ON PRIME ONE-SIDED ORDERED IDEALS OF ORDERED SEMIRINGS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. We define the notions of prime, semiprime, weakly prime and weakly semiprime properties of onesided ordered ideals of ordered semirings, study their connections and use them to characterize several kinds of regularities on ordered semirings.


INTRODUCTION
There are several different notions of prime, weakly prime, semiprime and weakly semiprime properties of ideals of semirings e.g. [1,2,3,5,6]. However, on an ordered semigroup S, Kehayopulu [7] defined an ideal T of S to be prime (resp. semiprime) if for any A, B ⊆ S, AB ⊆ T implies A ⊆ T or B ⊆ T (resp. A 2 ⊆ T implies A ⊆ T ). In sense of Kehayopulu, an ideal T of an ordered semigroup S is weakly prime if for any ideals A and B of S, AB ⊆ T implies A ⊆ T or B ⊆ T . Kehayopulu also showed that an ideal of an ordered semigroup is prime if and only if it is both semiprime and weakly prime. Particularly, if the ordered semigroup is commutative, then the concepts of prime and weakly prime ideals coincide. Then Kehayopulu use prime and weakly prime ideal to characterize intra-regular and fully idempotent ordered semigroups, respectively.
In this work, we study prime, weakly prime, semiprime and weakly semiprime properties in a similar way of results of Kehayopulu [7] but generalize them to the case of one-sided (either left or right) ordered ideals and study them on ordered semirings. Moreover, we have that there is a one-sided ordered ideal of an ordered semiring which is both semiprime and weakly prime but not prime. This means that there is a condition which is true in case of two-sided ordered ideals but not true in case of one-sided ordered ideals. Finally, we use prime, weakly prime, semiprime and weakly semiprime one-sided ordered ideals to characterize several kinds of regularities on ordered semirings.

PRELIMINARIES
An ordered semiring [4] is an algebraic structure (S, +, ·, ≤) such that (S, +, ·) is a semiring and (S, ≤) is a poset satisfying the property; for any a, b, c ∈ S, if a ≤ b, then a + c ≤ b + c, c + a ≤ c+b, ac ≤ bc and ca ≤ cb. An ordered semiring (S, +, ·, ≤) is called additively commutative if a + b = b + a for all a, b ∈ S. An element 0 of an ordered semiring S is said to be an absorbing zero if a + 0 = a = 0 + a and a0 = 0 = 0a for all a ∈ S. Throughout this work, we simply write S instead of an ordered semiring (S, +, ·, ≤) and always assume that it is additively commutative together with an absorbing zero.
For any nonempty subsets A and B of an ordered semiring S, we denote that (A] = {x ∈ S | x ≤ a for some a ∈ A} and ΣA = ∑ i∈I a i ∈ S | a i ∈ A and I is a finite index set . In a particular case of a ∈ S, we write Σa and (a] instead of Σ{a} and ({a}], respectively. If I = / 0, then we set ∑ i∈I a i = 0 for all a i ∈ S. We review some basic properties of the operator ( ] and the finite sums Σ of nonempty subsets of ordered semirings which occur in [8,9,10]  (1) A ⊆ ΣA and Σ(ΣA) = ΣA; (2) if A ⊆ B, then ΣA ⊆ ΣB; (3) Σ(A + B) ⊆ ΣA + ΣB; A nonempty subset A of an ordered semiring S is called a left (right) ordered ideal of S if If A is either left or right ordered ideal of S, then A is also called a one-sided ordered ideal of S. Moreover, if A is both a left and a right ordered ideal of S, then A is called an ordered ideal (also called a two-sided ordered ideal) of S [4]. is a left (resp. right) ordered ideals of S as well.
Let a ∈ S. We denote L(a) and R(a) to be the smallest left ordered ideal and right ordered ideal of S containing a, respectively. We recall their constructions which occur in [8,9] as follows.  (resp. a ∈ (a 2 S]) for all a ∈ S.
A generalization of a left regular (resp. right regular) ordered semiring is defined as follows.
Remark 2.6. If S is a left (resp. right) regular ordered semiring, then S is also left (resp. right) weakly regular. In consequences of Remark 2.6 and Lemma 2.7, we obtain the following corollary.
Using the condition (2.1) and Remark 2.1, we obtain that Using the condition (2.2) and Remark 2.1, we obtain that Therefore, S is left weakly regular.
We obtain the following lemma as a duality of Lemma 2.10.
Lemma 2.10. Let S be an ordered semiring. Then the following conditions are equivalent: (1) S is right weakly regular; (2) A ∩ B ⊆ (ΣAB] for any right ordered ideals A and B of S; (3) R = (ΣR 2 ] for every right ordered ideal R of S;

MAIN RESULTS
Now, we give the notions of prime and semiprime subsets of an ordered semiring as follows.
The following remark can be directly obtained by Definition 3.1.

Remark 3.2. A nonempty subset T of an ordered semiring S is prime if and only if for any nonempty subsets
The following remark can be directly obtained by Definition 3.3.
It is obvious that every prime subset of an ordered semiring is semiprime. The converse of this condition is not generally true as shown by the following example. Define a binary relation ≤ on S by Then (S, +, ·, ≤) is an additively commutative ordered semiring with an absorbing zero 0. It is easy to see that {0, b} is a left ordered ideal of S which is not a right ordered ideal. We have that {0, b} is a semiprime left ordered ideal but not prime because ca = b ∈ {0, b} and c, a ∈ {0, b}. Now, we define the weakly prime property of one-sided ordered ideals of an ordered semiring as follows.
Definition 3.6. A left (resp. right) ordered ideal T of an ordered semiring S is called weakly prime if for any left (resp. right) ordered ideals A and B of S, AB ⊆ T implies A ⊆ T or B ⊆ T .
It is clear that every prime one-sided ordered ideal is weakly prime. The converse is not true as shown by the following example In an ordered semigroup S, Kehayopulu [7] proved that if S is commutative, then the concepts of weakly prime (two-sided) ideals and prime (two-sided) ideals coincide. It is not difficult to prove that this condition is true in case of one-sided ordered ideals on a commutative ordered semiring as well. However, we are able to use a condition which is more general than the commutative condition as follows.
Remark 3.8. If an ordered semiring S is commutative, then xSy ⊆ Sxy for all x, y ∈ S.
We give the following example to show that the condition xSy ⊆ Sxy for all x, y ∈ S, is a generalization of the commutative condition.  Proof. Assume that xSy ⊆ Sxy for all x, y ∈ S. Let T be a weakly prime left ordered ideal of S.
We show that T is also prime. Let a, b ∈ S such that ab ∈ T . Then by Remark 2. Since T is weakly prime, we have that a ∈ L(a) ⊆ T or b ∈ L(b) ⊆ T . Therefore, T is prime.
As a duality of Remark 3.8 and Theorem 3.10, we obtain the following remark and theorem.
Remark 3.11. If an ordered semiring S is commutative, then xSy ⊆ xyS for all x, y ∈ S. Theorem 3.12. Let S be an ordered semiring. If xSy ⊆ xyS for all x, y ∈ S, then prime right ordered ideals and weakly prime right ordered ideals coincide.
We give the following two examples to show that the concepts of semiprime and weakly prime properties of one-sided ordered ideals are independent.  In consequences of Example 3.13 and 3.14, we are able to say that the concepts of semiprime property and weakly prime property of one-sided ordered ideals are independent.
In ordered semigroups, Kehayopulu [7] proved that if a two-sided ideal is both weakly prime and semiprime, then it is prime as well. However, this condition fails in cases of one-sided ordered ideals. On an ordered semiring, we refer the weakly prime left ordered ideal {0, c, e} defined in Example 3.7 again. It is easy to see that {0, c, e} is also semiprime but not prime. In case of an ordered semigroup, we give the following example. is not difficult to verify that {c} is a left ideal of S which is both weakly prime and semiprime but not prime. Now, we define the weakly semiprime property of one-sided ordered ideals of an ordered semiring as follows.
Definition 3.16. A left (resp. right) ordered ideal T of an ordered semiring S is called weakly semiprime if for each left (resp. right) ordered ideal A of S, It is clear that every semiprime one-sided ordered ideal is weakly semiprime. In general, the converse is not true as a consequence of Example 3.14. Now, we show that semiprime and weakly semiprime properties of left ordered ideals of an ordered semiring are coincidence without the commutative condition. Proof. Assume that xSy ⊆ Sxy for all x, y ∈ S. Let T be a weakly semiprime left ordered ideal of S. We show that T is also semiprime. Let a ∈ S such that a 2 ∈ T . Then by Remark 2.1 and Since T is weakly semiprime, we have that a ∈ L(a) ⊆ T . So, T is semiprime.
As a duality of Theorem 3.17, we obtain the following theorem.   It is easy to verify that the ordered semiring (S, +, ·, ≤) defined in Example 3.13 is left weakly regular. Moreover, we have that the left ordered ideal {0, b} of S is weakly semiprime but not weakly prime. This shows that the left weakly regular condition of S is not sufficient to get that every left ordered ideal of S is weakly prime. We add a condition to obtain it as the following theorem.
Theorem 3.20. Let S be an ordered semiring and the set of all left (resp. right) ordered ideals of S be a chain. Then S is left (resp. right) weakly regular if and only if every left (resp. right) ordered ideal of S is weakly prime.
Proof. Assume that S is left weakly regular. Let A, B and L be left ordered ideals of S such that AB ⊆ L. Since the set of all left ordered ideals of S is a chain, we get that A ⊆ B or B ⊆ A. Conversely, assume that every left ordered ideal of S is weakly prime. Then we immediately have that every left ordered ideal of S is weakly semiprime as well and thus S is left weakly regular by Theorem 3.19.
Theorem 3.21. Let S be an ordered semiring. Then S is left (resp. right) regular if and only if every left (resp. right) ordered ideal of S is semiprime.
Proof. Assume that S is left regular. Let L be a left ordered ideal of S and a ∈ S such that a 2 ∈ L.
Conversely, assume that every left ordered ideal of S is semiprime. Let x ∈ S. We have that and so x ∈ (Sx 2 ]. Therefore, S is left regular. Hence, the proof is done.  Conversely, assume that every left ordered ideal of S is prime. Accordingly, we now have that every left ordered ideal of S is semiprime as well. By Theorem 3.21, we get that S is left regular. As a duality of Lemma 3.22 and Theorem 3.23, we obtain the following lemma and theorem. PRIME ONE-SIDED ORDERED IDEALS OF ORDERED SEMIRINGS 3805 Lemma 3.24. Let S be an ordered semiring. If S is right regular and xSy ⊆ xyS for all x, y ∈ S, then R(x) ∩ R(y) ⊆ R(xy) for all x, y ∈ S. Theorem 3.25. Let S be an ordered semiring, the set of all right ordered ideals of S be a chain and xSy ⊆ xyS for all x, y ∈ S. Then S is right regular if and only if every right ordered ideal of S is prime.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.