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ON HOMOMORPHISMS of KRASNER HYPERRINGS 1. Introduction a Hyperoperation on a Nonempty Set H Is a Function

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ON HOMOMORPHISMS of KRASNER HYPERRINGS 1. Introduction a Hyperoperation on a Nonempty Set H Is a Function ANALELE S¸TIINT¸IFICE ALE UNIVERSITAT¸II˘ \AL.I. CUZA" DIN IAS¸I (S.N.) MATEMATICA,˘ Tomul LVII, 2011, f.2 DOI: 10.2478/v10157-011-0023-2 ON HOMOMORPHISMS OF KRASNER HYPERRINGS BY W. PHANTHAWIMOL, Y. PUNKLA, K. KWAKPATOON and Y. KEMPRASIT Abstract. By a homomorphism from a Krasner hyperring (A; +; ·) into a Krasner 0 0 0 0 hyperring (A ; + ; · ) we mean a function f : A ! A satisfying f(x+y) ⊆ f(x)+f(y) and 0 f(x · y) = f(x) · f(y) for all x; y 2 A: The kernel of f, ker f, is defined by ker f = fx 2 0 0 0 0 0 A j f(x) = 0 g where 0 is the zero of (A ; + ; · ). In fact, ker f may be empty. In this paper, some general properties of a Krasner hyperring homomorphism with nonempty kernel are given. Various examples are also provided. Mathematics Subject Classification 2000: 20N20. Key words: Krasner hyperring, homomorphism. 1. Introduction A hyperoperation on a nonempty set H is a function ◦ : H × H ! P(H)nf;g where P(H) is the power set of H and (H; ◦) is called a hyper- groupoid. For nonempty subsets A; B of H and x 2 H; let [ A ◦ B = a ◦ b; A ◦ x = A ◦ fxg and x ◦ A = fxg ◦ A: a2A b2B An element e 2 H is called an identity of (H; ◦) if x 2 x ◦ e \ e ◦ x, for all x 2 H and it is called a scalar identity of (H; ◦) if x ◦ e = e ◦ x = fxg, for all x 2 H. If e is a scalar identity of (H; ◦), then e is the unique identity of (H; ◦): The hypergroupoid is said to be commutative if x ◦ y = y ◦ x, for all x; y 2 H: A hypergroupoid (H; ◦) is called a semihypergroup if (x◦y)◦z = x◦(y◦z); for all x; y; z 2 H: 240 W. PHANTHAWIMOL, Y. PUNKLA, K. KWAKPATOON, Y. KEMPRASIT 2 A semihypergroup is called a hypergroup if H ◦ x = x ◦ H = H; for all x 2 H: An element y of a hypergroupoid (H; ◦) is called an inverse of x 2 H if (x ◦ y) \ (y ◦ x) contains at least one identity. A hypergroup (H; ◦) is said to be regular if every element of (H; ◦) has at least one inverse. A regular hypergroup (H; ◦) is said to be reversible if for all x; y; z 2 H; 0 0 0 x 2 y ◦ z ) z 2 y ◦ x and y 2 x ◦ z , for some inverse y of y and some 0 inverse z of z. A hypergroup (H; ◦) is called a canonical hypergroup if: 1. (H; ◦) is commutative; 2. (H; ◦) has a scalar identity; 3. every element of (H; ◦) has a unique inverse; 4. (H; ◦) is reversible. A canonical subhypergroup of a canonical hypergroup (H; ◦) is a non- empty subset K of H containing the scalar identity of (H; ◦) which forms a canonical hypergroup under the hyperoperation ◦ on H. A Krasner hyperring is a triple (A; +; ·) such that 1. (A; +) is a canonical hypergroup, 2. (A; ·) is a semigroup with zero 0 where 0 is the scalar identity of (A; +), 3. for all x; y; z 2 A; x · (y + z) = x · y + x · z and (y + z) · x = y · x + z · x: We call 0 the zero of the Krasner hyperring (A; +; ·): For x 2 A; let −x denote the unique inverse of x in (A; +). Then −(−x) = x, for all x 2 A: In addition, we have (x + y) · (z + w) ⊆ x · z + x · w + y · z + y · w; (−x) · y = x · (−y) = −(x · y); for all x; y; z; w 2 A; ([1], p.167). A subhyperring of a Krasner hyperring (A; +; ·) is a nonempty subset S of A which forms a Krasner hyperring containing 0 under the hyperoperation + and the operation · on A, that is, S is a canonical subhypergroup of (A; +) and S · S ⊆ S: Then a nonempty subset S of A is a subhyperring of (A; +; ·) if and only if, for all x; y 2 S; x + y ⊆ S, −x 2 S and x · y 2 S: A nonempty subset I of A is called a hyperideal of (A; +; ·) if I is a canonical subhypergroup of (A; +) and A · I [ I · A ⊆ I (see [1], p.168 or [2], p.20), that is, x + y ⊆ I and −x 2 I, for all x; y 2 I and xy; yx 2 I, for all x 2 I and y 2 A. 3 ON HOMOMORPHISMS OF KRASNER HYPERRINGS 241 Let I be a hyperideal of A and A=I = fx + I j x 2 Ag: We can see that for x 2 I; x + I = I and for x; y 2 A; (x + I) \ (y + I) =6 ; implies that x + I = y + I: Define (x + I) + (y + I) = fz + I j z 2 x + yg; (x + I) · (y + I) = x · y + I; for all x; y 2 I: Then (A=I; +; ·) is a Krasner hyperring ([1], p.169). It may be called the quotient hyperring of A by I. Notice that 0 + I is the zero of the Krasner hyperring (A=I; +; ·) and for every x 2 A; −x + I is the unique inverse of x + I in the canonical hypergroup (A=I; +). A homomorphism from a Krasner hyperring (A; +; ·) into a Krasner 0 0 0 0 0 hyperring (A ; + ; · ) is a function f : A ! A such that f(x + y) ⊆ f(x) + 0 f(y) and f(x · y) = f(x) · f(y); for all x; y 2 A: A homomorphism f from 0 0 0 (A; +; ·) into (A ; + ; · ) is said to be a good homomorphism if f(x + y) = 0 f(x) + f(y); for all x; y 2 A: 0 0 0 An isomorphism from (A; +; ·) into (A ; + ; · ) is a bijective good homo- 0 0 0 morphism from (A; +; ·) onto (A ; + ; · ). The Krasner hyperrings (A; +; ·) 0 0 0 ∼ 0 0 0 and (A ; + ; · ) are said to be isomorphic, and we write (A; +; ·) = (A ; + ; · ) 0 0 0 if there is an isomorphism from (A; +; ·) onto (A ; + ; · ). Notice that if f is 0 0 0 an isomorphism from (A; +; ·) onto (A ; + ; · ), then f −1 is an isomorphism 0 0 0 from (A ; + ; · ) onto (A; +; ·). Example 1.1. Let (A; ·) be a semigroup with zero 0 such that (Anf0g; ·) is a group. Define the hyperoperation + on A by 8 <> y + x = fxg; if y = 0; x + y = Anfxg; if x = y =6 0; :> fx; yg; if x; y 2 Anf0g and x =6 y: Then (A; +; ·) is a Krasner hyperring ([1], p.170). Notice that 0 is the zero of (A; +; ·) and for every x 2 A; x is the inverse of x in (A; +). Since Ax = A, for all x 2 Anf0g, it follows that f0g and A are the only hyperideals of (A; +; ·). Example 1.2. Define the hyperoperation ⊕ on the unit interval [0; 1] by ( fmax fx; ygg; if x =6 y; x ⊕ y = [0; x]; if x = y: Then ([0; 1]; ⊕; ·) is a Krasner hyperring where · is the usual multiplication ([3]). We can see that 0 is the zero of ([0; 1]; ⊕; ·) and for every x 2 [0; 1]; x is the inverse of x in ([0; 1]; ⊕). 242 W. PHANTHAWIMOL, Y. PUNKLA, K. KWAKPATOON, Y. KEMPRASIT 4 Define f : [0; 1] ! [0; 1] by f(x) = 1, for all x 2 [0; 1]: Since 1 ⊕ 1 = [0; 1] and 1 · 1 = 1, it follows that f is a homomorphism from the Krasner hyperring ([0; 1]; ⊕; ·) into itself. Observe that ker f = ;. It is easily seen that for a 2 (0; 1], [0; a] and [0; a) are nonzero hyper- ideals of ([0; 1]; ⊕; ·). Let I be an hyperideal of ([0; 1]; ⊕; ·). Then 0 2 I. Let a be the supremum of I. If a 2 I; then [0; a] ⊇ I ⊇ a ⊕ a = [0; a]; so I = [0; a]: Assume that a2 = I. Then a > 0 and I ⊆ [0; a): Let N be the set 2 N − 1 of positive integers. Let N be such that a N > 0. Since a = sup I, it follows that for every [n ≥ N; there] exists an element xn 2 I such that − 1 ≤ − 1 ⊆ ⊕ ⊆ ≥ a n xn < a. Hence 0; a n [0; xn] = xn xn I, for all n N. Thus [ [ 1 ] [0; a) = 0; a − ⊆ I: n n2N n≥N It follows that I = [0; a): This shows that f[0; a] j a 2 [0; 1]g [ f[0; a) j a 2 (0; 1]g is the set of all hyperideals of the Krasner hyperring ([0; 1]; ⊕; ·). Notice from the proof that it is also the set of all canonical subhypergroups of the canonical hypergroup ([0; 1]; ⊕). For a 2 (0; 1) and x; y 2 [0; 1]; we have ( ( [0; a]; if x ≤ a; [0; a); if x < a; x ⊕ [0; a] = ; x ⊕ [0; a) = fxg; if x > a; fxg; if x ≥ a; { } [0; 1]=[0; a] = fz ⊕ [0; a] j z 2 [0; 1]g = f[0; a]g [ fzg j z 2 (a; 1] ; { } [0; 1]=[0; a) = fz ⊕ [0; a) j z 2 [0; 1]g = f[0; a)g [ fzg j z 2 [a; 1] ; (x ⊕ [0; a]) ⊕ (y ⊕ [0; a]) = fz ⊕ [0; a] j z 2 x ⊕ yg 8{ } > f g < x ; { } if x > y > a; = f[0; a]g [ fzg j z 2 (a; x] ; if x = y > a; :> f[0; a]g if x; y ≤ a; (x ⊕ [0; a)) ⊕ (y ⊕ [0; a)) = fz ⊕ [0; a) j z 2 x ⊕ yg 8{ } > f g ≥ < x ; { } if x > y a; = f[0; a)g [ fzg j z 2 [a; x] ; if x = y ≥ a; :> f[0; a)g; if x; y ≤ a; 5 ON HOMOMORPHISMS OF KRASNER HYPERRINGS 243 ( [0; a]; if xy ≤ a; (x ⊕ [0; a]) · (y ⊕ [0; a]) = xy ⊕ [0; a] = fxyg; if xy > a; ( [0; a); if xy < a; (x ⊕ [0; a)) · (y ⊕ [0; a)) = xy ⊕ [0; a) = fxyg; if xy ≥ a: 2.
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