Ordered Quasi(BI)--Ideals in Ordered -Semirings

Abstract

In this paper, we have defined ordered quasi--ideals and ordered bi--ideals in ordered -semirings by defining the relation “” in ordered semiring as if for any By using this relation we have shown that ordered quasi--ideals and ordered bi--ideals in ordered -semirings are generalization of quasi-ideals and bi-ideals in ordered semirings. Properties of many types of ordered -ideals including (semi)prime, (strongly) irreducible, and maximal ordered quasi--ideals and ordered bi--ideals in ordered -semirings have been studied.

1. Introduction

The notion of semiring was introduced by Vandiver [1] in 1934 whereas -semiring was introduced and studied by Rao [2] in 1995 as a generalization of the notion of -rings as well as of semirings. It is well known that semiring is a generalization of a ring and -semiring is a generalization of semiring but interestingly ideals of semirings do not coincide with ideals of rings. Applications of semirings are not limited to theoretical computer science, graph theory, optimization theory, automata, coding theory, and formal languages but in other branches of sciences and technologies as well.

Quasi-ideal was first introduced by O. Steinfeld for semigroups [3] and then for rings by the same author O. Steinfeld [4] where it is seen that quasi-ideal is a generalization of left and right ideals. Kiyoshi Iseki [5] introduced quasi-ideals for semirings without zero and showed results based on them. Donges studied quasi-ideals in semirings in [6] whereas Shabir et al. [7] characterized semirings by using quasi-ideals. Minimal quasi-ideals for -semiring was studied by Iseki [5].

The concept of bi-ideal for semigroups was given by Good and Hughes [8] in 1952. Generalized bi-ideal was then introduced for rings in 1970 by Szasz [9, 10] and then for semigroups by Lajos [11]. Many types of ideals on the algebraic structures were characterized by several authors such as the following. In 2000, Dutta and Sardar studied the characterization of semiprime ideals and irreducible ideals of -semirings [12]. In 2004, Sardar and Dasgupta [13] introduced the notions of primitive -semirings and primitive ideals of -semirings. In 2008, Kaushik et al. [14] introduced and studied bi--ideals in -semirings. In 2008, Pianskool et al. [15] introduced and studied valuation -semirings and valuation -ideals of a -semiring.

In 2008, Chinram studied properties of quasi-ideals in -semirings [16], whereas, in 2009, quasi-ideals and minimal quasi-ideals in -semiring were studied by Jagatap and Pawar [17]. In 2014, Jagatap [18] studied prime -Bi-ideals in -semirings. In 2015, Lavanya et al. [19] studied prime biternary -ideals in ternary -semirings. In 2016, Jagatap [20] defined the concept of bi-ideals in a -semiring and studied properties of bi-ideals in a regular -semiring. In 2018, Rao introduced ideals in ordered -semirings [21]. It is natural to define and study properties of ordered quasi--ideals and ordered bi--ideals in ordered -semirings. Therefore, we have extended the concept of ordered ideals in ordered -semirings to that of ordered quasi--ideals and ordered bi--ideals in ordered -semiring by considering different order than the order defined in [21].

2. Preliminaries and Basic Definitions

In this section we will define important terminologies based on those concepts which are being used in this paper.

Definition 1. Let and be commutative semigroups. Then we call a -semiring, if there exists a mapping is written as such that it satisfies the following axioms for all and :(i)(ii)(iii)(iv)

Example 2 (see [21]). Let be a semiring and denote the additive abelian semigroup of all matrices with identity element whose entries are from . Then is a -semiring with ternary operation is defined by as the usual matrix multiplication, where denotes the transpose of the matrix , for all and

Definition 3. A -semiring is called an ordered -semiring if it admits a compatible relation ; i.e., is a partial ordering on satisfying the following conditions.
If and then(i)(ii)(iii), and .

Definition 4. An ordered -semiring is said to be commutative ordered -semiring if , for all and

Definition 5 (see [21]). A nonempty subset of an ordered -semiring is called a sub--semiring of if is a subsemigroup of and for all and .

Definition 6 (see [21]). A nonempty subset of an ordered -semiring is called a left (right) ideal if is closed under addition, and if, for any , , then .

Definition 7. If is both a left and a right ideal of ordered -semirings , then is called an ideal of .

Definition 8. A right ideal of an ordered -semiring is called a right ideal if and such that , and then

A left ideal of is defined in a similar way.

Definition 9 (see [21]). A nonempty subset of an ordered -semiring is called a ideal if is an ideal and , ; then .

Definition 10. An ordered -semiring is called totally ordered if any two elements of are comparable.

Definition 11. Let and be ordered -semirings. A mapping is called a homomorphism if(i),(ii), for all

3. Ordered Quasi--Ideals in Ordered--Semirings

In this section we define ordered quasi--ideal, ordered quasi--ideal, ordered quasi--ideal, ordered prime quasi--ideal, ordered semiprime quasi--ideal, ordered maximal quasi--ideal, ordered irreducible, and ordered strongly irreducible quasi--ideal of an ordered -semiring We define the relation “” in as if for any For a nonempty subset of and , we say that in if for any

Definition 12. An ordered sub--semiring of an ordered -semiring is called an ordered quasi--ideal of if and if, for any , , then in .

Definition 13. An ordered quasi--ideals is called ordered quasi--ideal of if is an ordered sub--semiring of , and if and such that , then
Clearly every ordered quasi--ideal is an ordered quasi--ideal but converse is not true in general as shown in the following example.

Example 14. Let be the set of nonnegative integers and be additive abelian semigroups. Tennary operation is defined as , usual multiplication of integers. Then is an ordered -semirings. A subset of is an ordered quasi--ideal of but it is not an ordered quasi--ideal of .

Definition 15. An ordered quasi--ideal of an ordered -semiring is called ordered quasi--ideal of if , for ; then .

Theorem 16. Every ordered quasi--ideal of an ordered -semiring is an ordered quasi--ideal of .

Proof. Let be an ordered quasi--ideal of an ordered -semiring . Consider , , , and ; then Then , since is an ordered quasi--ideal. Hence is an ordered quasi--ideal of .

The following example shows that the converse of Theorem 16 is not true in general.

Example 17. Let be the set of all natural numbers. Then with usual ordering is an ordered semiring. If , then is an ordered -semiring. If then forms an ordered quasi--ideal but not ordered quasi--ideals of ordered -semiring .

Definition 18. An ordered -semiring is called band if every element of is an idempotent.

Theorem 19. Let be an ordered sub--semiring of an ordered -semiring in which semigroup is a band. Then is an ordered quasi--ideal of if and only if is an ordered quasi--ideals of .

Proof. Let be an ordered quasi--ideal of an ordered -semiring and , , and then as is an ordered quasi--ideal of Hence, is a an ordered quasi--ideal of .
Conversely suppose that is an ordered quasi--ideal of an ordered -semiring . Let and , since is an ordered quasi--ideal of an ordered -semiring . Hence is an ordered quasi--ideal of .

Definition 20. An ordered quasi--ideal of is called an irreducible ordered quasi--ideal if implies or , for any ordered quasi--ideal and of

Definition 21. An ordered quasi--ideal of is called strongly irreducible ordered quasi--ideal if, for any ordered quasi--ideal and of , implies or It is obvious that every strongly irreducible ordered quasi--ideal is an irreducible ordered quasi--ideal.

Definition 22. An ordered quasi--ideal of is called a prime ordered quasi--ideal if implies or , for any ordered quasi--ideals of

Definition 23. An ordered quasi--ideal of is called a prime ordered quasi--ideal if implies or , for any ordered quasi--ideals of

Definition 24. An ordered quasi--ideal of is called a strongly prime ordered quasi--ideal if implies that or for any ordered quasi--ideals , of

Definition 25. An ordered quasi--ideal of is called a semiprime ordered quasi--ideal if for any ordered quasi--ideals , of implies

It is clear that every strongly prime ordered quasi--ideal in is a prime ordered quasi--ideal and every prime ordered quasi--ideal in is a semiprime ordered quasi--ideal.

Definition 26. An ordered quasi--ideal of an ordered -semiring is said to be maximal ordered quasi--ideal if and for every ordered quasi--ideal of with , then either or .

Theorem 27. In an ordered -semiring , every maximal ordered quasi--ideal of is irreducible ordered quasi--ideal of .

Proof. Let be a maximal ordered quasi--ideal of an ordered -semiring . Suppose is not irreducible and and and a contradiction. Hence is irreducible ordered quasi--deal of an ordered -semiring .

Theorem 28. Let be an ordered quasi--ideal of an ordered -semiring .(i)If is a prime ordered quasi--ideal, then is a strongly irreducible ordered quasi--ideal.(ii)If is a strongly irreducible ordered quasi--ideal, then is an irreducible ordered quasi--ideal.

Proof. Let be an ordered quasi--ideal of an ordered -semiring .
(i) Suppose is a prime ordered quasi--ideal, J and K are ordered quasi--ideal of an ordered -semiring such that . Then or . Hence, is a strongly irreducible ordered quasi--ideal.
(ii) Suppose is a strongly irreducible ordered quasi--ideal; and are ordered quasi--ideal of an ordered -semiring such that . Then certainly or Hence or .
Therefore, is an irreducible ordered quasi--ideal of .

Corollary 29. Let be an ordered -semiring. If is a prime ordered quasi--ideal of , then is an irreducible ordered quasi--ideal of .

Theorem 30. Let be a homomorphism of ordered -semirings. If is an ordered quasi--ideal of , then is an ordered quasi--ideal of an ordered -semiring .

Proof. Suppose is an ordered quasi--ideal of , is a homomorphism of ordered -semirings, and , . If , . We need to show here that is an ordered quasi--ideal of ; i.e., . For that we suppose and for and , which is a quasi-ideal. Now and , as
Let such that . Hence is an ordered quasi--ideal of an ordered -semiring .

Theorem 31. Let be an ordered -semiring. If is an ordered quasi--ideal of , then is a maximal ordered quasi--ideal of .

Proof. Let be an ordered quasi--ideal of an ordered -semiring . Suppose is an ordered quasi--ideal of such that and . We have , since is an ordered quasi--ideal of . Therefore . Hence ordered quasi--ideal of is maximal ordered quasi--ideal of .

Theorem 32. Let be an ordered -semiring in which is cancellative semigroup and If , then is an ordered quasi--ideal of an ordered -semiring

Proof. Let and . Then . Similarly Suppose . Then . Suppose . Then . Let for and Next, In a similar way, we can show that Therefore, Hence is a quasi-ideal.
Therefore, is an ordered quasi--ideal of an ordered -semiring. Suppose , and . Then . Hence is an ordered quasi--ideal of .

Theorem 33. The intersection of family of prime (or semiprime) ordered quasi--ideal of is a semiprime ordered quasi--ideal.

Proof. Let be the family of prime ordered quasi--ideal of S and Since and Hence is a quasi--ideal.
Now for any such that Hence is an ordered quasi--ideal of . For any ordered quasi--ideal of , implies for all . As are semiprime ordered quasi--ideals, for all . Hence

Theorem 34. Every strongly irreducible, semiprime ordered quasi--ideal of is a strongly prime ordered quasi--ideal.

Proof. Let be a strongly irreducible and semiprime ordered quasi--ideal of . For any ordered quasi--ideal and of , let . is a quasi--ideal of , since Similarly we get Therefore As is a semiprime ordered quasi--ideal, . But is a strongly irreducible ordered quasi--ideal. Therefore, or . Thus is a strongly prime ordered quasi--ideal of .

Theorem 35. If is an ordered quasi--ideal of and such that , then there exists an irreducible ordered quasi--ideal of such that and .

Proof. Let be the family of ordered quasi--ideals of which contain but do not contain element . Then is nonempty as . This family of ordered quasi--ideals of S forms a partially ordered set under the set inclusion. Hence by Zorn’s lemma there exists a maximal ordered quasi--ideal say in . Therefore and . Now to show that is an irreducible ordered quasi--ideal of , let and be any two ordered quasi--ideals of such that . Suppose that and both contain properly. But is a maximal ordered quasi--ideal in . Hence we get and . Therefore which is absurd. Thus either or . Therefore, is an irreducible ordered quasi--ideal of .

Theorem 36. The following statements are equivalent in :(1)The set of ordered quasi--ideals of is totally ordered set under inclusion of sets.(2)Each ordered quasi--ideal of is strongly irreducible.(3)Each ordered quasi--ideal of is irreducible.

Proof. Suppose that the set of ordered quasi--ideals of is a totally ordered set under inclusion of sets. Let be any ordered quasi--ideal of . Then is a strongly irreducible ordered quasi--ideal of for that let and be any two ordered quasi--ideal of such that . But, by the hypothesis, we have either or . Therefore, or . Hence or . Thus is a strongly irreducible ordered quasi--ideal of .
Suppose that each ordered quasi--ideal of is strongly irreducible. Let be any ordered quasi--ideal of such that for any ordered quasi--ideal and of . But by hypothesis or . As and , we get or . Hence is an irreducible ordered quasi--ideal of .
Suppose that each ordered quasi--ideal of is an irreducible ordered quasi--ideal. Let and be any two ordered quasi--ideal of . Then is also ordered quasi--ideal of . Hence will imply or by assumption. Therefore either or . This shows that the set of ordered quasi--ideal of is totally ordered set under inclusion of sets.

Theorem 37. A prime ordered quasi--ideal of is a prime one sided ordered ideal of .

Proof. Let be a prime quasi--ideal of . Suppose is not a one sided ideal of . Therefore, and . As is a prime quasi--ideal, . which is a contradiction. Therefore, or . Thus is a prime one sided ideal of .

Theorem 38. An ordered quasi--ideal of is prime if and only if, for a right ordered ideal and a left ordered ideal of , implies or .

Proof. Suppose that an ordered quasi--ideal of is a prime ordered quasi--ideal of . Let be a right ordered ideal and be a left ordered ideal of such that . and are ordered quasi--ideal of . Hence or . Conversely, we have to show that an ordered quasi--ideal of is a prime ordered quasi--ideal of . Let and be any two ordered quasi--ideals of such that . For any and , and , where and denote the right ordered ideal and left ordered ideal generated by and , respectively. Thus . Hence by the assumption or which further gives that or . In this way we get that or . Hence is a prime ordered quasi--ideal of .

4. Ordered Bi--Ideals in Ordered--Semirings

In this section we define ordered bi--ideal, ordered bi--ideal, ordered bi--ideal, ordered prime bi--ideal, ordered semiprime bi--ideal, ordered maximal bi--ideal, ordered irreducible, and ordered strongly irreducible bi--ideal of an ordered -semiring Recall the relation in the previous section “” in as if for any Further for a nonempty subset of and , in if for any

Definition 39. An ordered sub--semiring of an ordered -semiring is called an ordered bi--ideal of if and if, for any , , in , then .

Definition 40. An ordered bi--ideals is called ordered bi--ideal of if is an ordered sub--semiring of , and if and such that , then
Clearly every ordered bi--ideal is an ordered bi--ideal but converse is not true in general as shown in the following example.

Example 41. Let be the set of nonnegative integers and be additive abelian semigroups. Tennary operation is defined as , usual multiplication of integers. Then is an ordered -semirings. A subset of is an ordered bi--ideal of but it is not an ordered bi--ideal of .

Definition 42. An ordered bi--ideal of an ordered -semiring is called ordered bi--ideal of if , for , and then .

Theorem 43. Every ordered bi--ideal of an ordered -semiring is an ordered bi--ideal of .

Proof. Let be an ordered bi--ideal of an ordered -semiring . Consider , , and ; then Then , since is an ordered bi--ideal. Hence is an ordered bi--ideal of .

The following example shows that the converse of Theorem 43 is not true in general.

Example 44. Let be the set of all natural numbers. Then with usual ordering is an ordered semiring. If , then is an ordered -semiring. If then forms an ordered bi--ideal but not ordered bi--ideals of ordered -semiring .

Definition 45. An ordered -semiring is called band if every element of is a -idempotent.

Theorem 46. Let be an ordered sub--semiring of an ordered -semiring in which semigroup is a band. Then is an ordered bi--ideal of if and only if is an ordered bi--ideals of .

Proof. Let be an ordered bi--ideal of an ordered -semiring and , Then as is an ordered bi--ideal of Hence is a an ordered bi--ideal of .
Conversely suppose that is an ordered bi--ideal of an ordered -semiring . Let and in for any , since is an ordered bi--ideal of an ordered -semiring . Hence is an ordered bi--ideal of .

Definition 47. An ordered bi--ideal of is called an irreducible ordered bi--ideal if implies or , for any ordered bi--ideal and of

Definition 48. An ordered bi--ideal of is called strongly irreducible ordered bi--ideal if, for any ordered bi--ideal and of , implies or It is obviously that every strongly irreducible ordered bi--ideal is an irreducible ordered bi--ideal.

Definition 49. An ordered bi--ideal of is called a prime ordered bi--ideal if implies or , for any ordered bi--ideals of

Definition 50. An ordered bi--ideal of is called a prime ordered bi--ideal if implies or , for any ordered bi--ideals of

Definition 51. An ordered bi--ideal of is called a strongly prime ordered bi--ideal if implies that or for any ordered bi--ideals , of

Definition 52. An ordered bi--ideal of is called a semiprime ordered bi--ideal if for any ordered bi--ideals of implies

It is clear that every strongly prime ordered bi--ideal in is a prime ordered bi--ideal and every prime ordered bi--ideal in is a semiprime ordered bi--ideal.

Definition 53. An ordered bi--ideal of an ordered -semiring is said to be maximal ordered bi--ideal if and for every ordered bi--ideal of with , then either or .

Theorem 54. In an ordered -semiring , every maximal ordered bi--ideal of is irreducible ordered bi--ideal of .

Proof. Let be a maximal ordered bi--ideal of an ordered -semiring . Suppose is not irreducible and and and a contradiction. Hence is irreducible ordered bi--deal of an ordered -semiring .

Theorem 55. Let be an ordered bi--ideal of an ordered -semiring .(i)If is a prime ordered bi--ideal, then is a strongly irreducible ordered bi--ideal.(ii)If is a strongly irreducible ordered bi--ideal, then is an irreducible ordered bi--ideal.

Proof. Let be an ordered bi--ideal of an ordered -semiring .
(i) Suppose is a prime ordered bi--ideal; J and K are ordered bi--ideal of an ordered -semiring such that . Then or . Hence is a strongly irreducible ordered bi--ideal.
(ii) Suppose is a strongly irreducible ordered bi--ideal; and are ordered bi--ideal of an ordered -semiring such that . Then certainly or . Hence or .
Therefore, is an irreducible ordered bi--ideal of .

Corollary 56. Let be an ordered -semiring. If is a prime ordered bi--ideal of , then is an irreducible ordered bi--ideal of .

Theorem 57. Let be a homomorphism of ordered -semirings. If is an ordered bi--ideal of then is an ordered bi--ideal of an ordered -semiring .

Proof. Suppose is an ordered bi--ideal of , be a homomorphism of ordered -semirings and . If . Now for and Since is an ordered bi--ideal, so we have that is a bi--ideal of .
Let such that in for any in . Hence is an ordered bi--ideal of an ordered -semiring .

Theorem 58. Let be an ordered -semiring. If is an ordered bi--ideal of , then is a maximal ordered bi--ideal of .

Proof. Let be an ordered bi--ideal of an ordered -semiring . Suppose is an ordered bi--ideal of such that and . We have , since is an ordered bi--ideal of . Therefore, . Hence ordered bi--ideal of is maximal ordered bi--ideal of .

Theorem 59. Let be an ordered -semiring in which is cancellative semigroup and If , then is an ordered and bi--ideal of an ordered -semiring

Proof. Let and . Then Therefore, . Similarly Suppose . Then . Suppose . Then . Further and , and then
Therefore, is an ordered bi--ideal of an ordered -semiring. Suppose Then Hence is an ordered bi--ideal of an ordered -semiring Suppose and . Then . Hence is an ordered bi--ideal of .

Theorem 60. The intersection of family of prime (or semiprime) ordered bi--ideal of is a semiprime ordered bi--ideal.

Proof. Let be the family of prime ordered bi--ideal of S and For Now for any such that Hence P is an ordered bi--ideal of . For any ordered bi--ideal of , implies for all . As are semiprime ordered bi--ideals, for all . Hence

Theorem 61. Every strongly irreducible, semiprime ordered bi--ideal of is a strongly prime ordered bi--ideal.

Proof. Let be a strongly irreducible and semiprime ordered bi--ideal of . For any ordered bi--ideal and of , let . is a bi--ideal of . Since Similarly we get Therefore, As is a semiprime ordered bi--ideal, . But is a strongly irreducible ordered bi--ideal. Therefore or . Thus is a strongly prime ordered bi--ideal of .

Theorem 62. If is an ordered bi--ideal of and such that , then there exists an irreducible ordered bi--ideal of such that and .

Proof. Let be the family of ordered bi--ideals of which contain but do not contain element . Then is nonempty as . This family of ordered bi--ideals of S forms a partially ordered set under the set inclusion. Hence by Zorn’s lemma there exists a maximal ordered bi--ideal say in . Therefore and . Now to show that is an irreducible ordered bi--ideal of . Let and be any two ordered bi--ideals of such that . Suppose that and both contain properly. But is a maximal ordered bi--ideal in . Hence we get and . Therefore which is absurd. Thus either or . Therefore is an irreducible ordered bi--ideal of .

Theorem 63. Following statements are equivalent in : (1)The set of ordered bi--ideals of is totally ordered set under inclusion of sets.(2)Each ordered bi--ideal of is strongly irreducible.(3)Each ordered bi--ideal of is irreducible.

Proof. Suppose that the set of ordered bi--ideals of is a totally ordered set under inclusion of sets. Let be any ordered bi--ideal of . Then is a strongly irreducible ordered bi--ideal of for that let and be any two ordered bi--ideal of such that . But by the hypothesis, we have either or . Therefore, or . Hence or . Thus is a strongly irreducible ordered bi--ideal of .
Suppose that each ordered bi--ideal of is strongly irreducible. Let be any ordered bi--ideal of such that for any ordered bi--ideal and of . But by hypothesis or . As and , we get or . Hence is an irreducible ordered bi--ideal of .
Suppose that each ordered bi--ideal of is an irreducible ordered bi--ideal. Let and be any two ordered bi--ideal of . Then is also ordered bi--ideal of . Hence will imply or by assumption. Therefore either or . This shows that the set of ordered bi--ideal of is totally ordered set under inclusion of sets.

Theorem 64. A prime ordered bi--ideal of is a prime one sided ordered ideal of .

Proof. Let be a prime bi--ideal of . Suppose is not a one sided ideal of . Therefore, and . As is a prime bi--ideal, . =, which is a contradiction. Therefore, or . Thus is a prime one sided ideal of .

Theorem 65. An ordered bi--ideal of is prime if and only if, for a right ordered ideal and a left ordered ideal of , implies or .

Proof. Suppose that an ordered bi--ideal of is a prime ordered bi--ideal of . Let be a right ordered ideal and be a left ordered ideal of such that . and are ordered bi--ideal of . Hence or . Conversely, we have to show that an ordered bi--ideal of is a prime ordered bi--ideal of . Let and be any two ordered bi--ideals of such that . For any and , and , where and denote the right ordered ideal and left ordered ideal generated by and , respectively. Thus . Here by the assumption we get that or which gives that or . Thus or . Hence is a prime ordered bi--ideal of .

5. Conclusion

In this work we have given many types of ordered , -ideals mainly ordered -(quasi, bi)--ideal, ordered (quasi, bi)--ideal, ordered (semi)prime, -(semi)prime (quasi, bi)--ideals, ordered maximal-(quasi, bi)--ideals, ordered (strongly) irreducible (quasi, bi)--ideals. We have studied properties of these different ideals and some of their relations in ordered -semirings. Let be a homomorphism of ordered -semirings. We have defined homomorphism and prove that if is an ordered quasi--ideal of , then is an ordered quasi--ideal of an ordered -semiring . The same is true in case of ordered bi--ideals. For further research work the extended identical study can be taken under consider for other algebraic structures.

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Conflicts of Interest

The authors declare that they have no conflicts of interest.