Generalized k-Ideals in Semirings using Soft

International Journal of Algebra and Statistics Volume 4: 1(2015), 20–38

Published by Modern Science Publishers Available at: http://www.m-sciences.com

Generalized k-Ideals in Semirings using Soft Intersectional Sets Tahir Mahmooda , Usman Tariqa a Department

of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan. (Received: 7 July 2015; Accepted: 13 July 2015)

Abstract. To discuss the problems involving uncertainties using soft sets and semirings is very common now a days. Keeping in view this point in this paper we discuss soft intersectional sets in semirings using kideals of the algebraic structure. We introduce the concept of (X,Y)-SI-k-subsemirings, (X,Y)-SI-k-bi-ideals and (X,Y)-SI-k-quasi-ideals of semirings. We discuss (X,Y)-soft-union-intersection sum and (X,Y)-softunion-intersection product and investigate some related results. Finally, we characterize k-semisimple semirings, k-regular semirings and k-intra-regular semirings by using their (X,Y)-SI-k-ideals.

1. Introduction The concept of soft set was introduced by Molodsov [28], to mathematically model the element of uncertainty in different areas, that could not be handled through already defined soft computing models, such as rough theory, vague theory, fuzzy theory, see [15, 34, 36, 37]. Evaluating the need and importance of the soft sets in applied field, the names of Maji [26], Ali [6, 8, 9] and, Sezgin and Atagun [30], are note worthy among the researchers who accelerated the preliminaries of the very work. On this, the logic defined in soft set started to be used in programming the various complex problems in the fields of decision making [10, 11, 13, 14, 18, 25, 29], approximation and data analysis [33, 40], description logic [19], etc., into formal languages. Henriksen [17], defined a restricted class of ideals in semirings, named as k-ideal. These ideals have an interesting property that if a semiring S is ring, then a complex in S is a k-ideal if and only if it is a ring ideal. k-ideals in semirings are discussed in [16, 20, 22, 31, 32], in terms of fuzziness, that behaves almost similar to soft ideals in soft intersection semirings. h-ideal in h-semisimple hemirings established by Yin et. at [35]. It is the fact that the role of ideals in the formation of semirings is of pivotal importance, however usually they do not coexist with general ring ideals, that is the reason, the usage of ideals in semirings is on limited scale as various outcomes in the rings obviously have no relevance in semirings using only ideals. Feng Feng and Y. B. Jun [12] defined the soft semirings and soft ideals in soft semirings. The study of soft groups by Aktas and Cagman [5], opened the doors of progress in the mean of algebraic structure. This progress lead the researchers to detailed study of soft semirings [12], soft rings [4], soft BCK/BCI algebra [21] and much more things in soft sets in the new directions, see [1, 2, 23, 24, 38]. Recently, (M,N)-soft-intersection and (M,N)-soft-union ideals in (M,N)-soft-intersection subhemirings and (M,N)-soft-union subhemirings are defined by J. Zhan [23, 39], respectively. 2010 Mathematics Subject Classification. 16Y60, 08A72, 03G25, 03E72. Keywords. Soft set, soft-union-intersection product, (X,Y)-SI-k-subsemiring, (X,Y)-SI-k-bi-ideal, (X,Y)-SI-k-quasi-ideal, (X,Y)-SIinterior k-ideal, k-semisimple semiring, k-regular semiring, k-intra-regular semiring. Email addresses: [email protected] (Tahir Mahmood), [email protected] (Usman Tariq)

T. Mahmood and U. Tariq / Int. J. of Algebra and Statistics 4 (2015), 20–38

21

In this paper, we present the soft analog of the fuzzy methodology of [27]. we define the (X,Y)-soft intersection k-subsemirings, (X,Y)-SI-k-ideals, (X,Y)-SI-k-bi-ideals, (X,Y)-SI-k-quasi-ideals, (X,Y)-SI-k-interiorideals. In Section-4, we discuss k-semisimple semirings in terms of (X,Y)-SI-k-ideals. Next Section-5 contains the discussion of k-regular semiring in soft intersection. Finally, we investigate some characterizations of k-intra-regular semirings by means of (X,Y)-SI-k-ideals in Section-6. 2. Preliminaries A nonempty set S is known as semiring under the binary operations ”+” and ”·” such that (S, +) and (S, ·) are semigroups, with the algebraic property that ”·” distributes over ”+” from both sides, then it can be written as (S, +, ·). If (S, ·) has an element e satisfying u · e = u = e · u ∀ u ∈ S, then the element e is known as identity (must be unique) of semiring. If (S, +) has an element e´ satisfying u + e´ = e´ = e´ + u and u · e´ = e´ = e´ · u ∀ u ∈ S, such element is called zero or absorbing element of semiring. A semiring S is regular, if for each u ∈ S there exist some 1 ∈ S such that u = u1u. A semiring S is intra-regular, if for each u ∈ S there exist n P some 1i , hi ∈ S such that u = 1i u2 hi . If Φ , M ⊆ S satisfying the M + M ⊆ M and MM ⊆ M, then M is i=1

called subsemiring of S. If Φ , I ⊆ S satisfying the I + I ⊆ I and SI ⊆ I (IS ⊆ I), then I is called left(right) ideal of S. If Φ , I ⊆ S is both left and right ideal of S, then it is simply said to be ideal of S. If Φ , Q ⊆ S satisfying Q + Q ⊆ Q and SQ ∩ QS ⊆ Q, then Q is called quasi-ideal of S. A subsemiring B of S satisfying BSB ⊆ B is called bi-ideal of S. Every left right ideal of S is also quasi-ideal and every quasi-ideal is also bi-ideal, but the converse is not true in general. For Φ , K ⊆ S, the k-closure of K is denoted and defined as b = u ∈ S | u + 1 = h for some 1, h ∈ K . K b⊆b c =G bb For a semiring S, G ⊆ F ⇒ G F and GF F for all G, F ⊆ S. Definition 2.1. A subsemiring(left ideal, right ideal, ideal, bi-ideal) K of S is called an k-subsemiring(left k-ideal, right k-ideal, k-ideal, k-bi-ideal) of S, respectively, if u + 1 = h implies u ∈ K for each u ∈ S and 1, h ∈ K, and will be represented by kS (kLi , kRi , kI , kBi ) c ∩ QS c ⊆ Q and u + 1 = h implies u ∈ Q for any Definition 2.2. A quasi-ideal Q of S is called k-quasi-ideal of S if SQ u ∈ S and 1, h ∈ Q, and will be represented by kQi . c ⊆ G ∩ F. Lemma 2.3. If G and F are kRi and kLi of a semiring S, respectively, then GF Definition 2.4. A semiring S is said to be k-regular if for each u ∈ S there exist 1, h ∈ S such that u + u1u = uhu, and will be represented by kR-S . c = G ∩ F for any G and F as kR and kL of S, respectively. Lemma 2.5. A semiring S is kR-S ⇔ GF i i Lemma 2.6. For a semiring S, the following are equivalent: (i) S is kR-S . d for every subset B as kB of S. (ii) B = BSB i [ for every subset Q as kQ of S. (iii) Q = QSQ i

Definition 2.7. A semiring S is said to be k-intra-regular if for each u ∈ S there exist 1i , 1í , h j , h´j ∈ S such that P P u + 1i u2 1í = h j u2 h´j , and will be represented by kIR-S . c for G and F as kR and kL of S, respectively. Lemma 2.8. A semiring S is kIR-S ⇔ G ∩ F ⊆ GF i i


22

Lemma 2.9. For a semiring S, the following are equivalent: (i) S is kR-S and kIR-S . (ii) B = Bb2 for every subset B as kBi of S. c2 for every subset Q as kQ of S. (iii) Q = Q i c2 , then A is called idempotent. Definition 2.10. For A ⊆ S, if A = A Definition 2.11. A semiring S is known as k-semisimple if every k-ideal of S is idempotent, and will be represented by kSS-S . Lemma 2.12. A semiring S is kSS-S if and only if for each u ∈ S there exist ei , fi , 1i , hi , e´j , f´j , 1´j , h´j ∈ S such that m m P P u + ei u fi 1i uhi = e´j u f´j 1´j uh´j . i=1

i=1

A soft set theory was introduced by Molodtsov [28] and Cagman and Enginoglu [11] provided new definitions and various results on soft set theory. For the next exploration, we take E = S (semiring), where E be the set of parameters and U be the initial set of universe. Let P (U) denote the power set of U and A, B, C, ... ⊆ E = S. Definition 2.13. An ordered set ξG = {(u, ξG (u)) : u ∈ E, ξG (u) ∈ P (U)} over U, is known as a soft set, where ξG : E → P (U) such that ξG (u) = Φ if u < A. ξG is called approximate function. Note that, the collection of all soft sets of the subscript set of ξ, over U will be denoted by CS (U). Definition 2.14. Let ξG , ξF ∈ CS (U), then ˜ F , if ξA (u) ⊆ ξB (u), ∀ u ∈ E. (i) ξG ⊆ξ ˜ F = ξG∪F , where ξG∪F (u) = ξG (u) ∪ ξF (u), ∀ u ∈ E. (ii) ξG ∪ξ ˜ F = ξG∩F , where ξG∩F (u) = ξG (u) ∩ ξF (u), ∀ u ∈ E. (iii) ξG ∩ξ Definition 2.15. The upper inclusion set of ξG for α ∈ P (U) with α , Φ is defined as U (ξG ; α) = {x ∈ G | ξG (u) ⊇ α} . Definition 2.16. For ξS , ηS ∈ CS (U) . Then soft union-intersection sum is defined by ( ∪u+(11 +h1 )=(12 +h2 ) ξS 11 ∩ ξS 12 ∩ ηS (h1 ) ∩ ηS (h2 ) , ξS ⊕ ηS (u) = Φ if u can not be expressed as u + 11 + h1 = 12 + h2 , ∀ u ∈ S. Definition 2.17. For ξS , ηS ∈ CS (U) . Then ξS ~ ηS is defined by ( !)  m n   ´ ´  ∩ ξS 1i ∩ ηS (hi ) ∩ ∩ ξS 1 j ∩ ηS h j ,   ∪u+ Pm 1i hi = Pn 1´ h´ i=1  j=1 j j  i=1 j=1 ξS ~ ηS (u) =   m n P P    Φ if u can not be expressed as u + 1 h = 1´j h´j ,  i i  i=1

j=1

∀ u ∈ S. Definition 2.18. Let S be a semiring and Φ , G ⊆ S, then characteristic soft set is denoted and defined by ( U if x ∈ G, CG (x) = Φ if x ∈ S \ G.


23

˜ The soft set CS ∈ CS (U) is called the identity soft set and denoted by C. Lemma 2.19. Let S be a kS and G, F ⊆ S. Then we have: ˜ F. (i) G ⊆ F ⇔ CG ⊆C ˜ F = CG∩F . (ii) CG ∩C (iii) CG ~ CF = CGF c. Definition 2.20. A soft set ξS ∈ CS (U) is called soft intersection k-subsemiring (brie f ly, SI-k-subsemiring), if (IC1 ) ξS (u + v) ⊇ ξS (u) ∩ ξS (v), ∀ u, v ∈ S, (IC2 ) ξS (uv) ⊇ ξS (u) ∩ ξS (v), ∀ u, v ∈ S, (IC3 ) u + 1 = h ⇒ ξS (u) ⊇ ξS 1 ∩ ξS (h), ∀ u, 1, h ∈ S, and denoted by ξSk-SS . Definition 2.21. A soft set ξS ∈ CS (U) is called soft intersection left right k-ideal (brie f ly, SI-left right k-ideal), if it satisfies (IC1 ), (IC3 ) and (IC4 ) ξS (uv) ⊇ ξS (v) (ξS (uv) ⊇ ξS (u) ), ∀ u, v ∈ S, and will be denoted by ξSk-SLi and ξSk-SRi , respectively. A soft set ξS ∈ CS (U) is simply called soft intersection k-ideal (brie f ly, SI-k-ideal), if it is ξSk-SLi as well as ξSk-SRi , and denoted by ξSk-SI . Definition 2.22. A ξSk-SS ∈ CS (U) is called soft intersection k-bi-ideal (brie f ly, SI-k-bi-ideal), if it satisfies (IC5 ) ξS (uvw) ⊇ ξS (u) ∩ ξS (w) , ∀ u, v, w ∈ S, and denoted by ξSk-SBi . Definition 2.23. A soft set ξS ∈ CS (U) is called soft intersection k-quasi-ideal (brie f ly, SI-k-quasi-ideal), if it satisfies (IC1 ) , (IC3 ) and ˜ ∩ ˜ ~ ξS , and denoted byξS . ˜ C (IC6 ) ξS ⊇ ξS ~ C k-SQi Definition 2.24. A ξSk-SS ∈ CS (U) is called soft intersection k-interior-ideal (brie f ly, SI-k-interior-ideal), if it satisfies (IC7 ) ξS (uvw) ⊇ ξS (v) , and denoted by ξSk-SIi , ∀ u, v, w ∈ S. Note that: If ξS is an ξSk-SS (ξSk-SI , ξSk-SBi , ξSk-SQi , ξSk-SIi ), then ξS (0) ⊇ ξS (u), ∀ u ∈ S. 3. (X,Y)-SI-k-Ideals of Semirings In this section, we will discuss (X,Y)-soft intersection k-soft ideals, (X,Y)-soft intersection k-bi-ideals, (X,Y)-soft intersection k-quasi-ideals, (X,Y)-soft intersection interior k-ideals and investigate some related properties. Our next discussion would be under the consideration Φ ⊆ X ⊂ Y ⊆ U. Definition 3.1. [23]A soft set ξS ∈ CS (U) is called (X,Y)-soft intersection k-subsemiring (brie f ly, (X,Y)-SI-ksubsemiring), if (XC1 ) ξS (u + v) ∪ X ⊇ ξS (u) ∩ ξS (v) ∩ Y, ∀ u, v ∈ S, (XC2 ) ξS (uv) ∪ X ⊇ ξS (u) ∩ ξS (v) ∩ Y, ∀ u, v ∈ S, (XC3 ) u + 1 = h ⇒ ξS (u) ∪ X ⊇ ξS 1 ∩ ξS (h) ∩ Y, ∀ u, 1, h ∈ S, and denoted by ξS . [X,Y]k-SS


Remark 3.2. For ξS ∈ CS (U), every ξSk-SS is an ξS

24

with arbitrary X ⊂ Y ⊆ U. Converse is not true, but, one

[X,Y]k-SS

can easily observe that ξS

[X,Y]k-SS

is an ξSk-SS with X = ∅ and Y = U, trivially.

Example 3.3. Let S = {0, s, t} with defined ”+” and ”·” as follows + 0 s t

0 0 s t

s s 0 t

t t t 0

and

· 0 s t

0 0 0 0

s 0 0 0

t 0 0 t

Define a soft set ξS of S over U = Z6 = {0, 1, 2, 3, 4, 5} such that ξS (0) = {1, 2, 3, 4}, ξS (s) = {0, 1, 2, 3} and ξS (t) = {1, 2, 3}, If X = {0, 1, 2} and Y = {0, 1, 2, 3}, then one can easily check that ξS is an ξS , but it is not [X,Y]k-SS

ξSk-SS , since ξS (0) + ξS (a).

Definition 3.4. A soft set ξS ∈ CS (U) is called (X,Y)-soft intersection left right k-ideal (brie f ly, (X,Y)-SI-left (right) k-ideal) if it satisfies (XC1 ), (XC3 ) and (XC4 ) ξS (uv) ∪ X ⊇ ξS (v) ∩ Y ( (XC5 ) ξS (uv) ∪ X ⊇ ξS (u) ∩ Y), ∀ u, v ∈ S, and denoted by ξS

and ξS

[X,Y]k-SL i

, respectively.

[X,Y]k-SR i

A soft set ξS ∈ CS (U) is called (X,Y)-soft intersection k-ideal (brie f ly, (X,Y)-SI-k-ideal), if it is ξS well as ξS

[X,Y]k-SR i

, and deonted by ξS

as

[X,Y]k-SL i

.

[X,Y]k-SI

Example 3.5. In Example 3.3, the defined soft set ξS over defined U is ξS

[X,Y]k-SI

but it is not ξSk-SI . Definition 3.6. An ξS

, with X = {0, 1, 2} and Y = {0, 1, 2, 3},

∈ CS (U) is called (X,Y)-soft intersection k-bi-ideal (brie f ly, (X,Y)-SI-k-bi-ideal) if it

[X,Y]k-SS

satisfies

(XC6 ) ξS (uvw) ∪ X ⊇ ξS (u) ∩ ξS (w) ∩ Y, ∀ u, v, w ∈ S, and denoted by ξS

.

[X,Y]k-SB i

Definition 3.7. A soft set ξS ∈ CS (U) is called (X,Y)-soft intersection k-quasi-ideal (brie f ly, (X,Y)-SI-k-quasi-ideal) if it satisfies (XC1 ) , (XC3 ) and ˜ (u) ∩ C ˜ ~ ξS (u) ∩ Y, ∀ u ∈ S, (XC7 ) ξS (u) ∪ X ⊇ ξS ~ C and denoted by ξS

.

[X,Y]k-SQ i

Definition 3.8. A ξS ideal) if it satisfies

∈ CS (U) is called (X,Y)-soft intersection k-interior-ideal (brie f ly, (X,Y)-SI-k-interior-

[X,Y]k-SS

(XC8 ) ξS (uvw) ∪ X ⊇ ξS (v) ∩ Y, ∀ u, v, w ∈ S, and denoted by ξS

[X,Y]k-SI i

.


Obviously, ξS (0) ∪ X ⊇ ξS (u) ∩ Y and (ξS (0) ∩ Y) ∪ X ⊇ (ξS (u) ∩ Y) ∪ X, ∀ u ∈ S. ˜ [X,Y] ηS ⇔ (ξS (u) ∩ Y) ∪ X ⊆ ηS (u) ∩ Y ∪ X, ∀ u ∈ S. Definition 3.9. Let ξS , ηS ∈ CS (U). Then ξS ⊆ ˜ [X,Y] ηS and ηS ⊆ ˜ [X,Y] ξS . Definition 3.10. Let ξS , ηS ∈ CS (U). Then ξS = ˜ [X,Y] ηS ⇔ ξS ⊆ Theorem 3.11. Let ξS ∈ CS (U). Then ξS is an ξS

[X,Y]k-SS

⇔ ξS satisfies (XC3 ) and

˜ [X,Y] ξS , (XC9 ) ξS ⊕ ξS ⊆ ˜ [X,Y] ξS . (XC10 ) ξS ~ ξS ⊆ Proof. Assume that ξS ∈ CS (U) is an ξS

. Let u ∈ S. Then

[X,Y]k-SS

((ξS ⊕ ξS ) (u) ∩ Y) ∪ X = ∪u+(11 +h1 )=(12 +h2 ) (ξS 11 ∩ ξS 12 ∩ ξS (h1 ) ∩ ξS (h2 )) ∩ Y ∪ X ( ) ( ξS 1 1 ∩ ξS (h1 ) ∩ Y ∩ = ∪u+(11 +h1 )=(12 +h2 ) ∪X ξS 12 ∩ ξS (h2 ) ∩ Y ) ∩ Y ( ) ( ξS 11 + h 1 ∪ X ∩ ⊆ ∪u+(11 +h1 )=(12 +h2 ) ∪X ξS 12 + h2 ∪ X ) ∩ Y = ∪u+(11 +h1 )=(12 +h2 ) ξS 11 + h1 ∩ ξS 12 + h2 ∩ Y ∩ Y ∪ X ⊆ ∪u+(11 +h1 )=(12 +h2 ) (ξS (u) ∩ Y) ∪ X = (ξS (u) ∩ Y) ∪ X. ˜ [X,Y] ξS . Thus, (XC9 ) holds. It follows that ξS ⊕ ξS ⊆ In the same pattern, one can prove (XC10 ). Conversely, Suppose that (XC3 ), (XC9 ), (XC10 ) hold. Then (ξS (u + v) ∩ Y) ∪ X ⊇

(ξS (u + v) ∩ Y) ∪ X

⊇

((ξS ⊕ ξS ) (u + v) ∩ Y) ∪ X ( ) (ξS 11 ∩ ξS 12 ∩ ∪(u+v)+(11 +h1 )=(12 +h2 ) ∪X ξS (h1 ) ∩ ξS (h2 )) ∩ Y

= ⊇

((ξS (0) ∩ ξS (u) ∩ ξS (v)) ∩ Y) ∪ X

=

(((ξS (0) ∪ X) ∩ (ξS (u) ∩ Y) ∩ (ξS (v) ∩ Y)) ∩ Y) ∪ X

⊇

(ξS (u) ∩ ξS (v) ∩ Y) ∪ X.

Hence, (XC1 ) holds. (XC2 ) is analogous. Thus, ξS is an ξS

.

[X,Y]k-SS

By using the same methodology of Theorem 3.11, we can also be prove the following. Theorem 3.12. Let ξS ∈ CS (U). Then ξS is an ξS

( ξS ) [X,Y]k-SL [X,Y]k-RL i i

⇔ ξS satisfies (XC3 ), (XC9 ) and

˜ ~ ξS ⊆ ˜⊆ ˜ [X,Y] ξS ξS ~ C ˜ [X,Y] ξS . (XC11 ) C Theorem 3.13. Let ξS ∈ CS (U). Then ξS is an ξS

[X,Y]k-SB i

˜ ~ ξS ⊆ ˜ [X,Y] ξS . (XC12 ) ξS ~ C

⇔ ξS satisfies (XC3 ), (XC9 ), (XC10 ) and

25


Theorem 3.14. Let ξS ∈ CS (U). Then ξS is an ξS

[X,Y]k-SQ i

26

⇔ ξS satisfies (XC3 ), (XC9 ) and

˜ ~ ξS ⊆ ˜ ∩ ˜ [X,Y] ξS . ˜ C (XC13 ) ξS ~ C Theorem 3.15. Let ξS ∈ CS (U). Then ξS is an ξS

[X,Y]k-SI i

⇔ ξS satisfies (XC3 ), (XC9 ), (XC10 ) and

˜ ~ ξS ~ C ˜⊆ ˜ [X,Y] ξS . (XC14 ) C Theorem 3.16. Let Φ , A ⊆ S. Then A is kS (kI , kBi , kQi , kIi ) of S ⇔ CA ∈ CS (U) is an CA CA , CA ), [X,Y]k-SQ [X,Y]k-SI i i

( CA , CA , [X,Y]k-SS [X,Y]k-SI [X,Y]k-SB i

with arbitrary X and Y under the given condition.

Proof. Proof is straightforward. Example 3.17. Let S = {0, 1, a, b, c} be a semiring with the following defined addition and multiplication:

+ 0 1 a b c

0 0 1 a b c

1 1 b 1 a 1

a a 1 a b a

b b a b 1 b

c c 1 a b c

and

· 0 1 a b c

0 0 0 0 0 0

1 0 1 a b c

a 0 a a a c

Then, A = {0, c} is a kI of S. One can easily show that CA ∈ CS (U) is an CA

b 0 b a 1 c

c 0 c c c 0

, by using the Theorem 3.16.

[X,Y]k-SI

Lemma 3.18. Let ξS ∈ CS (U). Then ξS is an ξS

if and only if each nonempty subset

[X,Y]k-SI

U (ξS ; α) = {u ∈ S | ξS (u) ⊇ α ∩ Y} is kI of S for each α ⊆ U under the condition α ⊇ X. Proof. Let ξS ∈ CS (U) be an ξS

[X,Y]k-SI

ξS (u + v) =

such that ξS (u) ⊇ X for every u ∈ S and u, v ∈ U (ξS ; α). Then

⊇

ξS (u + v) ∪ X ξS (u) ∩ ξS (v) ∩ Y

⊇

α ∩ Y,

which implies u + v ∈ U (ξS ; α). Next, we let u ∈ S and v ∈ U (ξS ; α). Then ξS (uv) = ⊇

ξS (uv) ∪ X ξS (v) ∩ Y

⊇

α ∩ Y,

⇒ uv ∈ U (ξS ; α). Similarly, we get uv ∈ U (ξS ; α) for v ∈ S and u ∈ U (ξS ; α). Now, let u ∈ S and 1, h ∈ U (ξS ; α) such that u + 1 = h. Then ξS (u) = ⊇

ξS (u) ∪ X ξS 1 ∩ ξS (h) ∩ Y

⊇

α ∩ Y,


27

⇒ u ∈ U (ξS ; α). Therefore, U (ξS ; α) is an kI of S. Conversely, Let each nonempty subset U (ξS ; α) be an kI of S. Then, for u, v ∈ S there are α1 , α2 ⊆ U such that α1 ⊇ X, α2 ⊇ X with ξS (u) = α1 and ξS (v) = α2 . Thus, ξS (u) ⊇ α ⊇ α ∩ Y and ξS (v) ⊇ α ⊇ α ∩ Y for α = α1 ∩ α2 ⊇ X. Hence, u, v ∈ U (ξS ; α). Next u + v ∈ U (ξS ; α) for u, v ∈ U (ξS ; α), since, U (ξS ; α) is an kI of S. Then ξS (u + v) ∪ X

= ξS (u + v) ⊇ α∩Y = α1 ∩ α2 ∩ Y = ξS (u) ∩ ξS (v) ∩ Y.

The verification of (XC1 ) is complete. Also, we have uv ∈ U (ξS ; α) for u ∈ S and v ∈ U (ξS ; α). Then ξS (uv) ∪ X

=

ξS (uv)

⊇

α∩Y

=

α1 ∩ α2 ∩ Y ξS (u) ∩ ξS (v) ∩ Y

=

ξS (u) ∩ α ∩ Y ξS (u) ∩ Y.

⊇ =

Similarly, we get ξS (uv) ∪ X ⊇ ξS (v) ∩ Y. The verification of (XC2 ) is complete. Now, to verify (XC3 ) we let ξS 1 = α1 , ξS (h) = α2 and u + 1 = h. Then ξS 1 ⊇ α1 ∩ α2 and ξS (h) ⊇ α1 ∩ α2 obviously. So, 1, h ∈ U (ξS ; α1 ∩ α2 ). Since, U (ξS ; α1 ∩ α2 ) is k-ideal, then u ∈ U (ξS ; α1 ∩ α2 ). Thus ξS (u) ∪ X

= ξS (u) ⊇ α1 ∩ α2 ∩ Y = ξS 1 ∩ ξS (h) ∩ Y.

Hence, ξS is an ξS

.

[X,Y]k-SI

Lemma 3.19. Let ξS ∈ CS (U). Then ξS is an ξS

( ξS , ξS , ξS ) [X,Y]k-SS [X,Y]k-SB [X,Y]k-SQ [X,Y]k-SI i i i

if and only if

U (ξS ; α) = {x ∈ A | ξS (u) ⊇ α ∩ Y} is kS (kBi , kQi , kIi ) of S. Proof. Similar to Lemma 3.18. Theorem 3.20. Let ξS ∈ CS (U). Then ξS is an ξS ( ξS

, ξS , ξS , ξS [X,Y]k-SI [X,Y]k-SB [X,Y]k-SQ [X,Y]k-SI i i i

( ξS , ξS , ξS , ξS ) [X,Y]k-SS [X,Y]k-SI [X,Y]k-SB [X,Y]k-SQ [X,Y]k-SI i i i

˜ (u) = ξS (u) ∩ Y, ∀ u ∈ S. ), where (ξS ∩Y)

Proof. Proof is straightforward. Lemma 3.21. Let ξS ∈ CS (U). Then every ξS

( ξS ) [X,Y]k-SL [X,Y]k-SR i i

is an ξS

[X,Y]k-SQ i

Proof. Proof is straightforward. Lemma 3.22. Let ξS ∈ CS (U). Then every ξS

[X,Y]k-SQ i

is an ξS

[X,Y]k-SB

. i

.

˜ is an ξS ⇔ ξS ∩Y

[X,Y]k-SS


28

Proof. Proof is straightforward.

Remark 3.23. The following examples shows that the converses of the both Lemma 3.21 and Lemma 3.22 are not true in general.

(

! ) ( ! ) a b a 0 + + Example 3.24. Let S = : a, b, c, d ∈ Z ∪ {0} and A = : a ∈ Z ∪ {0} . Then S is a c d 0 0 semiring under usual operations, and A is a kQi of S but A is neither kLi nor kRi of S. Then, by Theorem 3.16, CA ∈ CS (U) is an CA but not CA ( CA ). [X,Y]k-SQ i

[X,Y]k-SL [X,Y]k-SR i i

Example 3.25. Let

( S

a b

0 c

(

a b

0 c

(

a b

=

R = L

=

!

+

+

: a, b ∈ R , c ∈ Z

) ,

!

) : a, b ∈ R+ , c ∈ Z+ , a < b , ! ) 0 + + : a, b ∈ R , c ∈ Z , b > 3 . c

Then S is a semiring under usual operations and R is a kRi and L is a kLi of S. Next RL is a kBi of S and it is not a kQi of S. Then, by Theorem 3.16, CRL ∈ CS (U) is an CRL but not CRL . [X,Y]k-SB i

[X,Y]k-SQ i

Definition 3.26. Let ξS ∈ CS (U). Then ˜ [X,Y] ηS = (ξS ∩η ˜ S ) ∩ Y ∪ X. (i) ξS ∩ ˜ [X,Y] ηS = (ξS ∪η ˜ S ) ∩ Y ∪ X. (ii) ξS ∪ (iii) ξS ⊕[X,Y] ηS = (ξS ⊕ ηS ) ∩ Y ∪ X. (iv) ξS ~[X,Y] ηS = (ξS ~ ηS ) ∩ Y ∪ X.

Lemma 3.27. CA ⊕[X,Y] CB (u) = CA+B [ (u) ∩ Y ∪ X.

Lemma 3.28. If ξS , ηS ∈ CS (U) are ξS

[X,Y]k-SR i

and ηS [X,Y]k-SL i

˜ S∩ ˜ [X,Y] ηS . , respectively. Then ξS ~[X,Y] ηS ⊆ξ


29

˜ S∩ ˜ [X,Y] ηS . Otherwise, Proof. Let u ∈ S. If ξS ~ ηS (u) = Φ or ξS ~[X,Y] ηS (u) = X, then ξS ~[X,Y] ηS ⊆ξ ξS ~[X,Y] ηS (u)   m       (h ) ∩ ξ 1 ∩ η ∩   S i S i         i=1 !  n = ∪ Pm ∩ Y ∪X  P ´ ´    n     u+ 1i hi = 1i hi   ´ ´      ∩ (ξS (1 j ) ∩ ηS (h j ))   i=1 j=1 j=1   m     ∩ ξS 1i ∩ Y ∩ ηS (hi ) ∩ Y ∩          i=1    !   n = ∪ Pm ∩ Y   P   n  ∪X  u+ 1i hi = 1í hí   ´ ´    ∩ ((ξS (1 j ) ∩ Y) ∩ (ηS (h j ) ∩ Y))   i=1 j=1 j=1    m    ∩ ξS 1i hi ∪ X ∩ ηS 1i hi ∪ X ∩          i=1  !   n ∩ Y ∪X ⊆ ∪ Pm   P ´ ´    n   u+ 1i hi = 1i hi   ´ ´ ´ ´      ∩ ((ξS (1 j h j ) ∪ X) ∩ (ηS (1 j h j ) ∪ X))   i=1 j=1 j=1   m        ∩ ξ 1 h ∩ Y ∩ η 1 h ∩ Y ∩  S i i S i i      i=1     !   n ⊆ ∪ Pm ∩ Y   P   n  ∪X  u+ 1i hi = 1í hí   ´ ´ ´ ´    ∩ ((ξS (1 j h j ) ∩ Y) ∩ (ηS (1 j h j ) ∩ Y))   i=1 j=1 j=1 ! !    m m   P P     1i hi ∩ ηS 1i hi ∩   ξS         i=1 i=1       ∩ Y n ∪X ⊆ ∪ Pm    P   n n   ´ ´     P P  u+ 1i hi = 1i hi    ´ ´   ´ ´        j=1 i=1 ξ 1 h ∩ η 1 h     S   S  j j j j    j=1 j=1 ˜ [X,Y] ηS (u) . ⊆ ξS ∩ ˜ S∩ ˜ [X,Y] ηS . Thus, ξS ~[X,Y] ηS ⊆ξ 4. k-Semisimple Semirings In this section, we characterize k-semisimple semirings in terms of their (X,Y)-SI-k-ideals. Theorem 4.1. Let S be a kSS-S and ξS ∈ CS (U). Then ξS is an ξS

[X,Y]k-SI

Proof. Let suppose that ξS ∈ CS (U) be an ξS

⇔ ξS is an ξS

.

[X,Y]k-SI i

. Then we have

[X,Y]k-SI

ξS (uv) ∪ X ⊇ ξS (v) ∩ Y and ξS (uv) ∪ X ⊇ ξS (u) ∩ Y Next, for any u, v, w ∈ S, we can write ξS (uvw) ∪ X

=

(ξS ((uv) w) ∪ X) ∪ X

⊇

(ξS (uv) ∩ Y) ∪ X

=

((ξS (uv) ∪ X) ∩ Y) ∪ X

⊇

(ξS (v) ∩ Y) ∪ X

⊇

ξS (v) ∩ Y.

Conversely, Assume ξS ∈ CS (U) is an ξS that u +

n P

ei u fi 1i uhi =

j=1

uv +

n P j=1

m X i=1

[X,Y]k-SI i

. For any u, v ∈ S, there exist ei , fi , 1i , hi , e´j , f´j , 1´j , h´j ∈ S such

e´j u f´j 1´j uh´j . Since, S is kSS-S . So,

ei u fi 1i uhi v =

n X j=1

e´j u f´j 1´j uh´j v


30

Thus,

= ⊇

ξS (uv) ∪ X (ξS (uv) ∪ X) ∪ X       m n  X X          ξS  ei u fi 1i uhi v ∩ ξS  e´j u f´j 1´j uh´j v ∩ Y ∪ X     i=1

j=1

=

         n m  X       X      ´ ´ ´ ´       ξS  ei u fi 1i uhi v ∪ X ∩ ξS  e j u f j 1 j uh j v ∪ X ∩ Y ∪ X

⊇

(ξS (u) ∩ Y) ∪ X

⊇

ξS (u) ∩ Y.

i=1

j=1

This shows that ξS is an ξS

[X,Y]k-SR i

. Similarly, we can show that ξS is an ξS

[X,Y]k-SL i

˜ S for any ξS Theorem 4.2. A semiring S is kSS-S ⇔ ξS ~ ηS = ˜ [X,Y] ξS ∩η

[X,Y]k-SI

i

Proof. Let S be a kSS-S and, ξS

, ηS [X,Y]k-SI [X,Y] i k-SIi

∈ CS (U). Then,

ξS ~ ηS (u) ∩ Y ∪ X   m     ∩ ξS 1i ∩ ηS (hi ) ∩        i=1    !    n = ∪ Pm ∩ Y ∪X   P   n    u+ 1i hi = 1í hí   ´ ´     ∩ ξS 1 j ∩ ηS h j  j=1 i=1 j=1 !!   m  ξS 1i ∩ Y ∩          ∩ ∩       (h ) η ∩ Y   i=1        S i     n = ∪ Pm ∩ Y   ´ P     n  ξS 1 j ∩ Y ∩   ∪X u+ 1i hi = 1í hí         ∩   i=1 j=1     j=1  ηS h´ ∩ Y   j !!  m      ∩ (ξS (ai ubi ) ∪ X) ∩ ∩          i=1    (c ) η ud ∪ X S i i             n ∪X ∩ Y ⊆ ∪ Pm   ´ ´ P         ´ ´   ξ a ub ∪ X ∩ n S     u+ 1i hi = 1i hi     j j               i=1 j=1   ∩      j=1    ηS c´j ud´j ∪ X  ! !    m m   P    ξ P    a ub ∩ η c ud ∩     S i i S i i          i=1 i=1       n ∩ Y ⊆ ∪ Pm     P  n n ∪X      P P  u+ 1i hi = 1í hí    ´ ´   ´ ´        i=1 j=1 ξ a ub ∩ η c ud       S S     j j j j   j=1 j=1 ˜ S (u) ∩ Y ∪ X. ⊆ ξS ∩η ˜ [X,Y] ξS ∩η ˜ S. This implies that ξS ~ ηS ⊆

. Hence, ξS is ξS

[X,Y]k-SI

, ηS

∈ CS (U).

[X,Y]k-SI

i

.


As, S is kSS-S . Then by Theorem 4.1, ξS

, ηS [X,Y]k-SI [X,Y] i k-SIi

are behave as ξS

, ηS . [X,Y]k-SI [X,Y] k-SI

31 Then we can write

ξS ~ ηS (u) ∩ Y ∪ X   m     ∩ ξS 1i ∩ ηS (hi ) ∩         i=1     !   n = ∪ Pm ∩ Y ∪X   P   n    u+ 1i hi = 1í hí    ´ ´     ∩ ξS 1 j ∩ ηS h j  i=1 j=1 j=1 m    m    ∩ ξS 1i ∩ ∩ ηS (hi ) ∩         i=1  i=1  ! !   n ∩ Y ∪X = ∪ Pm   P ´ ´    n n   u+ 1i hi = 1i hi   ´ ´       ∩ ξS 1 j ∩ ∩ ηS h j  i=1 j=1 j=1 j=1    ´ ´      (a ) ξ ub ∩ ξ a ub ∩  S i i S    j j   n ⊇ ∪ Pm ∩ Y ∪X   P     ´ ´ ´ ´   u+ 1i hi = 1i hi   (c ) η ud ∩ η c ud S i i S j j i=1 j=1 ˜ S (u) ∩ Y ∪ X. ⊇ ξS ∩η ˜ S⊆ ˜ [X,Y] ξS ~ ηS . Hence, ξS ~ ηS = ˜ S. ⇒ ξS ∩η ˜ [X,Y] ξS ∩η Conversely, Let A be a kI of S. Then A is kIi of S as well. Then, by Theorem 3.16, CA ∈ CS (U) is an CA . Then

[X,Y]k-SI i

˜ A = CA ~ CA = C c = Cc2 CA = CA ∩C AA A c2 ⇒A=A Hence, S is kSS-S . 5. k-Regular Semirings In this section, we discuss k-regular semirings in terms of (X,Y)-SI-k-ideals, (X,Y)-SI-k-quasi-ideals and (X,Y)-SI-k-bi-ideals of S over U. ˜ [X,Y] ηS = ξS ~[X,Y] ηS , for every ηS Theorem 5.1. A semiring S is kR-S ⇔ ξS ∩

, ηS [X,Y]k-SR [X,Y]k-SL i i

∈ CS (U).

Proof. Let we suppose S is kR-S and u ∈ S. Then there exists 1, 1´ ∈ S such that u + u1u = u1ú. Now ξS ~[X,Y] ηS (u) m   m     ∩ ξS 1i ∩ ∩ ηS (hi ) ∩         i=1 i=1   ! !    n = ∪ Pm ∩ Y   P   n n  ∪X  u+ 1i hi = 1í hí   ´ ´     ∩ ξS 1 j ∩ ∩ ηS h j  i=1 j=1 j=1 j=1 ⊇ ξS u1 ∩ ξS u1´ ∩ ηS (u) ∩ Y ∪ X ⊇ ξS (u) ∩ ηS (u) ∩ Y ∪ X ˜ [X,Y] ηS (u) . = ξS ∩ ˜ S∩ ˜ [X,Y] ηS and by Lemma 3.28, we also have ξS ~[X,Y] ηS ⊆ξ ˜ S∩ ˜ [X,Y] ηS . Thus, ξS ~[X,Y] ηS ⊇ξ ˜ [X,Y] ηS = ξS ~[X,Y] ηS . Hence, ξS ∩ Conversely, Let R and L be kRi and kLi of S, respectively. Then, by Theorem 3.16, CR and CL are CR ∈ CS (U), respectively. Then by hypothesis

and CL

[X,Y]k-SL

i

˜ [X,Y] CB CA ~[X,Y] CB = CA ∩

[X,Y]k-SR i


32

⇒ CAB c ∩ Y ∪ X = (CA∩B ∩ Y) ∪ X c =A∩B ⇒ AB ⇒ S is kR-S . Theorem 5.2. For a semiring S, following are equivalent: (i) S is kR-S . ˜ ~[X,Y] ξS for every ξS ˜ S ~[X,Y] C (ii) (ξS ∩ Y) ∪ X⊆ξ

∈ CS (U).

[X,Y]k-SB

˜ ~[X,Y] ξS for every ξS ˜ S ~[X,Y] C (iii) (ξS ∩ Y) ∪ X⊆ξ

i

∈ CS (U).

[X,Y]k-SQ i

Proof. (i) ⇒ (ii) Let ξS ∈ CS (U) is an ξS

[X,Y]k-SB

Now

. Then for any u ∈ S there exist 1, 1´ ∈ S such that u + u1u = u1ú. i

˜ [X,Y] ξS (u) ξS ~[X,Y] C   m  m   ˜     (h ) C 1 ∩ ∩ ξ ∩ ξ ~  i S i ∩  S [X,Y]       i=1  i=1  ! !  n m ∪ P ∩ Y ∪X  P ´ ´    n n     u+ 1i hi = 1 j h j   ´ ´   ˜ 1 ∩ ∩ ξS h     ∩ ξS ~[X,Y] C  j=1 i=1 j j j=1 j=1 ! ! ) ( n m ˜ u1´ ∩ ξS (u) ∩ Y ∪ X ˜ u1 ∩ ∩ ξS ~[X,Y] C ∩ ξS ~[X,Y] C

=

⊇

i=1

j=1

    m                 ∩ ξ 1 ∩     S i         m     i=1     !          m n ∩ ∪ ∩    P P ´ ´      n         u1+ 1 h = 1 h     i=1 ´     i i j j         ∩ ξ 1     S i=1 j=1   j     j=1     m   ∩ Y ∪X =                       ∩ ξ 1 ∩     S i         n         i=1 !     n   ∩ ∪ ´ Pm   P  ∩ ξS (u)  n    ´ h´    1 1 h = u1 +   j=1 ´   i i     j j     ∩ ξ 1     S j=1 i=1   j j=1   n o  # " ´      u ∩ u1 + u1u1 = u1ú1 and  n ξS u1u ∩ ξS u1   o ∪ X, because ⊇  ∩ Y     u1´ + u1u1´ = u1ú1´  ξS u1u ∩ ξS u1ú ∩ ξS (u)   ⊇ (ξS (u) ∩ Y) ∪ X. (ii) ⇒ (iii) Straightforward. (iii) ⇒ (i) Let Q be any kQi of S. Then by Theorem 3.16, CQ is an ξS

∈ CS (U). Now by using the

[X,Y]k-SQ i

given condition, we write ˜ ~[X,Y] CQ = C [ CQ ∩ Y ∪ X ⊆ CQ ~[X,Y] C QCQ [ ˜ ⇒Q⊆Q CQ. d d ˜ Q ⊆C ˜ Q∩Q ˜ = Q. Thus, Q = Q[ ˜ Q. Therefore, by reference of Lemma 2.5 S is Also, we know Q[ C C C kR-S . Theorem 5.3. For a semiring S, following are equivalent: (i) S is kR-S . ˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS ~[X,Y] ξS for every ξS (ii) ξS ∩ , ηS ˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS ~[X,Y] ξS for (iii) ξS ∩

∈ CS (U).

[X,Y]k-SB [X,Y] i k-SI every ξS , ηS [X,Y]k-SQ [X,Y] i k-SI

∈ CS (U).


33

Proof. By using Theorem 5.2, one can easily prove.

Theorem 5.4. For a semiring S, following statements are equivalent: (i) S is kR-S . ˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS for every ξS (ii) ξS ∩

[X,Y]k-SQ

, ηS

i

∈ CS (U).

[X,Y]k-SL i

˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS for every ξS (iii) ξS ∩

, ηS [X,Y]k-SB [X,Y] i k-SLi

∈ CS (U).

˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS for every ξS (iv) ξS ∩

∈ CS (U).

˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS for every ξS (v) ξS ∩

∈ CS (U).

, ηS [X,Y]k-SR [X,Y] i k-SQi

, ηS [X,Y]k-SR [X,Y] i k-SQi

˜ [X,Y] ηS ∩ ˜ [X,Y] ζS ⊆ξ ˜ S ~[X,Y] ηS ~[X,Y] ζS for every ξS (vi) ξS ∩

, ηS , ζS [X,Y]k-SR [X,Y] [X,Y]k-SL i i k-SQi

∈ CS (U).

˜ [X,Y] ηS ∩ ˜ [X,Y] ζS ⊆ξ ˜ S ~[X,Y] ηS ~[X,Y] ζS for every ξS (vii) ξS ∩

∈ CS (U).

, ηS , ζS [X,Y]k-SR [X,Y] [X,Y]k-SL i i k-SBi

Proof. (i) ⇒ (iii) Let ξS , ηS ∈ CS (U) be any ξS

[X,Y]k-SB

and ηS

i

u ∈ S there exist v, w ∈ S such that u + uvu = uwu. Now,

= ⊇ ⊇ =

ξS ~[X,Y] ηS (u)  m   ∩ ξS 1i ∩ ηS (hi ) ∩     i=1 ! n ∪ Pm  n P ´ ´ u+ 1i hi = 1i hi   ´ ´    ∩ ξS 1 j ∩ ηS h j j=1 i=1 j=1 ξS (u) ∩ ηS (vu) ∩ ηS (wu) ∩ Y ∪ X ξS (u) ∩ ηS (u) ∩ Y ∪ X ˜ [X,Y] ηS (u) . ξS ∩

[X,Y]k-SL

, respectively. Since, S is kR-S , then for any i

         ∩ Y ∪X        [because u + uvu = uwu]

˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS . Hence, ξS ∩ (iii) ⇒ (ii) is easy to prove by using Lemma 3.22. (ii) ⇒ (i) Let ξS , ηS ∈ CS (U) be any ξS

[X,Y]k-SR i

and ηS

. By Lemma 3.21, ξS

[X,Y]k-SL i

[X,Y]k-SR i

is an ξS

. Then by

[X,Y]k-SQ i

˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS . hypothesis, we have ξS ∩ ˜ S∩ ˜ [X,Y] ηS . Hence, ξS ~[X,Y] ηS = ξS ∩ ˜ [X,Y] ηS . Thus, by Theorem Also, by Lemma 3.28, we have ξS ~[X,Y] ηS ⊆ξ 5.1, S is kR-S . (i) ⇔ (iv) ⇔ (v) are straightforward. (i) ⇒ (vii) Let ξS , ηS , ζS ∈ CS (U) be any ξS

, ηS and ζS , [X,Y]k-SR [X,Y] [X,Y]k-SL i i k-SBi

respectively. Since, S is kR-S , then


34

for any u ∈ S there exist v, w ∈ S such that u + uvu = uwu. Now, ξS ~[X,Y] ηS ~[X,Y] ζS (u) m   m        (h ) ∩ ξ ~ η 1 ∩ ∩ ζ  S [X,Y] S i S i ∩        i=1  i=1  ! !  n = ∪ Pm ∩ Y ∪X  P ´ ´    n n     u+ 1i hi = 1 j h j   ´ ´       ∩ ξS ~[X,Y] ηS 1 j ∩ ∩ ζS h j  i=1 j=1 j=1 j=1 ⊇ ξS ~[X,Y] ηS (u) ∩ ζS (vu) ∩ ζS (wu) ∩ Y ∪ X   m          (h ) ∩ ξ 1 ∩ η ∩    S i S i            i=1 !     n (u) ⊇ ∪ Pm ∩ ζ ∩ Y ∪X   P ´ ´  S    n      u+ 1i hi = 1 j h j   ´ ´          ∩ ξS 1 j ∩ ηS h j  i=1 j=1 j=1 ⊇ ξS (uv) ∩ ηS (uw) ∩ ζS (u) ∩ Y ∪ X ⊇ ξS (u) ∩ ηS (u) ∩ ζS (u) ∩ Y ∪ X ˜ [X,Y] ηS ∩ ˜ [X,Y] ζS (u) . = ξS ∩ ˜ [X,Y] ηS ∩ ˜ [X,Y] ζS ⊆ξ ˜ S ~[X,Y] ηS ~[X,Y] ζS . Thus, ξS ∩ (vii) ⇒ (vi) This part is easy to prove, by using Lemma 3.22. (vi) ⇒ (i) Let ξS , ζS ∈ CS (U) be any ξS and ζS , respectively. Then [X,Y]k-SR i

˜ [X,Y] ζS ξS ∩

= ⊆

[X,Y]k-SL i

˜∩ ˜ [X,Y] C ˜ [X,Y] ζS ξS ∩ ˜ ~[X,Y] ζS ξS ~[X,Y] C

⊆ ξS ~[X,Y] ζS . ˜ S∩ ˜ [X,Y] ζS is always hold for any ξS But ξS ~[X,Y] ζS ⊆ξ

[X,Y]k-SR

Then by Theorem 5.1, S is kR-S .

, ζS i

[X,Y]k-SL i

˜ [X,Y] ζS . ∈ CS (U). Hence, ξS ~[X,Y] ζS = ξS ∩

6. k-Intra-Regular Semirings In this section, we discuss k-regular and k-intra-regular semirings and investigate some related properties in terms of (X,Y)-SI-k-ideals. Theorem 6.1. For a semiring S, following statements are equivalent: (i) S is kIR-S . ˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS , for every ξS (ii) ξS ∩ , ηS ∈ CS (U). [X,Y]k-SL [X,Y] i k-SRi

Proof. (i) ⇒ (ii) Let ξS , ηS ∈ CS (U) be any ξS

[X,Y]k-SL i m P

u ∈ S there exist 1i , 1í , h j , h´j ∈ S such that u +

= ⊇

and ηS

, respectively. Since, S is kIR-S , then for any

[X,Y]k-SR i n P 2 ´ 2 ´ 1i u 1i = h j u h j . i=1 j=1

ξS ~[X,Y] ηS (u)    m      (h ) ∩ ξ 1 ∩ η ∩   S i S i        i=1 !  m n ∩ Y ∪X ∪ P  P ´ ´    n     u+ 1i hi = 1i hi   ´ ´      ∩ ξS (1 j ) ∩ ηS (h j )   i=1 j=1 j=1 n o ξS 1i u ∩ ξS h j u ∩ ηS u1í ∩ ηS (uh´j ) ∩ Y ∪ X.

Then,


35

(ii) ⇒ (i) Suppose that R and L are kRi and kLi of S, respectively. By Theorem 3.16, CR , CL ∈ CS (U) are CR and CL , respectively. Then, by hypothesis, we have

[X,Y]k-SR

[X,Y]k-SL i

i

˜ [X,Y] CR ⊆ CL ~[X,Y] CR CL ∩ ⇒ (CL∩R ∪ Y) ∩ X ⊆ (CLR c ∪ Y) ∩ X c ⇒ L ∩ R ⊆ LR. Thus, by Lemma 2.8, S is kIR-S . Theorem 6.2. For a semiring S, the following are equivalent: (i) S is both kR-S and kIR-S . (ii) (ξS ∩ Y) ∪ X = ξS ~[X,Y] ξS for every ξS ∈ CS (U). [X,Y]k-SB

(iii) (ξS ∩ Y) ∪ X = ξS ~[X,Y] ξS for every ξS

i

∈ CS (U).

[X,Y]k-SQ i

Proof. (i) ⇒ (ii) Let ξS ∈ CS (U) be an ξS

[X,Y]k-SB i

a, b, ui , vi , r j , s j ∈ S such that x+

n X

n X xbr j x xs j ax + xar j x xs j bx

j=1

+ =

m X

j=1

(xaui x) (xvi ax) +

m X

(xbui x) (xvi bx)

i=1 m X

i=1 m X

i=1 n X

i=1 n X

(xbui x) (xvi ax) +

+

. Since, S is both kR-S and kIR-S , then for any x ∈ S there exist

(xaui x) (xvi bx)

xar j x xs j ax +

j=1

xbr j x xs j bx .

j=1

Then,

=

ξS ~[X,Y] ξS (x)  m   ∩ ξS 1i ∩ ξS (hi ) ∩    i=1  ! n ∪ Pm  P  n ´ x+ 1i hi = 1í hí  ´   ∩ ξ 1 ∩ ξ h  S S i=1 j=1 j j j=1

         ∩ Y ∪X       

!  n   ∩ ξS xhr j x ∩ ξS xs j 1x ∩ ξS x1r j x ∩ ξS xs j hx ∩    j=1 ⊇  m     (xhu (xv ∩ ξS x1ui x ∩ ξS xvi 1x ∩ ξS i x) ∩ ξS i hx) i=1

⊇ (ξS (x) ∩ Y) ∪ X. ˜ (ξS ∩ Y) ∪ X. ⇒ ξS ~[X,Y] ξS ⊇

          ∪ X ∩ Y       


On the other hand, if x +

m P

1i hi =

i=1

(ξS (x) ∩ Y) ∪ X

n P j=1

36

1í hí , then we have

= ({ξS (x) ∩ Y} ∪ X) ∪ X = ({ξS (x) ∪ X} ∩ Y) ∪ X         m n   X X          ´ ´    ⊇  ξ 1 h ∩ ξ 1 h ∩ Y    S i i S     i i   ∪X       i=1 j=1 ( !) ! m n ´ ´ ⊇ ∩ ξS 1i hi ∩ ∩ ξS 1i hi ∩Y ∪X i=1 j=1 m ⊇ ∩ ξS 1i ∩ ξS (hi ) ∩ ξS 1i ∩ ξS (hi ) ∩ Y ∪ X i=1

(because, ξS is an ξS

).

[X,Y]k-SB i

Thus, ξS ~[X,Y] ξS (x)  m    (h ) ∩ ξ 1 ∩ ξ ∩  S i S i    i=1 ! n = ∪ Pm  n P x+ 1i hi = 1í hí   ´ ´    ∩ ξ 1 ∩ ξ h  S S j=1 i=1 j j j=1

         ∩ Y ∪X       

⊆ (ξS (x) ∩ Y) ∪ X. Hence, ξS ~[X,Y] ξS = (ξS ∩ Y) ∪ X. (ii) ⇒ (iii) This part is obviously clear by Lemma 3.22. (iii) ⇒ (i) Let Q be a kQi of S. Then, by Theorem 3.16, CQ ∈ CS (U) is an CQ [X,Y]k-SQ i

have

CQ ∩ Y ∪ X

= CQ ~[X,Y] CQ n o = CQ ~ CQ ∩ Y ∪ X = CQ c2 ∩ Y ∪ X

c2 ⇒Q=Q Thus, by Lemma 2.9, S is both kR-S and kIR-S . Theorem 6.3. For a semiring S, following are equivalent: (i) S is both kR-S and kIR-S . ˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS , for all ξS (ii) ξS ∩ , ηS ∈ CS (U). [X,Y]k-SB [X,Y] i k-SBi

˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS , for every ξS (iii) ξS ∩

, ηS [X,Y]k-SB [X,Y] i k-SQi

∈ CS (U).

˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS , for every ξS (iv) ξS ∩

∈ CS (U).

˜ [X,Y] ηS ⊆ξ ˜ S ~[X,Y] ηS , for all ξS (v) ξS ∩

, ηS [X,Y]k-SQ [X,Y] i k-SBi

, ηS [X,Y]k-SQ [X,Y] i k-SQi

∈ CS (U).

Proof. One can prove by working in the same pattern as in Theorem 6.2.

. Now, by hypothesis we


37

Conclusion. Soft set theory is an effective tool to deals in uncertainties. In this paper, we investigate the semiring in terms of (X,Y)-SI-k-ideals as a soft version of the fuzzy methodology in [27], which may be useful for a number of complex problems containing uncertainties. Next, we can work in the same context, of soft rings and its generalization. We can apply the above idea to other algebraic structures to deal in decision making, data analysis, information technology and artificial intelligence. References [1] S. Abdullah, M. Aslam and K. Ullah, Bipolar fuzzy soft sets and its applications in decision making problem, J. Intell. Fuzzy Syst., 27:2 (2014) 729-742. [2] S. Abdullah, M. Aslam and K. Hila, A new generalization of fuzzy bi-ideals in semigroups and its applications in fuzzy finite state machines, J. Multiple-Valued Log. Soft Comput., 27:2 (2015) 599-623. [3] S. Abdullah and N. Amin, Analysis of S-box Image encryption based on generalized fuzzy soft expert set, Nonlinear Dyn., 79 (2015) 1679-1692. [4] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Comput. Math. Appl., 59 (2010) 3458-3463. [5] H. Aktas and N. Cagman, Soft sets and soft groups, Inform. Sci., 177 (2007) 2726-2735. [6] M.I Ali, F. Feng, X. Liu, W.K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl., 57 (2009) 1547-1553. [7] M. I. Ali, M. Shabir and K. P. Shum, On soft ideals over semigroups, Southeast Asian Bull. Math., 34 (2010) 595-610. [8] M.I. Ali, M. Shabir and M. Naz, Algebraic structures of soft sets associated with new operations, Comput. Math. Appl., 61 (2011) 2647-2654. [9] M.I. Ali, Another view on reduction of parameters in soft sets, Appl. Soft Comput., 12 (2012) 1814-1821. [10] N. Cagman and S. Enginoglu, Soft matrix theory and its decision making, Comput. Math. Appl., 59 (2010) 3308-3314. [11] N. Cagman and S. Enginoglu, Soft set theory and uni-int decision making, Eur. J. Oper. Res., 207 (2010) 848-855. [12] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl., 56 (2008) 2621-2628. [13] F. Feng, Y.B. Jun, X. Liu and L. Li, An adjustable approach to fuzzy soft set based decision making, J. Comput. Appl. Math., 234 (2010) 10-20. [14] F. Feng, C. Li, B. Davvaz and M.I. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Comput., 14 (2010) 899-911. [15] F. Feng, X.Y. Lin, V. Leoreanu-Fotea and Y.B. Jun, Soft sets and soft rough sets, Inform. Sci., 181 (2011) 1125-1137. [16] S. Ghosh, fuzzy k-ideals of semirings, Fuzzy Sets Syst., 95 (1998) 103-108. [17] M. Henriksen, Ideals in semirings with commutative addition, Am. Math. Soc., (1958) Notices 6:321 [18] Y. Jiang, Y. Tang and Q. Chen, An adjustable approach to intuitionistic fuzzy soft sets based decision making, Appl. Math. Model., 35 (2011) 824-836. [19] Y. Jiang, Y. Tang, Q. Chen, H. Liu and J. Tang, Extending Fuzzy soft sets with fuzzy description logic, Knowledge-Based Syst., 24 (2011) 1096-1107. [20] Y. B. Jun, H. S. Kim and M.A. Ozurk, Fuzzy k-ideals in semirings, J. Fuzzy Math., 13 (2005) 351-364. [21] Y.B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl., 56 (2008) 1408-1413. [22] C. B. Kim and M. Park, k-fuzzy ideals in semirings, Fuzzy Sets Syst., 81 (1996) 281-286. [23] X. Ma and J. Zhan, Applications of a new soft set to h-hemiregular hemirings via (M,N)-SI-h-ideals, J. Intell. Fuzzy Syst., 26 (2014) 2515-2525. [24] X. Ma and J. Zhan, Characterizations of hemiregular hemirings via a kind of new soft union sets, J. Intell. Fuzzy Syst., 27 (2014) 2883-2895. [25] P.K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (2002) 1077-1083. [26] P.K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003) 555-562. [27] T. Mehmood and M. Aslam, On interval-valued ∈γ , ∈γ ∨qδ -fuzzy k-ideals in hemirings, Neural Comput. Appl. 21 (2012) S231-S244. [28] D. Molodtsov, Soft set theory-First results, Comput. Math. Appl., 37 (4-5) (1999) 19-31. [29] A. R. Roy and P. K. Maji, A fuzzy soft set theoreties approach to decision making problems, J. Comput. Appl. Math., 203 (2007) 214-418 [30] A. Sezgin and A. O. Atagun, On operations of soft sets, Comput. Math. Appl., 61 (2011) 1457-1467. [31] M. Shabir and R. Anjum, On k-regular and k-intra-regular hemirings, Inst. Adv. Sci. Research, 4 (2012) 157-184. [32] M. Shabir and R. Anjum, Characterizations of hemirings by their k-ideals, Appl. Math., 4 (2013) 753-768. [33] Z. Xiao, K. Gong, S. Xia and Y. Zou, Exlusive disjunctive soft sets, Comput. Math. Appl., 59 (2010) 2128-2137. [34] X.B. Yang, T.Y. Lin, J.Y. Yang and D. Yu, Combination of interval-valued fuzzy set and soft set, Comput. Math. Appl., 58 (2009) 521-527. [35] Y. Yin, X. Huang, D. Xu and H. Li, The characterizations of h-semisimple hemirings, Int. J. Fuzzy Syst., 11 (2009) 116-122. [36] Y. Yin, Y.B. Jun and J. Zhan, Vague soft hemirings, Comput. Math. Appl., 62 (2011) 199-213. [37] Y. Yin and J. Zhan, The characterizations of hemirings in terms of fuzzy soft h-ideals, Neural. Comput. Appl., 21 (2012) S43-S57. [38] J. Zhan, N. Cagman and A. S. Sezer, Applications of soft union sets to hemirings via SU-h-ideals, J. Intell. Fuzzy Syst., 26 (2014) 1363-1370.


38

[39] J. Zhan, W. A. Dudek and J. Neggers, A new soft union set: characterizations of hemirings via the uncertainty modelling, submitted [40] Y. Zou and Z. Xiao, Data analysis approachs of soft sets under incomplete information, Knowledge-Based Syst., 21 (2008) 941-945.