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Journal of Intelligent & Fuzzy Systems 26 (2014) 1363–1370 DOI:10.3233/IFS-130822 IOS Press
Applications of soft union sets to hemirings via SU-h-ideals Jianming Zhana,∗ , Naim C ¸ aˇgmanb and Aslihan Sezgin Sezerc a Department
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of Mathematics, Hubei University for Nationalities, Enshi, Hubei Province, P.R. China of Mathematics, Gaziosmanpasa University, Tokat, Turkey c Department of Mathematics, Amasya University, Amasya, Turkey
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b Department
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Abstract. The aim of this article is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, we introduce the concepts of soft union hemirings (soft union h-ideals) of hemirings by soft intersection-union product and obtain some related results. Finally, we investigate some characterizations of h-hemiregular hemirings by soft union h-ideals. Keywords: Soft set, soft intersection-union product, soft union hemiring, soft union h-ideal, h-hemiregular hemiring
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1. Introduction
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There are three theories: probability theory, fuzzy set theory and rough set theory which we can consider as mathematical tools for dealing with uncertainties. However, all of them have their advantages as well as inherent limitations in dealing with uncertainties. To overcome these difficulties, Molodtsov [23] introduced the concept of soft sets as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Nowadays, works on the soft sets are progressing rapidly. Maji [21] discussed further soft set theory. Ali et al. [4, 5] proposed some new operations on soft sets. Qin [24] investigated soft equality. At the same time, this theory has been proven useful in many different fields such as decision making [7– 9, 13, 14, 22], data analysis [29, 33], forecasting and so on. It also is interesting to see that soft sets are closely ∗ Corresponding
author. Jianming Zhan, Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei Province 445000, China. Tel: +86 0718-8437732. E-mails: zhanjianming@ hotmail.com;
[email protected] (N. C¸aˇgman); aslihan.sezgin@ amasya.edu.tr (A. Sezgin Sezer).
related to many other soft computing models such as rough sets and fuzzy sets. Feng et al. [14] first considered the combination of soft sets, fuzzy sets and rough sets. Using soft sets as the granulation structures, Feng et al. [10] initiated soft approximation spaces and soft rough sets, which extended Pawlak’s rough set model using soft sets. In some cases Feng’s soft rough set model could provide better approximations than classical rough sets. Recently, the algebraic structures of soft sets have been studied increasingly, such as, soft rings [1, 6], soft groups [3], soft semirings [10], soft hemirings [20], soft BCK/BCI-algebras [16, 18], soft ordered semigroups [17], soft BCH-algebras [19], soft near-rings [26, 27]. Feng et al. [12] investigated five different types of soft subsets and considered the free soft algebras associated with soft product operations. It has been shown that soft sets have some non-classical algebraic properties which are distinct from those of crisp sets or fuzzy sets. We note that the ideals of semirings play a crucial role in the structure theory, ideals in semirings do not in general coincide with the ideals of a ring. For this reason, the usage of ideals in semirings is somewhat limited. By a hemiring, we mean a special semiring
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J. Zhan et al. / Soft union hemirings
2. Preliminaries
a(b + c) = ab + ac and (a + b)c = ac + bc
A soft set over U can be represented by the set of ordered pairs fA = {(x, fA (x)|x ∈ E, fA (x) ∈ P(U)}. It is clear to see that a soft set is a parameterized family of subsets of U. Note that the set of all soft sets over U will be defined by S(U). Definition 2.2. [8, 27] (i) Let fA , fB ∈ S(U). Then, fA ˜ B if is called soft subset of fB and denoted by fA ⊆f ˜ fA (x)⊆fB (x) for all x ∈ E. (ii) Let fA , fB ∈ S(U). Then, union of fA and fB , ˜ B = fA∪B denoted by fA ∪f ˜ , where fA∪B ˜ (x) = fA (x) ∪ fB (x) for all x ∈ E. (iii) Let fA be a soft set over U and α ⊆ U. Then, lower α-inclusion of fA , denoted by L(fA ; α), is defined as L(fA ; α) = {x ∈ A|fA (x) ⊆ α}. Definition 2.3. [25] Let X be a subset of S. We denote by SXC the soft characteristic function of the complement of X and define as ∅ if x ∈ X, SXC (x) = U if x ∈ S\X.
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A semiring is an algebraic system (S, +, ·) consisting of a non-empty sets S together with two binary operations on S called addition and multiplication (denoted in the usual manner) such that (S, +) and (S, ·) are semigroups and the following distributive laws:
Definition 2.1. [23] A soft set fA over U is defined as fA : E → P(U) such that fA (x) = ∅ if x ∈ / A. Here fA is also called an approximate function.
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with a zero and with a commuatire addition. The properties of h-ideals were thoroughly investigated by Torre [28] and by using h-ideals, Torre established some analogous ring theorems for hemirings. In particular, the h-hemiregular hemirings were described by Zhan et al. [32]. Some characterizations of the h-semisimple and h-intra-hemiregular hemirings were investigated by Yin et al. [2, 30, 31]. In [25], Sezgin made a new approach to the classical ring theory via soft set theory with the concept of soft union rings. By this new idea, in this paper, we introduce the concept of soft union h-ideals of hemirings and obtain some related results. Finally, we investigate some characteristics of h-hemiregular hemirings by soft union h-ideals.
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1364
AU
TH
are satisfied for all a, b, c ∈ S. By zero of a semiring (S, +, ·) we mean an element 0 ∈ S such that 0 · x = x · 0 = 0 and 0 + x = x + 0 = 0 for all x ∈ S. A semiring with zero and a commutative semigroup (S, +) is called a hemiring. For the sake of simplicity, we shall write ab for a · b(a, b ∈ S). A subhemiring of a hemiring S is a subset A of S closed under addition and multiplication. A subset A of S is called a left (resp., right) ideal of S if A is closed under addition and SA ⊆ A (resp., AS ⊆ A). Further, A is called an ideal of S if it is both a left ideal and a right ideal of S. A subhemiring(left ideal, right ideal, ideal) A of S is called an h-subhemiring(left h-ideal, right h-ideal, h-ideal) of S, respectively, if for any x, z ∈ S, a, b ∈ A, and x + a + z = b + z implies x ∈ A. The h-closure A of a subset A of S is defined as A ={x ∈ S|x + a + z = b + z for some a, b ∈ A, z ∈ S}.
From now on, S is a hemiring, U is an initial unervise, E is a set of parameters, P(U) is the power set of U and A, B, C ⊆ E.
Proposition 2.4. [25] Let X, Y ⊆ S. Then the following hold: ˜ YC ; (1) Y ⊆ X ⇒ SXC ⊆S ˜ Y C = SXC ∪Y C . (2) SXC ∪S
3. SU-hemirings In this section, we introduce the concept of soft union hemirings and investigate some related properties. Definition 3.1. A soft set fS over U is called a soft union hemiring(briefly, SU-hemiring) of S if (SU1 ) fS (x + y) ⊆ fS (x) ∪ fS (y) for all x, y ∈ S, (SU2 ) fS (xy) ⊆ fS (x) ∪ fS (y) for all x, y ∈ S, (SU3 ) fS (x) ⊆ fS (a) ∪ fS (b) with x + a + z = b + z for all x, a, b, z ∈ S. It is clear that fS (0) ⊆ fS (x) for all x ∈ S. Example 3.2. (i)Let S = Z6 = {0, 1, 2, 3, 4, 5} be the hemiring of non-negative integers module 6. Assume that U = Z5 is the universal set.
J. Zhan et al. / Soft union hemirings
i=1
Then, one can easily check that fS is an SU-hemiring of S over U.
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Remark 3.3. It is easy to see that if fS (x) = ∅ for all x ∈ S, then fS is an SU-hemiring of S over U. we denote ˜ It is obvious that such a kind of SU-hemiring by θ. θ˜ = SS C .
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Definition 3.4. Let fS and gS be two soft sets over U. Then (1) Soft intersection-union sum fS ⊕ gS is defined by (fS ⊕ gS )(x) (fs (a1 ) ∪ fs (a2 ) ∪ gs (b1 ) ∪gs (b2 )) = x+a1 +b1 +z =a2 +b2 +z
and (fS ⊕ gS )(x) = U if x cannot be expressed as x + a1 + b1 + z = a2 + b2 + z. (2) Soft intersection-union product fS ♦gS is defined by )(x) (fS ♦gS = (fS (ai ) ∪ fS (aj ) ∪ gS (bi ) ∪ gS (bj )) m ai bi +z
x+
j=1
Theorem 3.5. Let fS be a soft set over U. Then fS is an SU-hemiring of S over U if and only if it satisfies (SU3 ) and ˜ S; (SU4 ) fS ⊕ fS ⊇f ˜ S. (SU5 ) fS ♦fS ⊇f
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Proof. Assume that fS is an SU-hemiring of S over U. Let x ∈ S. If (fS ⊕ fS )(x) = U, then (fS ⊕ fS )(x) ⊇ ˜ S . Otherwise, let x + a1 + fS (x). Thus, fS ⊕ fS ⊇f b1 + z = a2 + b2 + z for some a1 , a2 , b1 , b2 , z ∈ S. Thus (fS ⊕ fS )(x) = (fS (a1 ) ∪ fS (a2 ) ∪ fS (b1 ) ∪ fS (b2 )) x+a1 +b1 +z =a2 +b2 +z
⊇
⊇
OR
fS (2) = fS (4) 11 22 33 44 = , , , , 10 20 30 40 11 22 33 fS (1) = fS (5) = , , , 10 20 30 11 44 fS (3) = , . 10 40
and (fS ♦gS )(x) = U if x cannot be expressend m n as x + ai b i + z = aj bj + z.
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Define a soft set fS over U by fS (0) = {1}, fS (1) = fS (5) = {1, 2, 3, 4}, fS (2) = fS (4) = {1, 2, 3} and fS (3) = {1, 4}. Then, one can easily check that fS is an SU-hemiring of S over U. xx (ii) Assume that U = |x ∈ Z5 , 2 × 2 x0 matrices with Z5 terms, is the universal set. Let S = Z6 = {0, 1, 2, 3, 4, 5} be the hemiring of non-negative integers module 6. Define a soft set fS over U by 11 fS (0) = , 10
1365
x+a1 +b1 +z =a2 +b2 +z
x+a1 +b1 +z =a2 +b2 +z
(fS (a1 + b1 ) ∪ fS (a2 ∪ b2 )) (fS (x))
= fS (x),
˜ S . Thus, (SU4 ) holds. which implies, (fS ⊕ fS )⊇f Let x ∈ S. If (fS ♦fS )(x) = U, then (fS ♦fS )(x) ⊇ ˜ S Otherwise, let fS (x). Thus, fS ♦fS ⊇f m n x+ ai bi + z = aj bj + z for all i = 1, 2, ..., m; i=1
j=1
j = 1, 2, ..., n. Thus, (fS ♦fS )(x) = (fS (ai ) ∪ fS (a j ) ∪ fS (bi ) ∪ fS (bj )) m ai bi +z
x+
i=1 n
=
aj bj +z
j=1
⊇
m
x+
n
ai bi +z=
i=1
fS
j=1
m i=1
ai b i ∪
n j=1
x+
i=1
aj bj +z
fS ⊇
m
n
ai bi +z=
aj bj
,
(fS (x)) = fS (x), aj bj +z
j=1
i=1
n
=
aj bj +z
j=1
for all i = 1, 2, ..., m; j = 1, 2, ..., n
˜ S . Thus, (SU5 ) holds. which implies, fS ♦fS ⊇f Conversely, assume that the conditions (SU3 ), (SU4 ) and (SU5 ) hold. Then
J. Zhan et al. / Soft union hemirings
x+y+a1 +b1 +z =a2 +b2 +z
.
⊆ fS (x) ∪ fS (y) ∪ fS (0) = fS (x) ∪ fS (y). Then, (SU1 ) holds. fS (xy) ⊆ (fS ♦fS )(xy) = (fS (ai ) ∪ fS (aj ) ∪ fS (bi ) ∪ fS (bj )) m ai bi +z
xy+
i=1
n
=
aj bj +z
j=1
⊆ fS (x) ∪ fS (y) ∪ fS (0) = fS (x) ∪ fS (y). Then, (SU2 ) holds. Hence, fS is an SU-hemiring of S over U.
Corollary 3.7. Let A be a non-empty subset of S. Then A is an h-subhemiring of S if and only if SAC is an SU-hemiring of S over U. Theorem 3.8. (i) Let fS be a soft set over U and α ⊆ U such that α ∈ Im (fS ). If fS is an SU-hemiring of S over U, then L(fS ; α) is an h-subhemiring of S. (ii) Let fS be a soft set over U, L(fS ; α) a lower h-subhemiring of fS for each α ⊆ U and Im (fS ) an ordered set by inclusion. Then fS is an SU-hemiring of S over U. Proof. (i) Since fs (x) = α for some x ∈ S, then ∅= / L(fS ; α) ⊆ S. Let x, y ∈ L(fs ; α), then fS (x) ⊆ α and fS (y) ⊆ α. Then fS (x + y) ⊆ fS (x) ∪ fS (y) ⊆ α ∪ α = α, fS (xy) ⊆ fS (x) ∪ fS (y) ⊆ α ∪ α = α, which implies, x + y, xy ∈ L(fS ; α). Now, let x, z ∈ S and a, b ∈ L(fS ; α) with x + a + z = b + z. Then fS (a) ⊆ α and fS (b) ⊆ α. Thus fS (x) ⊆ fS (a) ∪ fS (b) = α ∪ α = α, which implies, x ∈ L(fS ; α). Thus, L(fS ; α) is an h-subhemiring of S. (ii) Let x, y ∈ S be such that fS (x) = α1 and fS (y) = α2 , where α1 ⊆ α2 . Then x ∈ L(fS ; α1 ) and y ∈ L(fS ; α2 ), and so x ∈ L(fS ; α2 ). Since L(fS ; α) is an h-subhemiring of S for all α ⊆ U, then x + y ∈ L(fS ; α2 ) and xy ∈ L(fS ; α2 ). Hence fS (x + y) ⊆ α2 = α1 ∪ α2 = fS (x) ∪ fS (y) and fS (xy) ⊆ α2 = α1 ∪ α2 = fS (x) ∪ fS (y). Now, let x, z, a, b ∈ S with x + a + z = b + z be such that fS (a) = α1 and fS (b) = α2 , where α1 ⊆ α2 , then a ∈ L(fS ; α1 ) and b ∈ L(fS ; α2 ), and so a ∈ L(fS ; α2 ). Since L(fS ; α) is an h-subhemiring of S for each α ⊆ U, then x ∈ L(fS ; α2 ). Thus fS (x) ⊆ α2 = α1 ∪ α2 = fS (a) ∪ fS (b). Therefore fS is an SUhemiring of S over U.
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OR
Proposition 3.6. A non-empty subset A of S is an hsubhemiring of S if and only if the soft subset fS defined by α if x ∈ S\A, fS (x) = β if x ∈ A, is an SU-hemiring of S over U, where α, β ⊆ U such that α ⊇ β.
fS (a) ∪ fS (b) = β, and so fS (x) = β. Thus, x ∈ A. Hence A is an h-subhemiring of S. The following is a consequence of the above proposition.
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fS (x + y) = (fS ⊕ fS )(x + y) = (fS (a1 ) ∪ fS (a2 ) ∪ fS (b1 ) ∪ fS (b2 ))
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Proof. Let A be an h-subhemiring of S and x, y ∈ S.
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(i) If x, y ∈ A, then xy, x + y ∈ A. Hence fS (x + y) = fS (xy) = fS (x) = fS (y) = β, and so fS (x + y) ⊆ fS (x) ∪ fS (y) and fS (xy) ⊆ fS (x) ∪ fS (y). (ii) If either one if x and y does not belong to A, then x + y ∈ A or x + y ∈ / A and xy ∈ A or xy ∈ / A. In any case, fS (x + y) ⊆ fS (x) ∪ fS (y) = α and fS (xy) ⊆ fS (x) ∪ fS (y) = α. (iii) Now, let a, b, x, z ∈ S be such that x + a + z = b + z. (1) If a, b ∈ A, then x ∈ A, and so fS (x) = fS (a) ∪ fS (b) = β. (2) If a ∈ / A or b ∈ / A, then fS (x) ⊆ fS (a) ∪ fS (b) = α. Thus, fS is an SUhemiring of S over U.
Conversely, assume that fS is an SU-hemiring of S over U. (i) Let x, y ∈ A, then fS (x + y) ⊆ fS (x) ∪ fS (y) = β and fS (xy) ⊆ fS (x) ∪ fS (y) = β, which implies, x + y, xy ∈ A. (ii) Let x, z ∈ S and a, b ∈ A be such that x + a + z = b + z. Then fS (x) =
4. SU-h-ideals In this section, we investigate the properties of soft union h-ideals of hemirings. Definition 4.1. A soft set fS over U is called a soft union left(right) h-ideal of S over U(briefly, SU-left(right) h-ideal) if it satisfies (SU1 ), (SU3 ) and
J. Zhan et al. / Soft union hemirings
(SU6 ) fS (xy) ⊆ fS (y)(fS (xy) ⊆ fS (x)) for all x, y ∈ S.
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=
m
xy+
˜ i ) ∪ θ(a ˜ ) ∪ fS (bi ) ∪ fS (b )) (θ(a j j
ai bi +z
i=1
Theorem 4.3. Let fS be a soft set over U. Then fS is an SU-left(right) h-ideal of S over U if and only if it satisfies (SU3 ), (SU4 ) and ˜ S ⊇f ˜ S (fS ♦θ˜ ⊇f ˜ S ). (SU7 ) θ♦f
i=1
j=1
aj bj +z
j=1
⊇
m
x+
∅ ∪ fS
m
ai bi
i=1
ai bi +z
i=1
n
=
aj bj +z
j=1
⊇
(by SU1 and SU6 ) m
x+
n
ai bi +z=
i=1
fS (x)
∪ fS
n
aj bj
AU
=
TH
ai bi +z
i=1 n
= fS (y), which implies, (SU6 ) holds. Thus, fS is an SU-left h-ideal of S over U. Similarly, we can prove the case for SU-right hideals. Proposition 4.4. A non-empty subset A of S is a left(right) h-ideal of S if and only if the soft subset fS defined by α if x ∈ S\A, fS (x) = β if x ∈ S, is an SU-left(right) h-ideal of S. Proof. It is similar to the proof of proposition 3.6.
j=1
Corollary 4.5. Let A be a non-empty subset of S. Then A is a left(right) h-ideal of S if and only if SAC is an SU-left(right) h-ideal of S over U.
˜ S )(x) (θ♦f ˜ i ) ∪ θ(a ˜ ) ∪ fS (bi ) ∪ fS (b )) = (θ(a j j m x+
˜ ∪ fS (y) ⊆ θ(x) = ∅ ∪ fS (y)
OR
Proof. Let fS be an SU-left h-ideal of S over U. If ˜ S )(x) = U, then it is clear that θ♦f ˜ S ⊇f ˜ S . Other(θ♦f m n
ai bi + z = wise, let x + aj bj + z. Then
aj bj +z
j=1
PY
Example 4.2. Let S = Z6 = {0, 1, 2, 3, 4, 5} be the hemiring of non-negative integers module 6 and U = Z6 . Define a soft set fS over U by fS (0) = {1}, fS (1) = fS (5) = {1, 2, 3}, fS (2) = fS (4) = {1, 2} and fS (3) = {1, 3}. Then, one can easily check that fS is an SU-h-ideal of S over U.
n
=
CO
A soft set over U is called a soft union h-ideal of S(briefly, SU-h-ideal) if it is both an SU-left h-ideal and an SU-right h-ideal of S.
(by SU3 )
aj bj +z
j=1
= fS (x), ˜ S ⊇f ˜ S . This proves that (SU7 ) which implies, θ♦f holds. Conversely, assume that the conditions (SU3 ), (SU4 ) and (SU7 ) hold. Then fS (xy) ˜ S )(xy) ⊆ (θ♦f
Theorem 4.6. (i) Let fS be a soft set over U and α ⊆ U such that α ∈ Im (fS ). If fS is an SU-left(right) h-ideal of S over U, then L(fS ; α) is a left(right) h-ideal of S. (ii) Let fS ba a soft set over U, L(fS ; α) a lower left(right) h-ideal of fS for each α ⊆ U and Im (fS ) an ordered set by inclusion. Then fS is an SU-left(right) h-ideal of S over U.
Proof. It is similar to the proof of Theorem 3.8.
Proposition 4.7. Let fS and hS be two SU-left(right) ˜ S. h-ideals of S over U, then so is fS ∪h Proof. Let fS and hS be two SU-left h-ideals of S over U. For any x, y ∈ S, then ˜ S )(x + y) (fS ∪h = fS (x + y) ∪ hS (x + y) ⊆ fS (x) ∪ fS (y)) ∪ (hS (x) ∪ hS (y) = (fS (x) ∪ (hS (x)) ∪ (fS (y) ∪ hS (y)) ˜ S )(x) ∪ (fS ∪h ˜ S )(y). = (fS ∪h
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J. Zhan et al. / Soft union hemirings
˜ S )(xy) (fS ∪h = fS (xy) ∪ hS (xy)
=
m
x+
⊆ fS (y) ∪ hS (y) ˜ S )(y). = (fS ∪h
ai bi +z
i=1 n
=
aj bj +z
j=1
Now, let x, z, a, b ∈ S with x + a + z = b + z. Then ˜ S )(x) = fS (x) ∪ hS (x) (fS ∪h
=
m
xy+
= (fS (a) ∪ (hS (a)) ∪ (fS (b) ∪ hS (b)) ˜ S )(a) ∪ (fS ∪h ˜ S )(b). = (fS ∪h
.
Proposition 4.8. Let fS and hS be two SU-left(right) h-ideals of S over U, then so is fS ♦hS . Proof. Let fS and hS be two SU-left h-ideals of S over U and x, y ∈ S. Then
(fS (ai ) ∪ fS (aj ) ∪ hS (bj ) ∪ hS (bi )) ⊇ m n
(xai )bi +xz=
ai bi +z1
i=1 n
TH
aj bj +z1
j=1
(fS (ci ) ∪ fS (cj ) ∪ hS (di ) ∪ hS (dj ))
p
y+
ci di +z2
cj dj +z2
j=1
=
m
x+
n
ai bi +z1 =
i=1
AU
i=1 q
=
aj bj +z1
j=1
p
y+
q
ci di +z2 =
i=1
k
x+y+
(xaj )bj +xz
j=1
(fS (xai ) ∪ fS (xaj ) ∪ hS (bi ) ∪ hS (bj )) = (fS ♦hS )(xy).
(iii) Now, let x, z, a, b ∈ S with x + a + z = b + z. Then it is similar to check that (fS ♦hS )(a) ∪ (fS ♦hS )(b) ⊇ (fS ♦hS )(x). Thus, fS ♦hS is an SU-left h-ideal of S over U. Similarly, we can prove that fS ♦hS is an SU-right h-ideal of S over U.
l
xi yi +z1 +z2 =
i=1
cj dj +z2
Theorem 4.9. Let fS and gS be an SU-right h-ideal and an SU-left h-ideal of S over U, respectively, then ˜ S ∪g ˜ S. fS ♦gS ⊇f Proof. Let fS and gS be an SU-right h-ideal and an SU-left h-ideal of S over U, respectively, then ˜ S ♦θ˜ ⊇f ˜ S and by Theorem 4.3, we have fS ♦gS ⊇f ˜ ˜ ˜ ˜ ˜ fS ♦gS ⊇θ♦gS ⊇gS . Thus, fS ♦gS ⊇fS ∪gS . 5. h-hemiregular hemirings In this section, we characterize h-hemiregular hemirings in terms of SU-h-ideals.
j=1
(fS (ai ) ∪ fS (aj ) ∪ hS (bi ) ∪ hS (bj ) ∪fS (ci ) ∪ fS (cj ) ∪ hS (di ) ∪ hS (d j )) ⊇
i=1
OR
(i)(fS ♦hS )(x) ∪ (fS ♦hS )(y) = (fS (ai ) ∪ fS (aj ) ∪ hS (bi ) ∪ hS (bj ))∪ m
(xaj )bj +xz
j=1
CO
˜ S is an SU-left h-ideal of S over U. Thus, fS ∪h ˜ S is an SU-right h-ideal Similarly, we can prove fS ∪h of S over U.
n
(xai )bi +xz=
i=1
xy+
=
PY
⊆ (fS (a) ∪ fS (b)) ∪ (hS (a) ∪ hS (b))
x+
(fS (ai ) ∪ fS (aj ) ∪ hS (bi ) ∪ hS (bj ))
xj yj +z1 +z2
j=1
(fS (xi ) ∪ fS (xj ) ∪ hS (yi ) ∪ hS (yj ))
Definition 5.1. [32] A hemiring S is called hhemiregular if for each a ∈ S, there exist x1 , x2 , z ∈ S such that a + ax1 a + z = ax2 a + z. Lemma 5.2. [32] If A and B, are respectively, a right h-ideal and a left h-ideal of S, then AB ⊆ A ∩ B.
(k = max{m, p}, l = max{n, q}, xi yi = ai bi + ci di , xj yj = a j bj + cj dj )
Lemma 5.3. [32] A hemiring S is h-hemiregular if and only if for any right h-ideal A and left h-ideal B, we have AB = A ∩ B.
= (fS ♦hS )(x + y). (ii)(fS ♦hS )(y)
Theorem 5.4. For any hemiring S, then the following are equivalent:
J. Zhan et al. / Soft union hemirings
Proof. (1) ⇒ (2) Let S be an h-hemiregular hemiring. fS and gS an SU-right h-ideal and an SU-left h-ideal of S over U, respectively. By Theorem 4.9, we have ˜ S. fS ♦gS = fS ∪g Let x ∈ S, then there exist a, a , z ∈ S such that x + xax + z = xa x + z since S is h-hemiregular. Thus, we have (fS ♦gS )(x) = (fS (ai ) ∪ fS (aj ) ∪ gS (bi ) ∪ gS (bj )) m
Proof. Let S be an h-hemiregular hemiring and hS an SU-h-ideal of S, we have ˜ S. ˜ S ♦θ˜ ⊇h hS ♦hs ⊇h Let x ∈ S, then there exist a, a , z ∈ S such that x + xax + z = xa x + z since S is h-hemiregular. Thus, we have (hS ♦hS )(x) = (hS (ai ) ∪ hS (a j ) ∪ hS (bi ) ∪ hS (bj )) m i=1 n
=
m
(hS (xa) ∪ hS (xa ) ∪ hS (x))
ai bi +z
i=1 n
=
aj bj +z
aj bj +z
CO
aj bj +z
x+
i=1 n
=
j=1
⊆
ai bi +z
x+
ai bi +z
x+
PY
(1) S is h-hemiregular; ˜ S for any SU-right h-ideal fS (2) fS ♦gS = fS ∪g and any SU-left h-ideal gS of S over U.
1369
j=1
j=1
⊆ hS (x),
⊆ fS (xai ) ∪ fS (xa ) ∪ gS (x) ∪ gS (x) ˜ S )(x), ⊆ fS (x) ∪ gS (x) = (fS ∪g
AU
TH
OR
˜ S ∪g ˜ S Thus, which implies, fS ♦gS ⊆f ˜ S. fS ♦gS = fS ∪g (2) ⇒ (1) Let R and L be any right h-ideal and left h-ideal of S, respectively. Then by Corollary 4.5, SRC and SLC are an SU-right h-ideal and an SU-left h-ideal of S over U, respectively. Moreover, by Lemma 5.2, we have RL ⊆ R ∩ L. Let a ∈ R ∩ L, then a ∈ R and and a ∈ L, and so SRC (a) = SLC (a) = ∅. Thus, ˜ LC )(a) = SRC (a) ∪ (SRC ♦SLC )(a) = (SRC ∪S SLC (a) = ∅, which implies, SRC (a1 ) ∪ SLC (b1 ) ∪ SRC (a2 ) ∪ SLC (b2 ) = ∅ for some a1 , a2 , b1 , b2 ∈ S satisfying a + a1 b1 + z = a2 b2 + z, and so SRC (a1 ) = SLC (b1 ) = SRC (a2 ) = SLC (b2 ) = ∅, which implies, a1 , a2 ∈ R and b1 , b2 ∈ L. Hence a ∈ RL. Thus, R ∩ L ⊆ RL, and so R ∩ L = RL. By Lemma 5.3, we know that S is hhemiregular.
˜ S . Then, hS ♦hS = hS . which implies, hS ♦hS ⊆h 2 Thus (hS ) = hS ♦hS = hS . Conversely, assume that hS and kS are SU-h-ideals ˜ S ∪k ˜ S. of S over U. By Theorem 4.9, we have hS ♦ks ⊇h ˜ Moreover, by Proposition 4.7, hS ∪kS is an SU-h-ideal ˜ S )(x) = of S over U. By the assumption, we have (hS ∪k 2 ˜ (hS ∪kS ) (x) for all x ∈ S. Thus, ˜ S )(x) (hS ∪k
Corollary 5.5. For any hemiring S, then the following are equivalent: (1) S is h-hemiregular; ˜ S for any SU-h-ideals fS and gS (2) fS ♦gS = fS ∪g of S over U.
˜ S )2 (x) = (hS ∪k ˜ S )(ai ) ∪ (hS ∪k ˜ S )(aj )∪ = ((hS ∪k m ai bi +z
x+
i=1 n
=
aj bj +z
j=1
˜ S ˜ S )(bj )) (hS ∪k )(bi ) ∪ (hS ∪k (hS (ai ) ∪ hS (aj )∪ ⊇ m
,
ai bi +z
x+
i=1 n
=
aj bj +z
j=1
kS (bi ) ∪ kS (bj )) = (hS ♦kS )(x), ˜ S ⊇h ˜ S ♦kS . Thus, hS ∪k ˜ S= which implies, hS ∪k hS ♦kS . It follows from Corollary 5.5 that S is hhemiregular. 6. Conclusion
Theorem 5.6. A hemiring S is h-hemiregular if and only if every SU-h-ideal of S is idempotent.
In this paper, we discuss soft union hemirings(soft union h-ideals) of hemirings and obtain some related
J. Zhan et al. / Soft union hemirings
(1) To describe some new kinds of soft union hideals; (2) To establish some new kinds of soft hhemiregular hemirings; (3) To apply soft hemirings to some applied fields, such as decision making, data analysis and forecasting and so on.
Acknowledgments
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References
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This research is partially supported by a grant of National Natural Science Foundation of China (61175055), Innovation Term of Higher Education of Hubei Province, China (T201109), Natural Science Foundation of Hubei Province (2012FFB01101).
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properties. Finally, we give some characterizations of h-hemiregular hemirings by means of SU-h-ideals. We believe that the research along this direction can be continued, and in fact, some results in this paper have already constituted a foundation for further investigation concerning the further development of hemirings. In the future study of soft hemirings, perhaps the following topics are worth to be considered:
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