A Thus here exiss s [ ] ha s A and [ ] such ha A assumpion Hence ( A) y Then [ ] [ ], if i is non empy, is a prime ideal of S such x s x y A This conradics our 52 Theorem Le be a complee pseudoorder on an ordered semigroup S and A a prime ideal of S Then ( A) is a prime ideal of S Proof Le A be an ideal of S Then by Theorem 42, ( A) is an ideal of S Now le xy ( A) Then N( xy) A φ Since is a complee pseudoorder so N( xy) N( x) N( y) Thus here exiss c N( xy) A N( x) N( y) A = = Hence

17 c N( xy) = N( x) N( y) N( y) and c A This implies c s Since A is a prime ideal of S, so c= s A implies s A or A N x A φ N y A φ Hence Thus, or x ( A) or y ( A) Thus, ( A) is a prime ideal of S 53 Definiion =, for some s N( x) and A bi-ideal B of an ordered semigroup S is called prime bi-ideal of S if xy B implies x B or y B, for all x, y S 54 Theorem Le be a complee pseudoorder on an ordered semigroup S If B is a prime bi-ideal of S, hen i is an upper rough prime bi-ideal of S Proof The proof is similar o he proof of Theorem Theorem If B is a prime bi- Le be a complee congruence on an ordered semigroup S such ha ideal of S, hen ( B), if i is non empy, is a prime bi-ideal of S Proof The proof is similar o he proof of Theorem 51 5 Rough fuzzy prime ideals in ordered semigroup 61 Definiion Le be a pseudoorder on an ordered semigroup S Le f be a fuzzy subse of S Then he fuzzy subses and y N x f x = f y y N x f x = f y of S are called he -lower and -upper approximaions of he fuzzy se f, respecively ( f ) = ( f ), ( f ) is called a rough fuzzy subse of S wih respec o if ( f ) ( f ) 100

18 62 Definiion Le f be a fuzzy subse of S and [ 0,1] { : } f = x S f x and s { : } f = x S f x > Then he ses are called -level se and srong -level se of he fuzzy se f The following heorem is proved in many papers 63 Theorem Le f be a fuzzy subse of an ordered semigroup S Then f is a fuzzy lef (righ, wo sided ideal) of S if and only if [ 0,1] f, if i is non empy, a lef (righ, wo sided) ideals of S for every 64 Theorem Le be a pseudoorder on an ordered semigroup S and f be a fuzzy subse of S If f is a fuzzy lef (righ, wo sided) ideal of S, hen ( f ) is also a fuzzy lef (righ, wo sided) ideal of S Proof Le f be a fuzzy lef ideal of S and, ( f )( y) = f ( z) f ( xz) (if z N( y) z N y z N y hen yz xyxz xz N( xy) ) a N xy f a = f xy Thus ( f )( xy) ( f )( y) Now, le x, y S be such ha x y x y S Then Since x y, so x y If a N( y), hen y a and so x a This implies a N( x) N( y) N( x) As ( f )( y) = f ( z) f ( a) = ( f )( x), hus ( f )( x) ( f )( y) z N y a N x So ( f ) is a fuzzy lef ideal of S 65 Theorem Le be a complee congruence on an ordered semigroup S such ha 101 Hence Le f be a fuzzy

19 subse of S If f is a fuzzy lef (righ, wo sided) ideal of S, hen ( f ) is a fuzzy lef (righ, wo sided) ideal of S Proof Le f be a fuzzy lef ideal of S and x, y S Then ( f )( y) = f ( y ) f ( xy ) [ ] [ ] y y y y Since is a complee congruence and y [ ], so xy [ ] [ ] [ xy] ( xy ) ( f )( xy) [ ] y y y y xy f y = f y f xy = f = Now, le x, y S be such ha x y Since, so x y f y = f y = f y = f x [ ] [ ] y y y x y So ( f )( x) ( f )( y) Hence ( f ) is also a fuzzy lef ideal of S Similarly we can prove ha 102 xy Thus his implies [ x] [ y] = 66 Theorem (1) Le be a pseudoorder on an ordered semigroup S and f a fuzzy bi-ideal of S Then ( f ) is also a fuzzy bi-ideal of S (2) Le be a complee congruence on an ordered semigroup S such ha Le f be a fuzzy bi-ideal of S Then ( f ) is a fuzzy bi-ideal of S 67 Definiion A fuzzy ideal f of an ordered semigroup S is called a fuzzy prime ideal if f ( xy) = f ( x) or f ( xy) = f ( y) for all x, y S 68 Theorem Le f be a fuzzy subse of an ordered semigroup S Then f is a fuzzy prime ideal of S if and only if f, if i is non-empy, is a prime ideals of S for every [ 0,1] Proof Assume ha f is a fuzzy prime ideal of S Since f is a fuzzy ideal of S, so by Theorem 63, f is an ideal of S Now le ab f for any ab, or b f S Then f ( ab) his implies f ( a) or f ( b), so a f

20 Conversely, assume ha f, if i is non-empy, is a prime ideal of S for every [ 0,1] x, y S such ha neiher f ( xy) = f ( x) nor f ( xy) f ( y) such ha f ( xy) > f ( x) and f ( xy) > f ( y) Thus xy f bu neiher x f y Which is a conradicion, so f ( xy) = f ( x) or f ( xy) f ( y) S Le = This implies here exis [0,1] nor f = Hence f is a fuzzy prime ideal of 69 Theorem Le be a pseudoorder on an ordered semigroup S If f is a fuzzy subse of S and [ 0,1] hen ( 1 ) ( ( f )) = ( f ) s s ( f ) ( 2 ) ( f ) = Proof ( 1 ) Le x ( ( f )), hen ( f )( x) Thus f ( x ), so for all x N ( x ), f ( x ) This implies N ( x ) f, ha is x ( f ) x N x, Conversely, if x ( f ) hen N( x) f This implies f ( x ) for all x N( x) f ( x ), his implies ( f )( x ) and so x ( ( f )) x N x ( 2 ) The proof is similar o he proof of par 610 Theorem Le f be a fuzzy prime ideal of an ordered semigroup S ( 1 ) If is a complee congruence on S such ha, and ( f ) 1 Hence φ, hen f is a lower rough fuzzy prime ideal of S 2 If is a complee pseudoorder on an ordered semigroup S, hen f is an upper rough fuzzy prime ideal of S Proof ( 1 ) Since f is a fuzzy prime ideal, so by Theorem 68, f ( [ 0,1] ), if i is nonempy, is a prime ideal of S Then by Theorem 51, ( f ), if i is non-empy, a prime ideal of S Now by Theorem 69, ( ( f )) is a prime ideal of S Thus by Theorem 68, ( f ) is a fuzzy prime ideal of S ( 2 ) The proof is similar o he proof of par ( 1 ) References 1 Pawlak, Z, 1982 Rough Ses In J Inf Comp Sci, 11: Kryszkiewicz, M, 1998 Rough se approach o incomplee informaion sysems 103

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