On Prime and Fuzzy Prime Ideals of a Subtraction Algebra

International Mathematical Forum, 4, 2009, no. 47, 2345-2353 On Prime and Fuzzy Prime Ideals of a Subtraction Algebra P. Dheena and G. Mohanraaj Department of Mathematics, Annamalai University Annamalainagar - 608 002, India dheenap@yahoo.com, gmohanraaj@gmail.com Abstract In this paper we introduce the notion of m-system,fuzzy ideal,fuzzy prime ideal and fuzzy m-system in a subtraction algebra. We have shown that an ideal in a subtraction algebra X is prime if and only if its complement is a m-system. We have also shown that in a subtraction algebra X, for any ideal A which does not intersect a m-system M,then there exits a prime ideal P containing A that does not intersect M.We have also shown that a fuzzy ideal is fuzzy prime if and only if its image contains two elements and its level sets are prime ideals. Mathematics Subject Classification: 06F35, 16Y30 Keywords and Phrases: subtraction algebra, prime ideal,m-system,fuzzy ideal,fuzzy prime ideal,fuzzy m-system Introduction B.M. Schein [5] considered the systems of the form (φ;, \)where φ is a set of functions closed under the composition of functions (and hence (φ; )is a function semigroup) and the set theoretic subtraction \ and (hence(φ; \) is a subtraction algebra in the sense of [1]). B. Zelinka [7] discussed a problem proposed by B.M. Schein concerning the structure of multiplication in a subtraction semigroup.he solved the problem for subtraction algebra of a special type, called the atomic subtraction algebras.y.b.jun,h.s.kim and E.H.Roh [2] introduced the notion of ideals in subtraction algebras and obtained significant results.y.b.jun and K.H.Kim [3] introduced the notion of prime ideals of a subtraction algebra and gave a characterization of a prime ideal. In this paper we introduce the notion of m-system,fuzzy ideal, fuzzy prime ideal and fuzzy m-system of a subtraction algebra. Similar to Fuzzy set theory, we have obtained significant results in a subtraction algebra.

2346 P. Dheena and G. Mohanraaj Preliminaries A nonempty set X together with a binary operation is said to be a subtraction algebra if it satisfies the following: 1. x (y x) =x. 2. x (x y) =y (y x). 3. (x y) z =(x z) y, for every x, y, z X. Example 1. Let A be any nonempty set. Then (P(A), \) is a subtraction algebra, where P (A) denotes the power set of A and \ denotes the set theoretic subtraction. Example 2. Let X = 0, a,b,c} in which is defined by Then (X, ) is a subtraction algebra. 0 a b c 0 0 0 0 0 a a 0 a 0 b b b 0 0 c c b a 0 The subtraction determines an order relation on X :a b a b = 0,where 0 = a a is an element that does not depend on the choice of a X.The ordered set (X, ) is a semi-boolean algebra in the sense of [1],that is, it is a meet semilattice with zero 0 in which every interval [0,a] is a Boolean algebra with respect to the induced order.here a b = a (a b), the complement of an element b [0,a]isa b and if b, c [0,a],then b c = (b c ) = a ((a b) (a c)) = a ((a b) ((a b) (a c))) In a subtraction algebra the following holds : 1. x 0=x and 0 x =0. 2. (x y) x =0. 3. (x y) y = x y. 4. x (x y) y. 5. (x y) (y x) =x y.

Prime and fuzzy prime ideals 2347 6. x (x (x y)) = x y. 7. (x y) (z y) x z. 8. x y if and only if x = y w for some w X 9. x y implies x z y z and z y z x for all z X 10. x, y z implies x y = x (z y). Definition 3. A nonempty set A of a subtraction algebra X is called an ideal of X if it satisfies 1. 0 A 2. y A and x y A imply x A for all x, y X. Definition 4. A nonempty subset S of a subtraction algebra X is said to be a subalgebra of X, if x y S, whenever x, y S. Definition 5. An ideal P of a subtraction algebra X is said to be a prime ideal of X, if x y P, implies eitherx P or y P Lemma 6. [3] Let A be an ideal of a subtraction algebra X. If x y and y A, then x A. Lemma 7. [2] Let A be a nonempty subset of a subtraction algebra X. Then A is an ideal of X if only if (1) for all a A and for all x X, a x A (2) a b A whenever a b exits for all a, b A. Lemma 8. [2] Let A be an ideal of a subtraction algebra X. If y / A, then Q = x X x y A} is the least ideal containing A and y Theorem 9. [3] Let P be an ideal of a subtraction algebra X. Then P is a prime ideal if only if for any ideals A and B of X,A B P implies A P or B P where A B = a b a A, b B} Lemma 10. If A and B are the ideals of a subtraction algebra X,then A B = A B. Proof : If x A B, then x = a b for some a A,b B. Then a b a and a A implies a b A.Similarly a b B. Thusa b A B. On the other hand, if x A B, then x = x x A B. Therefore A B = A B. Definition 11. A nonempty subset M of a subtraction algebra X is said to be a m-system of X if x, y M implies x y M.

2348 P. Dheena and G. Mohanraaj Theorem 12. Let P be an ideal of a subtraction algebra X. Then P is a prime ideal if and only if X P is a m-system of X Proof : Let P be a prime ideal of a subtraction algebra X. Let x, y X P. x/ P and y / P imply x y/ P. Thus x y X P. Therefore X P is a m-system of X. On the other hand,let x y P. If x/ P and y/ P, then x, y X P. Since X P is a m-system of X, x y X P. This contradicts x y P. Hence P is a prime ideal of X Example 13. Let X = 0, a,b,c,d} in which is defined by 0 a b c d 0 0 0 0 0 0 a a 0 a a 0 b b b 0 b 0 c c c c 0 c d d b a d 0 Then P = 0, a, b, d} is a prime ideal. A = 0, c} is an ideal, but not a prime ideal since a b =0 A,with a/ P and b/ P. Theorem 14. Let A be an ideal and M be a m-system of a subtraction algebra X such that A M = φ. Then there is a prime ideal P of X such that A P with P M = φ. Proof : A = I is an ideal of X A I, I M = φ}. Since A A,A is nonempty. Note that every chain of A has an upper bound. By Zorn s lemma A has a maximal element P. We assert that P is a prime ideal of X. If there exits y, z X such that y z P with y / P and z / P, then Q = x X x y P } and R = x X x z P } are the ideals that contain P, y and P, z respectively. By the maximality of P, Q and R intersect M. Let a Q M and b R M. Then a b M. Since a Q and a a b, a b Q. Similarly a b R. Then a b Q R. Clearly P Q R. If x Q R then x y, x z P. Clearly x y, x z [0,x]. Thus (x y) (x z) = x ((x (x y)) (x (x z))) = x [(x (x y)) ((x (x y)) (x (x z)))] = x [(x y) ((x y) (x z))] = x [(x y) (x z)] = x [x (y z)] Since y z P, x (y z) P.By Lemma 7. and x y, x z P imply (x y) (x z) =x [x (y z)] P. x [x (y z)] P and x (y z) P imply x P. Hence Q R = P. Then a b P contradicts P M = φ. Therefore P is a prime ideal of X.

Prime and fuzzy prime ideals 2349 Definition 15. Let X be a a subtraction algebra. A mapping μ from X into [0,1] is called a fuzzy subset of X. Definition 16. Let μ be a fuzzy subset of a subtraction algebra X.μ is called fuzzy ideal of X if it satisfies 1. μ(0) μ(x) for all x X. 2. μ(x) min μ(x y),μ(y)} for all x, y X. Definition 17. Let μ be a fuzzy subset of a subtraction algebrax.then the level set of μ denoted by μ, t is defined as μ t =x X μ(x) t } for all t [0, 1] Theorem 18. Let μ be a fuzzy subset of a subtraction algebrax. Then μ is a fuzzy ideal of X if and only if for any t [0, 1], μ t is an ideal of X whenever μ t is nonempty. Proof. Let μ be a fuzzy ideal of X and let t [0, 1] such that μ t is nonempty. Let x μ t. Then μ(0) μ(x) t. Hence 0 μ t. Suppose x y, y μ t. Then μ(x) minμ(x y),μ(y)} t, therefore x μ t. Hence μ t is an ideal of X. Conversely, let x X and μ(x) =t for some t [0, 1]. Since μ t is an ideal, 0 μ t. Therefore μ(0) μ(x). For x, y X, let minμ(x y),μ(y)} = t. Then x y, y μ t imply x μ t. Then μ(x) t= minμ(x y),μ(y)}. Hence μ is a fuzzy ideal of X. Lemma 19. Let X be a subtraction algebra and μ be a fuzzy ideal of X. If x y, then μ(x) μ(y). Moreover μ(x y) maxμ(x),μ(y)} for all x, y X. Proof. If x y, then x y =0. μ(x) min μ(x y),μ(y)} = min μ(0), μ(y)} = μ(y) μ(x) μ(y) Since x y x implies μ(x y) μ(x) and x y y implies μ(x y) μ(y),thus μ(x y) maxμ(x),μ(y)}. Definition 20. Let μ be a fuzzy ideal of a subtraction algebra X. μ is called a fuzzy prime ideal of X if μ(x y) =maxμ(x),μ(y)} for all x, y X. Example 21. Let X = 0, a,b,c,d} in which is defined by

2350 P. Dheena and G. Mohanraaj 0 a b c d 0 0 0 0 0 0 a a 0 a a 0 b b b 0 b 0 c c c c 0 c d d b a d 0 μ(x) = 0.9 if x 0,a,b,d} 0.3 otherwise 0.8 if x 0,a} σ(x) = 0.6 if x b, d} 0.4 if x = c μ is a fuzzy prime ideal. σ is a fuzzy ideal but not a fuzzy prime ideal since σ(b c) =0.8 but σ(b) =0.6 and σ(c) =0.4 Theorem 22. Let μ be a fuzzy ideal of a subtraction algebra X. Then μ is a fuzzy prime ideal if and only if for any fuzzy ideals λ, σ of X, λ t1 σ t2 μ t3 implies λ t1 μ t3 or σ t2 μ t3, for all t 1,t 2,t 3 [0, 1]. Proof. Let μ be a fuzzy prime ideal of a subtraction algebra X. If there exits the fuzzy ideals λ and σ of X such that λ t1 σ t2 μ t3 but λ t1 μ t3 and σ t2 μ t3. Then there exits y, z X such that y λ t1 but y/ μ t3 and z σ t2 but z/ μ t3. Then μ(y) <t 3 and μ(z) <t 3.Therefore maxμ(y),μ(z)} <t 3. Since λ t1 σ t2 μ t3, y z μ t3, hence μ(y z) t 3.Thus μ(y z) maxμ(y),μ(z)}. This contradicts that μ is a fuzzy prime ideal of a subtraction algebra X. Conversely, if there exits a, b X such that μ(a b) =t 1 >μ(a) and μ(a b) > μ(b). For any x X, let λ(x) = t 1 if x a 0 otherwise σ(x) = t 1 if x b 0 otherwise Clearly λ, σ are fuzzy ideals of X. If x λ t1 σ t1 = λ t1 σ t1 then λ(x) t 1 and σ(x) t 1.Thusx a b. Since μ is a fuzzy ideal of a subtraction algebra X, μ(x) μ(a b) =t 1. Then x μ t1.thusλ t1 σ t1 μ t1 but λ t1 μ t1 and σ t1 μ t1. This is a contradiction to our assumption.therefore μ is a fuzzy prime ideal of a subtraction algebra X. Lemma 23. If μ is a fuzzy prime ideal of a subtraction algebra X, then Im μ = t 1,t 2 } where t 1,t 2 [0, 1]. Proof. Let μ be a fuzzy prime ideal of a subtraction algebra X. Suppose Im μ = t 1,t 2,t 3 } where 1 t 1 >t 2 >t 3 0. Then there exits a, b, c X such that μ(a) =t 1,μ(b) =t 2 and μ(c) =t 3. Clearly b / μ t1 and c / μ t2. If b a μ t1, then b a, a μ t1 implies b μ t1 which is a contradiction.

Prime and fuzzy prime ideals 2351 Therefore b a/ μ t1.thus μ(b a) <t 1. Similarly c b/ μ t2. Now b μ t2 implies b a μ t2,thusμ(b a) =t 2. Similarly μ(c b) =t 3. Clearly μ(0) = t 1. (b a) (c b) = (b a) [(b a) (c b)] = (b a) [(b (c b)) a] = (b a) [b a] =0 Thus μ((b a) (c b)) = μ(0) = t 1. But μ((b a) (c b)) μ(b a) =t 2 and μ((b a) (c b)) μ(c b) =t 3. This contradicts that μ is a fuzzy prime ideal. Hence Im μ = t 1,t 2 }. Lemma 24. If μ is a fuzzy prime ideal of a subtraction algebra X, then x X μ(x) =μ(0)} is a prime ideal of X. Proof. Let μ be a fuzzy prime ideal. Let P = x X μ(x) = μ(0) }. If x y P, then μ(x y) =μ(0).since μ is a fuzzy prime ideal, μ(x y) =μ(x) or μ(x y) = μ(y). Thus μ(x) = μ(0) or μ(y) = μ(0). Therefore x P or y P. Hence P is a prime ideal of X. Theorem 25. Let μ be a fuzzy ideal of a subtraction algebra X. Then μ is a fuzzy prime ideal of X if and only if (a) Imμ = t 1,t 2 } where 1 t 1 >t 2 0. (b) x X μ(x) = μ(0) } is a prime ideal of X. Proof. Let μ be a fuzzy prime ideal of X. Then (a)and (b) follow from Lemma 23. and 24. Conversely, let x, y X. If μ(x y) = t 2, then μ(x) =μ(y) =t 2 = μ(x y). Let P = x X μ(x) = μ(0) }. If μ(x y) = t 1, then x y P. Since P is a prime ideal of X, x P or y P.Thus μ(x y) = μ(x) or μ(x y) = μ(y). Hence μ is a fuzzy prime ideal of X. Definition 26. Let ν be a fuzzy subset of a subtraction algebra X. ν is said to be a fuzzy m-system if ν(x y) =minν(x),ν(y)} for all x, y X. Theorem 27. Let μ be a fuzzy ideal of a subtraction algebra X. Then μ is a fuzzy prime ideal of X if and only if 1 μ is a fuzzy m-system. Proof. Let μ be a fuzzy prime ideal of X. For any x, y X, μ(x y) = maxμ(x), μ(y)}. Thus μ(x y) = maxμ(x),μ(y)} = min μ(x), μ(y)}. Hence 1 μ(x y) = min1 μ(x), 1 μ(y)}. Thus (1 μ)(x y) = min(1 μ)(x), (1 μ)(y)}. Hence 1 μ is a fuzzy m-system of X. Conversely, for any x, y X, (1 μ)(x y) = min(1 μ)(x), (1 μ)(y)}. Hence 1 μ(x y) = min1 μ(x), 1 μ(y)}. Thus μ(x y) = min μ(x), μ(y)} = maxμ(x), μ(y)}. Hence μ(x y) = maxμ(x), μ(y)}. Therefore μ is a fuzzy prime ideal of X.

2352 P. Dheena and G. Mohanraaj Lemma 28. If ν is a fuzzy m-system of a subtraction algebra X then ν t is a m-system of X for all t [0, 1] whenever nonempty. Proof. Let ν be a fuzzy m-system of a subtraction algebra X. For any x, y ν t, ν(x y) =minν(x),ν(y)} t. Then x y ν t. Therefore ν t is a m-system. Remark 29. The converse of above Lemma 28. need not true as shown by the following example. Consider a subtraction algebra X as in Example 21. 0.8 if x =0 ν(x) = 0.3 otherwise Then the level sets of ν are 0} and X which are m-systems of X. But ν is not a fuzzy m-system since ν(a c) =ν(0) =.8 > minν(a),ν(c)} =.3 Definition 30. Let μ, λ be a fuzzy subset of a set S.λ μ =0if there exits t [0, 1) such that λ t μ t = φ. Example 31. Consider a subtraction algebra X as in Example 21. 0.9 if x 0,d} μ(x) = 0.7 if x b, c} 0.3 otherwise 0.8 if x 0,a} σ(x) = 0.6 if x b, d} 0.4 if x = c 0.7 if x = c θ(x) = 0.5 if x a, b, 0} 0.4 if x = d Clearly μ σ 0,σ is a fuzzy ideal and θ is a fuzzy m-system.σ is not a fuzzy prime ideal since σ(b c) =σ(0) =.8 >.6=maxσ(b),σ(c)}. σ θ = 0 since σ.65 θ.65 = φ. Theorem 32. Let λ be a fuzzy ideal and ν be a fuzzy m-system of a subtraction algebra X such that λ ν =0. Then there is a fuzzy prime ideal μ with λ μ and μ ν =0. Proof. Letλ ν =0. Then there is a t [0, 1) such that λ t ν t = φ. By Theorem 18. λ t is an ideal of X and by Lemma 28. ν t is a m-system. Then

Prime and fuzzy prime ideals 2353 by Theorem 14. there is a prime ideal P containing λ t such that P ν t = φ. We can find s [0, 1] such that 1 >s>t>0. Now, 1 if x P μ(x) = t otherwise By Theorem 25. μ is a fuzzy prime ideal.if x λ t then μ(x) =1 λ(x). x/ λ t then λ(x) <t.now μ(x) t>λ(x). Therefore μ λ. Now μ s = P. Since s > t, ν s ν t.p ν t = φ implies P ν s = φ. Then μ s ν s = φ. Thus μ ν =0. References [1] J.C. Abbott, Sets, Lattices and Boolean Algebras, Allyn and Bacon, Boston, 1969. [2] Y.B.Jun,H.S.Kim and E.H.Roh, Ideal theory of subtraction algebras,scientiae Mathematicae Japonicae,61,No.3 (2005),459-464. [3] Y.B.Jun and K.H.Kim, Prime and irreducible ideals in subtraction algebras,international mathematical forum,3,no.10(2008), 457-462. [4] E.H. Roh, K.H. Kim and Jong Geol Lee, On Prime and Semiprime ideals in Subtraction Semigroups, Scientiae Mathematicae Japonicae, 61, No.2(2005), 259-266. [5] B.M. Schein, Difference Semigroups, Communications in algebra, 20, (1992), 2153-2169. [6] L.A.Zadeh, Fuzzy sets, Inform. and control, 8 (1965), 338-353. [7] B. Zelinka, Subtraction Semigroups, Math. Bohemica, 120, (1995), 445-447. Received: March, 2009