On cartesian product of fuzzy prime and fuzzy semiprime ideals of ...

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64 Samit Kumar Majumder, Sujit Kumar Sardar 3. Fuzzy Prime and Fuzzy Semiprime Ideals Definition 3.1. [4] A fuzzy ideal µ of a semigroup S is called a fuzzy prime ideal of S if µ(xy) = max{µ(x), µ(y)}∀x, y ∈ S. Definition 3.2. [4] A fuzzy ideal µ of a semigroup S is called a fuzzy semiprime ideal of S if µ(x) ≥ µ(x 2 )∀x ∈ S. Theorem 3.1. Let S be a semigroup and µ be a non-empty fuzzy subset of S. Then the following are equivalent: (1) µ is a fuzzy prime ideal of S, (2) for any t ∈ [0, 1], the t-level subset µ t of µ(if it is non-empty) is a prime ideal of S. Proof. Let µ be a fuzzy prime ideal of S. Let t ∈ [0, 1] be such that µ t is non-empty. Let for x, y ∈ S, xy ⊆ µ t . Then µ(xy) ≥ t. Since µ is a fuzzy prime ideal of S, it follows that max{µ(x), µ(y)} ≥ t. So µ(x) ≥ t or µ(y) ≥ t. Consequently, x ∈ µ t or y ∈ µ t . Hence µ t is a prime ideal of S. Conversely, let every non-empty level subset µ t of µ be a prime ideal of S. Let x, y ∈ S and µ(xy) = t. Then µ(xy) ≥ t and xy ∈ µ t . So µ t is non-empty and xy ⊆ µ t . Since µ t is a prime ideal of S, x ∈ µ t or y ∈ µ t . So µ(x) ≥ t or µ(y) ≥ t. So max{µ(x), µ(y)} ≥ t, i.e., max{µ(x), µ(y)} ≥ µ(xy)......(1). Again since µ is a fuzzy ideal of S, so we have µ(xy) ≥ µ(x) and µ(xy) ≥ µ(y). Then µ(xy) ≥ max{µ(x), µ(y)}.....(2). Combining (1) and (2), we have µ(xy) = max{µ(x), µ(y)}. Hence µ is a fuzzy prime ideal of S. Theorem 3.2. Let S be a semigroup and µ be a non-empty fuzzy subset of S. Then the following are equivalent: (1) µ is a fuzzy semiprime ideal of S, (2) for any t ∈ [0, 1], the t-level subset µ t of µ(if it is non-empty) is a semiprime ideal of S. Proof. Let µ be a fuzzy semiprime ideal of S. Let t ∈ [0, 1] be such that µ t is nonempty. Let for x ∈ S, x 2 ∈ µ t . Then µ(x 2 ) ≥ t. Since µ is a fuzzy semiprime ideal of S, then µ(x) ≥ µ(x 2 ). It follows that µ(x) ≥ t. Consequently, x ∈ µ t . Hence µ t is a semiprime ideal of S. Conversely, let every non-empty level subset µ t of µ be a semiprime ideal of S. Let x ∈ S and µ(x 2 ) = t. Then µ(x 2 ) ≥ t and x 2 ∈ µ t . So µ t is non-empty. Since µ t is a semiprime ideal of S, x ∈ µ t . So µ(x) ≥ t ⇒ µ(x) ≥ µ(x 2 ). Hence µ is a fuzzy semiprime ideal of S. 4. Cartesian Product of Fuzzy Completely Prime and Fuzzy Completely Semiprime Ideals Definition 4.1. [1] Let µ and σ be two fuzzy subsets of a set X. Then the cartesian product of µ and σ is defined by (µ × σ)(x, y) = min{µ(x), σ(y)}∀x, y ∈ X. Lemma 4.1. Let µ and σ be two fuzzy subsets of a set X and t ∈ [0, 1]. Then (µ × σ) t = µ t × σ t .
On cartesian product of fuzzy prime and... 65 Proof. Let (x, y) ∈ µ t × σ t ⇔ x ∈ µ t and y ∈ σ t ⇔ µ(x) ≥ t and σ(y) ≥ t ⇔ min{µ(x), σ(y)} ≥ t ⇔ (µ × σ)(x, y) ≥ t ⇔ (x, y) ∈ (µ × σ) t . Hence (µ × σ) t = µ t × σ t . Proposition 4.1. Let µ and σ be two fuzzy left ideals(fuzzy right ideals, fuzzy ideals) of a semigroup S. Then µ × σ is a fuzzy left ideal(resp. fuzzy right ideal, fuzzy ideal) of S × S. Proof. Let µ and σ be two fuzzy left ideals of S and (a, b), (c, d) ∈ S × S. Then (µ × σ){(a, b)(c, d)} = (µ × σ)(ac, bd) = min{µ(ac), σ(bd)} ≥ min{µ(c), σ(d)} (since µ and σ are fuzzy left ideals of S) = (µ × σ)(c, d). Hence µ × σ is a fuzzy left ideal of S × S. Similarly we can prove the other cases also. Proposition 4.2. Let µ and σ be two fuzzy prime ideals of a semigroup S. Then µ × σ is a fuzzy prime ideal of S × S. Proof. By Proposition 4.3, µ×σ is a fuzzy ideal of S×S. Let (a, b), (c, d) ∈ S×S. Then (µ×σ){(a, b)(c, d)} = (µ×σ)(ac, bd) = min{µ(ac), σ(bd)} = min[max{µ(a), µ(c)}, max{ σ(b), σ(d)}]( since µ and σ are fuzzy prime ideals of S) = max[min{µ(a), σ(b)}, min{µ(c) , σ(d)}] = max{(µ × σ)(a, b), (µ × σ)(c, d)}. Hence (µ × σ) is a fuzzy prime ideal of S × S. Proposition 4.3. Let µ and σ be two fuzzy semiprime ideals of a semigroup S. Then µ × σ is a fuzzy semiprime ideal of S × S. Proof. By Proposition 4.3, µ×σ is a fuzzy ideal of S ×S. Let (a, b) ∈ S ×S. Then (µ× σ)(a, b) = min{µ(a), σ(b)} ≥ min{µ(a 2 ), σ(b 2 )}( since µ and σ are fuzzy semiprime ideals of S) = (µ × σ)(a 2 , b 2 ) = (µ × σ)(a, b) 2 . Hence (µ × σ) is a fuzzy semiprime ideal of S × S. Proposition 4.4. Let µ and σ be two fuzzy prime ideals of a semigroup S. Then the level subset (µ × σ) t , t ∈ Im(µ × σ) is a prime ideal of S × S. Proof. By Proposition 4.4, µ×σ is a fuzzy prime ideal of S ×S. Let for (x, y), (m, n) ∈ S × S, (x, y)(m, n) ∈ (µ × σ) t . Then (µ × σ){(x, y)(m, n)} ≥ t ⇒ (µ × σ)(xm, yn) ≥ t ⇒ min{µ(xm), σ(yn)} ≥ t ⇒ µ(xm) ≥ t and σ(yn) ≥ t ⇒ xm ∈ µ t and yn ∈ σ t ⇒ x ∈ µ t or m ∈ µ t and y ∈ σ t or n ∈ σ t (since µ t and σ t are prime ideals of S(cf. Theorem 3.3)). Hence (x, y) ∈ µ t × σ t or (m, n) ∈ µ t × σ t . Since by Lemma 4.2, (µ×σ) t = µ t ×σ t , we deduce that (x, y) ∈ (µ×σ) t or (m, n) ∈ (µ×σ) t . Consequently, (µ × σ) t is a prime ideal of S × S. Proposition 4.5. If the level subset (µ × σ) t , t ∈ Im(µ × σ) of µ × σ is a prime ideal of S × S then (µ × σ) is a fuzzy prime ideal of S × S. Proof. By Theorem 3.3, the proof follows immediately. Proposition 4.6. Let µ and σ be two fuzzy semiprime ideals of a semigroup S. Then the level subset (µ × σ) t , t ∈ Im(µ × σ) is a semiprime ideal of S × S.
Page 1: J o u r n a l of Mathematics and Ap

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