On the Prime Radical of a Hypergroupoid

In this study, we give definitions of a prime ideal, a s-semiprime ideal and a w-semiprime ideal for a hypergroupoid K. For an ideal A of K we show that radical of A (R(A)) can be represented as the intersection of all prime ideals of K containing A and we define a strongly A-nilpotent element. For any ideal A of K, we prove that R(A)=∩(s-semiprime ideals of K containing A)= ∩(w-semiprime ideals of K containing A)={strongly A nilpotent elements}. For an ideal B of K put B=B and B=(B). If a hypergroupoid K satisfies the ascending chain condition for ideals then (R(A))⊆A for some n. For an ideal A of K we give a definition of right radical of A (R+(A)). If K is associative then R(A)=R+(A)=R_(A).

K is a set and ⋅ is a binary operation on K. Definition 1.2 [1] : A complete -groupoid is a system (K, ⋅), where K is a complete lattice and ⋅ is a binary operation on K which satisfies the following conditions: a⋅ for all a, b t ∈K Let K be a set and denote by 2 K the set of all its subsets. Definition 1.3 [2] : A multivariable binary operation on K is a map ϑ:KxK→2 K . A hypergroupoid is a system (K, ϑ), where K is a set and ϑ is a multivariable operation on K. From now on, we write a⋅b instead of ϑ (a, b) Let (K, ⋅) be a hypergroupoid. For A, B ∈2 K . A≠∅, B≠∅, put A⋅B= and ∅⋅A=A⋅∅=∅ for all A∈2 K . Then (2 K , ⋅) is a complete -groupoid. Conversely, If (2 K , ⋅) is a complete -groupoid then a restriction of the binary operation of 2 K to K is a multivariable operation on K and K is a hypergroupoid, with respect to this operation. Let w be a ternary relation on K.
Conversely, let (K, ⋅) be a hypergroupoid. Denote by w the set (a, b, c)∈KxKxK such that a⋅b≠∅ and c∈a⋅b. Then w is a ternary relation on K.
Hypergroupoids contain the following two classes of algebraic systems.

1.
A partial binary operation ϑ on K is a map ϑ:A→K, where A is a subset of KxK. A partial groupoid is a system (K, ⋅), where ⋅ is a partial binary operation on K. Let (K, ⋅) be a partial groupoid and A is the definition domain of ⋅. For (a, b)∉A put a⋅b=∅. Then ⋅ is defined for all (a, b)∈KxK and (K, ⋅) is a hypergroupoid. 2. Let {k, ϑ V , v∈S} be a universal algebra such that every ϑ V is a binary operation on K. For (a, b)∈KxK put a⋅b={ϑ v (a, b), v∈S} then (K, ⋅) is a hypergroupoid.

Prime and Semiprime Elements of an Ordered
Gruopoid: Let (G, ⋅) be an ordered groupoid [1] , ch XIV). An ordered groupoid G is called o -groupoid if G is a complete lattice. Denote by 1 G the greatest element of G.
Definition 2.1 [1] : Let (G, ⋅) be an o -groupoid. An element p∈G is prime if p≠1 G and a⋅b≤p, for a, b∈G, then a≤p or b≤p. For a∈G, a≠1 G , denote by R G (a) the intersection of all prime elements of G containing a. Put R G (a)=1 G if there are not any element with this property.

Definition 2.2:
An element h∈G is s-semiprime if h≠1 G and a 2 ≤h, for a∈G, implies that a≤h.
For a∈G, a≠1 G , denote by r s G (a) the intersection of all s-semiprime elements of G containing a. Put r s G (a)=1 G if there are not any element with this property. For a∈G denote by <a> the groupoid generated by a. An element of the groupoid <a> will be denoted by f(a).

The Prime Radical of an Ideal
Definition3.1: Let K be a hypergroupoid. A right (left) ideal of K is a subset H such that ha⊆H (respectively a⋅h⊆H) for all a∈K, h∈H. An (two-side) ideal of K is a subset H such that ha⊆H and ah⊆H for all a∈K, h∈H. Denote by Id(K) (Id + (K), Id_(K)) the set of all ideals (respectively, right ideals, left ideals) of K. Put ∅∈Id(K), ∅∈Id + (K), ∅∈Id_(K). Then Id(K), Id + (K), Id_(K) are complete lattices with respect to the inclusion relation. The proof is clear. We next consider the multiplication operation A⋅B on 2 K .

Definition 3.3:
Hypersemigroup is a hypergroupoid K such that (A⋅B)⋅C=A⋅(B⋅C) for any A, B, C∈2 K .
If K is hypersemigroup then A⋅B∈Id(K) for any A, B∈Id(K). But there are a hypergroupoid K and A, B∈Id(K) such that A⋅B∉Id(K). Therefore for any hypergroupoid K we define a multiplication operation of ideals as follows: For A, B∈Id(K) denote by A⋅B the intersection of all ideals of K containing the set G={x|x=a⋅b, a∈A, b∈B} Multiplication operations on Id + (K) and Id_(K) are introduced similarly.

Proof:
We give a proof for Id(K) and the proofs for Id + (K) and Id_(K) are similar. Suppose A, B t ∈Id(K), t∈T. It is clear that A⋅ Conversely the ideal A⋅(∪B t ) is the smallest ideal containing all elements a⋅b, where a∈A, b∈ Since b∈B t for some t∈T then a⋅b∈A⋅B t . Therefore A⋅ Now, we apply the definitions and designations of the prime and semiprime elements of ordered groupoids to 2 K , Id(K), Id + (K), Id_(K). Put Remark: This proposition is known for semigroups [5] .
Every sequence {x o , x 1 ,..., x n ,...}, where x o =a, x n+1 ∈[x n ] 2 , will be called an s-sequence of the element a. Remark: This definition is similar to the definition of the n-sequence [6] .
Denote by n(A) the set of all strongly A-nilpotent elements of K. Assume that there exists an s-semiprime ideal of K containing A. Let a∈n(A) and S be s-semiprime ideal of K containing A. We first prove that a∈S. If a∉S, then By continuing in this manner we obtain an ssequence {x o , x 1 , ..., x n , ...} of the element a such that x n ∉S for all n. But this is a contradiction since every ssequence of the element a meets A. Thus a∈S and a∈r s (A) since S is any semiprime ideal containing A.

Hence n(A)⊆r s (A)⊆r w (A)⊆R(A). Now we prove that R(A)=n(A). If n(A)=K then n(A)=r s (A)=r w (A)=R(A)=K. Let n(A)≠K. Hence there exists b∈K such that b∉n(A).
Then there exists an ssequence X={x o , x 1 , ..., x n , ...} of the element b such that X∩A=∅. Denote by Σ the set of ideals M in K such that X∩M=∅, M⊇A. Σ is not empty since A∈Σ.
We can apply Zorn's lemma to the set Σ so there exists a maximal element P of Σ. We show that P is prime.
First, P is proper since b∉P. Let B, C∈Id(K), B⊄P, C⊄P. Then P∪B≠P and P∪C≠P. By the maximality of P in Σ. We have P∪B∉Σ and P∪C∉Σ. Hence there exist x m ∈X, x q ∈X such that x m ∈P∪B, x q ∈P∪C. Then [x m ]⊆P∪B, [x q ]⊆P∪C. Hence x m+1 ∈[x m ] 2 ⊆P∪B, x q+1 ∈[x q ] 2 ⊆P∪C. By continuing in this manner we find x m+1 ∈P∪B, x q+1 ∈P∪C for all t. Put n=max(m, q). Then x n ∈P∪B, x n ∈P∪C. Hence, x n+1 ∈[x n ] 2 ⊆(P∪B)⋅ (P∪C)⊆ P∪B⋅C by the Proposition 3.4. But x n+1 ∉P. Hence B⋅C⊄P. Therefore P is prime. Thus there exists a prime ideal P such that b∉P. Thus n(A)=r s (A)=r w (A)=R(A). From the Theorem 3.8, we obtain that every s-semiprime ideal of K is radical.
The ideal R(∅) will be called the prime radical of the hypergroupoid K and will be denoted by Pr. rad(K). The proof of this lemma follows from the Proposition 1.6 [7] .
(1) (2): If every ideal of K is radical then using the lemma we obtain A⋅B=R(A⋅B)=R(A)∩R(B)=A∩B.
(2) (3) ideal B∪P is A s -nilpotent. By the maximality of P we have B∪P=P. Hence B⊆P. This means that P is ssemiprime. Since P⊇A then R(A)⊆P by Theorem 3.8.

But P (n) ⊆A⊆R(A) for some n. Since R(A) is ssemiprime then P⊆R(A). Thus P=R(A)
Remark: This theorem is similar to the proposition for associative rings [9] .
Corollary 3.15: Let K be hypergroupoid satisfying the ascending chain condition for ideals. Then the following conditions are equivalent: 1. K (n) =∅ for some n. 2. K doesn't have a prime ideal; 3. K doesn't have a s-semiprime ideal. A proof follows from Theorem 3.14 and the definition of Pr. rad(K). Denote by Id r (K) the set of all radical ideals of K. Id r (K) is a complete lattice with respect to the inclusion relation. Denote by ∨ and ∧ the lattice operations in Id r (K). Theorem 3.16: Let K be a hypergroupoid. Then the lattice Id r (K) satisfies the infinite ∧-distributive condition: for any A, B t ∈Id r (K) Proof: The proof follows from Theorem 1.3 [7] . Theorem 3.17: Let K be a hypergroupoid satisfying the ascending chain candition for ideals. Then any radical ideal of K is an intersection of finite prime ideals and a such representation is unique.
Proof: First we prove the following lemma.

Lemma: H∈Id r (K) is prime ideal if and only if H is an
∧-indecomposable element of the lattice Id r (K).
Proof: Let A be a prime ideal of K and A=A 1 ∧A 2 , A 1 , A 2 ∈Id r (K). Then [7] . A 1 A 2 ⊆A 1 ∩A 2 ⊆R(A 1 ∩A 2 )=A 1 ∧A 2 =A. Hence A 1 ⊆A or A 2 ⊆A. Then A=A 1 or A=A 2 Let A be an ∧indecomposable element in Id r (K) and BC⊆A, B, C∈Id(K). Then R(B⋅C)⊆A. By the lemma 1.6 [7] we have R(B)∧R(C)=R(B⋅C)⊆A. By the distributivity Id r (K) we obtain A=A∨(R(B)∧ R(C))=(A∨R (B))∧ (A∨R(C)). Then A=A∨R(B) or A=A∨R(C) since A is ∧-indecomposable. This means that B⊆R(B)⊆A or C⊆R(C)⊆A.
Thus A is prime. The lemma is proved. By the lemma and the Corollary [1] we obtain that every radical ideal of K is an intersection of finite prime ideals and a such represantation is unique.    Proof: By Proposition 4.5 it is enough to prove that R + (A)⊆R(A). Denote by P(K) the set of all prime ideals of K and by P + (K) the set of all prime right ideals of K. We prove that P(K)⊆P + (K). Let Q∈P(K) and B⋅C⊆Q, B, C∈Id + (K). Then, (B∪K⋅B) (C∪K⋅C)= (B⋅C)∪ (B⋅(KC)) ∪((K⋅B)⋅C)∪(K⋅B)⋅(K⋅C)⊆Q.

The Right Prime Radical of an Ideal
Note that B∪KB and C∪KC are ideals of K. Indeed K⋅(B∪KB)=K⋅B∪(K⋅(K⋅B))⊆B∪KB. From (B∪KB) (C∪KC)⊆Q we obtain B⊆B∪KB⊆Q or C⊆C∪KC⊆Q since Q is prime. This means Q∈P + (K).

Remark:
The conditions of this theorem are satisfied for hypersemigroup. Therefore the same theorem is given for nonasociative hypergroupoid K and A∈Id(K) such that R(A)=R + (A) and R(A)≠R_(A). Let A∈Id + (K). For b∈K put b (o) =b, b (n+1) =(b (n) ) 2 .