The characterizations of h-hemiregular hemirings and h-intra-hemiregular hemirings

Information Sciences 178 (2008) 3451–3464

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Information Sciences

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The characterizations of h-hemiregular hemiringsand h-intra-hemiregular hemirings

Yunqiang Yin a, Hongxing Li b,*

a School of Mathematical Sciences, Beijing Normal University, Beijing 100875, Chinab School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116024, China

a r t i c l e i n f o

Article history:Received 19 April 2007Received in revised form 31 March 2008Accepted 15 April 2008

Keywords:HemiringsFuzzy h-idealsFuzzy h-bi-idealsFuzzy h-quasi-idealsh-Hemiregular hemiringsh-Intra-hemiregular hemirings

0020-0255/$ - see front matter � 2008 Published bdoi:10.1016/j.ins.2008.04.002

* Corresponding author. Tel.: +86 411 84706402;E-mail addresses: [email protected]

a b s t r a c t

In this paper, the concepts of fuzzy h-bi-ideals and fuzzy h-quasi-ideals of a hemiring areintroduced, and related properties are investigated. The notion of h-intra-hemiregularity ofa hemiring, which is a generalization of the notion of intra-regularity of a ring, is provided.Some characterizations of h-hemiregular hemirings, h-intra-hemiregular hemirings andhemirings that are both h-hemiregular and h-intra-hemiregular are derived in terms offuzzy left, fuzzy right h-ideals, fuzzy h-bi-ideals and fuzzy h-quasi-ideals.

� 2008 Published by Elsevier Inc.

1. Introduction

As a generalization of rings, semirings have been found useful for solving problems in different areas of applied mathe-matics and information sciences, since the structure of a semiring provides an algebraic framework for modelling and study-ing the key factors in these applied areas. They play an important role in studying optimization theory, graph theory, theoryof discrete event dynamical systems, matrices, determinants, generalized fuzzy computation, automata theory, formal lan-guage theory, coding theory, analysis of computer programs, and so on (see [3,4,6,8,10,23,24] for details). Ideals of semiringsplay a central role in the structure theory and are useful for many purposes. However, they do not in general coincide withthe usual ring ideals and, for this reason, their use is somewhat limited in trying to obtain analogues of ring theorems forsemirings. Indeed, many results in rings apparently have no analogues in semirings using only ideals. In order to overcomethis deficiency, Henriksen [11] defined a more restricted class of ideals in semirings, which is called the class of k-ideals, withthe property that if the semiring S is a ring then a complex in S is a k-ideal if and only if it is a ring ideal. A still more restrictedclass of ideals in hemirings has been given by Iizuka [12]. According to Iizuka’s definition, an ideal in any additivelycommutative semiring S can be given which coincides with a ring ideal provided S is a hemiring, and it is called h-ideal.The properties of h-ideals and k-ideals of hemirings were thoroughly investigated by La Torre [19] and by using the h-idealsand k-ideals, La Torre established some analogous ring theorems for hemirings.

The theory of fuzzy sets, proposed by Zadeh [25] in 1965, has provided a useful mathematical tool for describing thebehavior of systems that are too complex or illdefined to admit precise mathematical analysis by classical methods and tools.Extensive applications of fuzzy set theory have been found in various fields such as artificial intelligence, computer science,

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3452 Y. Yin, H. Li / Information Sciences 178 (2008) 3451–3464

control engineering, expert systems, management science, operations research, pattern recognition, robotics, and others. Itsoon invoked a natural question concerning a possible connection between fuzzy sets and algebraic systems. The study of thefuzzy algebraic structures has started in the pioneering paper of Rosenfeld [22] in 1971. Rosenfeld introduced the notion offuzzy groups and showed that many results in groups can be extended in an elementary manner to develop the theory offuzzy group. Since then the literature of various fuzzy algebraic concepts has been growing very rapidly. For example, Kuroki[20,21] introduced and studied the concept of fuzzy ideals of a semigroup. Subsequently, many authors fuzzified certainstandard concepts and results on rings and modules. In [15,16], Kehayopulu and Tsingelis applied the concept of fuzzy setsto the theory of ordered semigroups and characterized the regular ordered semigroups in terms of fuzzy ideals. The relation-ships between the fuzzy sets and semirings (hemirings) have been considered by Dutta, Baik, Ghosh, Jun, Kim, Zhan and oth-ers. The reader is refereed to [1,2,5–7,9,13,14,17,26].

Recently, Zhan et al. [27] introduced the concept of h-hemiregularity of a hemiring and gave a characterization of h-hemi-regular hemirings in terms of fuzzy right and fuzzy left h-ideals. As a continuation of the paper [27], we consider the char-acterizations of h-hemiregular hemirings and h-intra-hemiregular hemirings. We introduce the concepts of fuzzy h-bi-idealsand fuzzy h-quasi-ideals of a hemiring, and give some of their properties. We provide the notion of h-intra-hemiregularity ofa hemiring as a generalization of the notion of intra-regularity of a ring. Further, we investigate the characterizations of h-hemiregular hemirings, h-intra-hemiregular hemirings and hemirings that are both h-hemiregular and h-intra-hemiregularin terms of fuzzy left, fuzzy right h-ideals, fuzzy h-bi-ideals and fuzzy h-quasi-ideals.

2. Preliminaries

A semiring is an algebraic system ðS;þ; �Þ consisting of a non-empty set S together with two binary operations on S calledaddition and multiplication (denoted in the usual manner) such that ðS;þÞ and ðS; �Þ are semigroups and the following dis-tributive laws

aðbþ cÞ ¼ abþ bc and ðaþ bÞc ¼ ac þ bc

are satisfied for all a; b; c 2 S.By zero of a semiring ðS;þ; �Þ we mean an element 0 2 S such that 0 � x ¼ x � 0 ¼ 0 and 0þ x ¼ xþ 0 ¼ x for all x 2 S. A

semiring with zero and a commutative semigroup ðS;þÞ is called a hemiring. For the sake of simplicity, we shall write abfor a � b ða; b 2 SÞ.

A subset A in a hemiring S is called a left (resp. right) ideal of S if A is closed under addition and SA � A (resp. AS � A). Fur-ther, A is called an ideal of S if it is both a left ideal and a right ideal of S. A subset A in a hemiring S is called a bi-ideal if A isclosed under addition and multiplication satisfying ASA � A.

A left ideal A of S is called a left h-ideal if x; z 2 S; a; b 2 A, and xþ aþ z ¼ bþ z implies x 2 A [19]. Right h-ideals, h-idealsand h-bi-ideals are defined similarly.

The h-closure A of A in a hemiring S is defined as

A ¼ fx 2 Sjxþ a1 þ z ¼ a2 þ z for some a1; a2 2 A; z 2 Sg:
A subset A in a hemiring S is called a quasi-ideal of S if A is closed under addition and SA \ AS � A. A quasi-ideal A of S is calledan h-quasi-ideal of S if SA \ SA � A and xþ aþ z ¼ bþ z implies x 2 A for all x; z 2 S; a; b 2 A.
Lemma 2.1 [27]. For a hemiring S, we have

(1) A � A; 8A � S.(2) If A � B � S, then A � B.(3) A ¼ A; 8A � S.(4) AB ¼ AB and ABC ¼ ABC; 8A;B;C � S.(5) For any left (right) h-ideal, h-bi-ideal or h-quasi-ideal A of S, we have A ¼ A.

Note that every left h-ideal (resp. right h-ideal, h-bi-ideal, h-quasi-ideal) of a hemiring S is a left ideal (resp. right ideal, bi-ideal, quasi-ideal) of S. Also, every left (right) h-ideal of S is an h-quasi-ideal of S and every h-quasi-ideal of S is an h-bi-idealof S. However, the converse of these properties do not hold in general as shown in the following examples.

Example 2.2. Let S ¼ f0; a; b; cg be a set with a addition operation ðþÞ and a multiplication operation (�) as follows:

Then S is a hemiring. Let A ¼ f0; bg. Evidently A is an ideal of S and it is not an h-ideal of S, since aþ 0þ b ¼ 0þ b while a 62 A.

Y. Yin, H. Li / Information Sciences 178 (2008) 3451–3464 3453

Example 2.3. We denote by N0 the set of all non-negative integers and let S be the set of all 2� 2 matrices a11 a12

a21 a22

� �ðaij 2 N0Þ. Then S is a hemiring with respect to the usual addition and multiplication of matrices. Consider the set Q of all

matrices of the form a 00 0

� �ða 2 N0Þ. Evidently Q is an h-quasi-ideal of S and it is not a left (right) h-ideal of S.

Example 2.4. We denote by N and P the sets of all positive integers and positive real numbers. The set S of all matrices of the

form a 0b c

� �ða; b 2 P; c 2 NÞ together with 0 0

0 0

� �is a hemiring with respect to the usual addition and multiplication of

matrices. Let R and L be the sets of all matrices a 0b c

� �ða; b 2 P; c 2 N; a < bÞ together with 0 0

0 0

� �and

p 0q k

� �ðp; q 2 P; k 2 N;3 < qÞ together with 0 0

0 0

� �, respectively. It is easy to show that R and L are a right h-ideal and

a left h-ideal of S, respectively. Now the product RL is an h-bi-ideal of S and it is not an h-quasi-ideal of S. Indeed, the element

6 09 1

� �¼

6 03 1

� �1 02 1

� �1 04 1

� �� ¼

1 076 1

!24 04 1

� � !14 01 1

!

belongs to the intersection SðRLÞ \ ðRLÞS, but it is not an element of RL. Hence SðRLÞ \ ðRLÞS 6� RL.

Definition 2.5 [14]. Let l and m be fuzzy subsets in a hemiring S. Then the h-product of l and m is defined by

ðl�hmÞðxÞ ¼ supxþa1b1þz¼a2b2þz

ðminflðaiÞ; mðbiÞji ¼ 1;2gÞ

and ðl�hmÞðxÞ ¼ 0 if x cannot be expressed as xþ a1b1 þ z ¼ a2b2 þ z.

As a generalization, we introduce the h-intrinsic product of fuzzy subsets in a hemiring as follows.

Definition 2.6. Let l and m be fuzzy subsets in a hemiring S. Then the h-intrinsic product of l and m is defined by

ðl�hmÞðxÞ ¼ supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞ; mðbiÞ; mðb0jÞji ¼ 1; . . . ;m; j ¼ 1; . . . ;ngÞ

and ðl�hmÞðxÞ ¼ 0 if x cannot be expressed as xþPm

i¼1aibi þ z ¼Pn

j¼1a0jb0j þ z.

By directly calculation we obtain immediately the following result.

Proposition 2.7. Let S be a hemiring and l; m;x; k any fuzzy subsets in S. Then we have

(1) l�hm � l�hm.(2) If l � x and m � k, then l�hm � x�hk.

The inclusion symbol ‘‘�” in Proposition 2.7(1) may not be replaced by an equal sign as shown in the next example.

Example 2.8. Let S ¼ f0; a; bg be a set with a addition operation (+) and a multiplication operation (�) as follows:

Then S is a hemiring. Let r; s; t 2 ½0;1� be such that r 6 s < t. Define fuzzy subsets l and m in S by

lð0Þ ¼ r; lðaÞ ¼ t; lðbÞ ¼ r

and
mð0Þ ¼ s; mðaÞ ¼ t; mðbÞ ¼ s;
respectively. Then ðl�hmÞðaÞ ¼ s < t ¼ ðl�hmÞðaÞ.

For any subset A in a hemiring S, vA will denote the characteristic function of A.

Lemma 2.9. Let S be a hemiring and A;B � S. Then we have

(1) A � B if and only if vA � vB.(2) vA \ vB ¼ vA\B.(3) vA�hvB ¼ vAB.


Proof. The proof of (1) and (2) is straightforward. We show (3). Let x 2 S. If x 2 AB, then vABðxÞ ¼ 1 and xþPm

i¼1piqi þ z ¼Pnj¼1p0jq

0j þ z for some pi; p

0j 2 A, qi; q

0j 2 B and z 2 S. Thus we have

ðvA�hvBÞðxÞ ¼ supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminfvAðaiÞ; vAða0jÞ; vBðbiÞ; vBðb0jÞgÞP minfvAðpiÞ; vAðp0jÞ; vBðqiÞ; vBðq0jÞg ¼ 1;

and so ðvA�hvBÞðxÞ ¼ 1 ¼ vABðxÞ.If x 62 AB, then vABðxÞ ¼ 0. If possible, let ðvA�hvBÞðxÞ 6¼ 0. Then

ðvA�hvBÞðxÞ ¼ supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminfvAðaiÞ; vAða0jÞ; vBðbiÞ; vBðb0jÞgÞ 6¼ 0:

Hence there exist pi; qi; p0j; q0j; z 2 S such that

xþXm

i¼1

piqi þ z ¼Xn

j¼1

p0jq0j þ z

and

minfvAðpiÞ; vAðp0jÞ; vBðqiÞ; vBðq0jÞg 6¼ 0;

that is,

vAðpiÞ ¼ vAðp0jÞ ¼ vBðqiÞ ¼ vBðq0jÞ ¼ 1;

hence pi; p0j 2 A and qi; q

0j 2 B, and so x 2 AB, which contradicts vABðxÞ ¼ 0. Thus we have

ðvA�hvBÞðxÞ ¼ 0 ¼ vABðxÞ:

In any case, we have ðvA�hvBÞðxÞ ¼ vABðxÞ. This completes the proof. h

3. Fuzzy h-ideals in hemirings

It is well known that ideal theory plays a fundamental role in the development of hemirings. In [14], Jun et al. introducedthe concepts of fuzzy left and fuzzy right h-ideals of a hemiring. In this section, we define the notions of fuzzy h-bi-ideals andfuzzy h-quasi-ideals of a hemiring, and investigate some of their properties.

Definition 3.1 [14]. A fuzzy subset l in a hemiring S is called a fuzzy left h-ideal if for all x; y; z; a; b 2 S we have

(i) lðxþ yÞP minflðxÞ; lðyÞg,(ii) lðxyÞP lðyÞ,

(iii) xþ aþ z ¼ bþ z! lðxÞP minflðaÞ; lðbÞg.

Fuzzy right h-ideals are defined similarly. A fuzzy subset in a hemiring S is called a fuzzy h-ideal of S if it is both a fuzzy lefth-ideal and a fuzzy right h-ideal of S.

Definition 3.2. A fuzzy subset l in a hemiring S is called a fuzzy h-bi-ideal if for all x; y; z; a; b 2 S we have

(i) lðxþ yÞP minflðxÞ; lðyÞg,(ii) lðxyÞP minflðxÞ; lðyÞg,

(iii) lðxyzÞP minflðxÞ; lðzÞg,(iv) xþ aþ z ¼ bþ z! lðxÞP minflðaÞ; lðbÞg.

Definition 3.3. A fuzzy subset l in a hemiring S is called a fuzzy h-quasi-ideal if for all x; y; z; a; b 2 S we have

(i) lðxþ yÞP minflðxÞ; lðyÞg,(ii) ðl�hvSÞ \ ðvS�hlÞ � l,

(iii) xþ aþ z ¼ bþ z! lðxÞP minflðaÞ; lðbÞg.

Note that if l is a fuzzy left h-ideal (right h-ideal, h-bi-ideal, h-quasi-ideal), then lð0ÞP lðxÞ for all x 2 S.

Example 3.4. The set N0 of all non-negative integers is a hemiring with respect to the usual addition and multiplication. Letr; s 2 ½0;1Þ be such that r 6 s. Define a fuzzy subset l in N0 by

lðaÞ ¼s if a 2 h3i;r otherwise

�

for all a 2 N0. Then l is both a fuzzy h-bi-ideal and a fuzzy h-quasi-ideal of N0.


For any fuzzy subset in a set X and any t 2 ½0;1�, define a set Uðl; tÞ ¼ fx 2 XjlðxÞP tg, which is called a level subset of l. In[18], Kondo et al. introduced the Transfer Principle in fuzzy set theory, from which a fuzzy subset l can be characterized byits level subsets. For any algebraic system A ¼ ðX; FÞ, where F is a family of operations defined on X, the Transfer Principle canbe formulated as follows.

Lemma 3.5. A fuzzy subset defined on A has the property P if and only if all non-empty level subsets Uðl; tÞ have the property P.

As a direct consequence of the above Lemma, we may obtain the following results.

Lemma 3.6. Let S be a hemiring. Then the following conditions hold:

(1) l is a fuzzy left (resp. right) h-ideal of S if and only if all non-empty level subsets Uðl; tÞ are left (resp. right) h-ideals of S [14].(2) l is a fuzzy h-bi-ideal of S if and only if all non-empty level subsets Uðl; tÞ are h-bi-ideals of S.(3) l is a fuzzy h-quasi-ideal of S if and only if all non-empty level subsets Uðl; tÞ are h-quasi-ideals of S.

Lemma 3.7. Let S be a hemiring and A � S. Then the following conditions hold:

(1) A is a left (resp. right) h-ideal of S if and only if vA is a fuzzy left (resp. right) h-ideal of S [14].(2) A is an h-bi-ideal of S if and only if vA is a fuzzy h-bi-ideal of S.(3) A is an h-quasi-ideal of S if and only if vA is a fuzzy h-quasi-ideal of S.

Lemma 3.8. A fuzzy subset l in a hemiring S is a fuzzy left (resp. right) h-ideal of S if and only if for all x; y; z; a; b 2 S, we have

(1) lðxþ yÞP minflðxÞ; lðyÞg;(2) vS�hl � l (resp. l�hvS � l);(3) xþ aþ z ¼ bþ z! lðxÞP minflðaÞ; lðbÞg.

Proof. Assume that l is a fuzzy left h-ideal of S. It is sufficient to show that the condition (2) is satisfied. Let x 2 S. IfðvS�hlÞðxÞ ¼ 0, it is clear that ðvS�hlÞðxÞ 6 lðxÞ. Otherwise, there exist elements ai; bi; a0j; b

0j; z 2 S such that

xþPm

i¼1aibi þ z ¼Pn

j¼1a0jb0j þ z. Then we have

ðvS�hlÞðxÞ ¼ supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðbiÞ; lðb0jÞgÞ 6 supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaibiÞ; lða0jb0jÞgÞ

6 supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

min lXm

i¼1

aibi

!; l

Xn

j¼1

a0jb0j

!( ) !6 sup

xþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

lðxÞ ¼ lðxÞ:

This implies that vS�hl � l.Conversely, assume that the given conditions hold. It is sufficient to show that the condition (ii) of Definition 3.1 is valid.

Let x; y 2 S. Then we have

lðxyÞP ðvS�hlÞðxyÞ ¼ supxyþPm

i¼1aibi¼

Pn

j¼1a0

jb0j

ðminflðbiÞ; lðb0jÞgÞP lðyÞðsince xyþ 0yþ 0 ¼ xyþ 0Þ:

This implies that the condition (ii) of Definition 3.1 is valid and so l is a fuzzy left h-ideal of S. The case for fuzzy right h-idealcan be similarly proved. h

Lemma 3.9. Let l and m be a fuzzy right h-ideal and a fuzzy left h-ideal of a hemiring S, respectively. Then l \ m is a fuzzy h-quasi-ideal of S.

Proof. Let x and y be any elements of S. Then

ðl \ mÞðxþ yÞ ¼minflðxþ yÞ; mðxþ yÞgP minfminflðxÞ; lðyÞg;minfmðxÞ; mðyÞgg ¼minfminflðxÞ; mðxÞg;minflðyÞ; mðyÞgg¼minfðl \ mÞðxÞ; ðl \ mÞðyÞg:

Now let a; b; x; z 2 S such that xþ aþ z ¼ bþ z. Then

ðl \ mÞðxÞ ¼minflðxÞ; mðxÞgP minfminflðaÞ; lðbÞg;minfmðaÞ; mðbÞgg ¼ minfminflðaÞ; mðaÞg;minflðbÞ; mðbÞgg¼minfðl \ mÞðaÞ; ðl \ mÞðbÞg:

On the other hand, we have

ððl \ mÞ�hvSÞ \ ðvS�hðl \ mÞÞ � ðl�hvSÞ \ ðvS�hmÞ � l \ m:

This completes the proof. h


Lemma 3.10. Any fuzzy h-quasi-ideal of a hemiring S is a fuzzy h-bi-ideal of S.

Proof. Let l be any fuzzy h-quasi-ideal of S. It is sufficient to show that

lðxyzÞP minflðxÞ; lðzÞg and lðxyÞP minflðxÞ; lðyÞg for all x; y; z 2 S:

In fact, by the assumption, we have

lðxyzÞP ððl�hvSÞ \ ðvS�hlÞÞðxyzÞ ¼minfðl�hvSÞðxyzÞ; ðvS�hlÞðxyzÞg

¼ min supxyzþPm

i¼1aibiþz0¼

Pn

j¼1a0

jb0jþz0ðminflðaiÞ; lða0jÞgÞ; sup

xyzþPm

i¼1aibiþz0¼

Pn

j¼1a0

jb0jþz0ðminflðbiÞ; lðb0jÞgÞ

8<:

9=;

P minfminflð0Þ; lðxÞg;minflð0Þ; lðzÞgg ðsince xyzþ 00þ 0 ¼ xðyzÞ þ 0 and xyzþ 00þ 0 ¼ ðxyÞzþ 0Þ¼ minflðxÞ; lðzÞg:

Similarly, we can show that lðxyÞP minflðxÞ; lðyÞg for all x; y 2 S. This completes the proof. h

The following example shows that the converse of Lemma 3.10 is not true.

Example 3.11. Consider Example 2.3. It follows from Lemma 3.7 that the characteristic function vRL of RL is a fuzzy h-bi-idealof S and it is not a fuzzy h-quasi-ideal of S.

4. h-Hemiregular hemirings

The concept of h-hemiregularity of a hemiring was first introduced by Zhan et al. [27] as a generalization of the concept ofregularity of a ring. In this section, we concentrate our study on the characterizations of h-hemiregular hemirings. We startby formulating the following definition.

Definition 4.1 [27]. A hemiring S is said to be h-hemiregular if for each x 2 S, there exist a; a0; z 2 S such thatxþ xaxþ z ¼ xa0xþ z.

Lemma 4.2 [27]. A hemiring S is h-hemiregular if and only if for any right h-ideal R and any left h-ideal L of S we have RL ¼ R \ L:

Theorem 4.3. A hemiring S is h-hemiregular if and only if for any fuzzy right h-ideal l and any fuzzy left h-ideal m of S we havel�hm ¼ l \ m:

Proof. Let S be an h-hemiregular hemiring, l any fuzzy right h-ideal and m any fuzzy left h-ideal of S, respectively. Then byLemma 3.8, we have l�hm � l�hvS � l and l�hm � vS�hm � m. Thus l�hm � l \ m: To show the converse inclusion, let x be anyelement of S. Since S is h-hemiregular, there exist a; a0; z 2 S such that xþ xaxþ z ¼ xa0xþ z. Then we have


i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞ; mðbiÞ; mðb0jÞgÞP minflðxaÞ; lðxa0Þ; mðxÞgP minflðxÞ; mðxÞg ¼ ðl \ mÞðxÞ:

This implies that l�hm � l \ m. Therefore l�hm ¼ l \ m.Conversely, let R and L be any right h-ideal and any left h-ideal of S, respectively. Then by Lemma 3.7, the characteristic

functions vR and vL of R and L are a fuzzy right h-ideal and a fuzzy left h-ideal of S, respectively. Now, by the assumption andLemma 2.9, we have

vRL ¼ vR�hvL ¼ vR \ vL ¼ vR\L:

It follows from Lemma 2.9 that RL ¼ R \ L. Therefore S is h-hemiregular by Lemma 4.2. h

Combing Theorem 4.3 and Theorem 3.6 of [27] we obtain immediately the following result.

Corollary 4.4. Let S be an h-hemiregular hemiring. Then l�hm ¼ l�hm for every fuzzy right h-ideal l and every fuzzy left h-ideal mof S.

Lemma 4.5. Let S be a hemiring. Then the following conditions are equivalent.

(1) S is h-hemiregular.(2) B ¼ BSB for every h-bi-ideal B of S.(3) Q ¼ QSQ for every h-quasi-ideal Q of S.

Proof. (1)) (2) Assume that (1) holds. Let B be any h-bi-ideal of S and x any element of B. Then there exist a; a0; z 2 S suchthat xþ xaxþ z ¼ xa0xþ z. It is easy to see that xax; xa0x 2 BSB and so x 2 BSB. Hence B � BSB. On the other hand, since B is anh-bi-ideal of S, we have BSB � B and so BSB � B ¼ B by Lemma 2.1. Therefore B ¼ BSB.


(2)) (3) This is straightforward.(3)) (1) Assume that (3) holds. Let R and L be any right h-ideal and any left h-ideal of S, respectively. Then we have

ðR \ LÞS \ SðR \ LÞ � RS \ SL � R \ L ¼ R \ L;

and thus R \ L is an h-quasi-ideal of S. By the assumption and Lemma 2.1, we have

R \ L ¼ ðR \ LÞSðR \ LÞ � RSL � RL � R \ L � R \ L ¼ R \ L;

and thus RL ¼ R \ L. Therefore S is h-hemiregular by Lemma 4.2. h

Theorem 4.6. Let S be a hemiring. Then the following conditions are equivalent.

(1) S is h-hemiregular.(2) l � l�hvS�hl for every fuzzy h-bi-ideal l of S.(3) l � l�hvS�hl for every fuzzy h-quasi-ideal l of S.

Proof. (1)) (2) Assume that (1) holds. Let l be any fuzzy h-bi-ideal of S and x any element of S. Since S is h-hemiregular,there exist a; a0; z 2 S such that xþ xaxþ z ¼ xa0xþ z. Then we have

ðl�hvS�hlÞðxÞ ¼ supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminfðl�hvSÞðaiÞ; ðl�hvSÞða0jÞ; lðbiÞ; lðb0jÞgÞP minfðl�hvSÞðxaÞ; ðl�hvSÞðxa0Þ; lðxÞg

¼min supxaþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞgÞ; supxa0þPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞgÞ; lðxÞ

8<:

9=;

P minfminflðxaxÞ; lðxa0xÞg;minflðxaxÞ; lðxa0xÞg; lðxÞg ðsince xaþ xaxaþ za¼ xa0xaþ za and xa0 þ xaxa0 þ za0 ¼ xa0xa0 þ za0ÞP minflðxÞ; lðxÞ; lðxÞg ¼ lðxÞ:

This implies that l � l�hvS�hl.(2)) (3) This is straightforward by Lemma 3.10.(3)) (1) Assume that (3) holds. Let Q be any h-quasi-ideal of S. Then by Lemma 3.7, the characteristic function vQ of Q is a

fuzzy h-quasi-ideal of S. Now, by the assumption and Lemma 2.9, we have

vQ � vQ�hvS�hvQ ¼ vQSQ :

Then it follows from Lemma 2.9 that Q � QSQ . On the other hand, since Q is an h-quasi-ideal of S, we have QSQ �SQ \ QS � Q , and so QSQ ¼ Q . Therefore S is h-hemiregular by Lemma 4.5 h.


(1) S is h-hemiregular.(2) l \ m � l�hm�hl for every fuzzy h-bi-ideal l and every fuzzy h-ideal m of S.(3) l \ m � l�hm�hl for every fuzzy h-quasi-ideal l and every fuzzy h-ideal m of S.

Proof. (1)) (2) Assume that (1) holds. Let l and m be any fuzzy h-bi-ideal and any fuzzy h-ideal of S, respectively. Now let xbe any element of S. Since S is h-hemiregular, there exist a; a0; z 2 S such that xþ xaxþ z ¼ xa0xþ z. Then we have

ðl�hm�hlÞðxÞ ¼ supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminfðl�hmÞðaiÞ; ðl�hmÞða0jÞ;lðbiÞ; lðb0jÞgÞP minfðl�hmÞðxaÞ; ðl�hmÞðxa0Þ; lðxÞg

¼min supxaþPm

i¼1ai biþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ;lða0jÞ; mðbiÞ; mðb0jÞgÞ; supxa0þPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞ; mðbiÞ; mðb0jÞgÞ;lðxÞ

8<:

9=;

P minfminflðxÞ; mðaxaÞ; mða0xaÞg;minflðxÞ; mðaxa0Þ; mða0xa0Þg;lðxÞg ðsince xaþ xaxaþ za ¼ xa0xa

þ za and xa0 þ xaxa0 þ za0 ¼ xa0xa0 þ za0ÞP minfminflðxÞ; mðxÞg;minflðxÞ; mðxÞg; lðxÞg¼minflðxÞ; mðxÞg ¼ ðl \ mÞðxÞ:

This implies that l \ m � l�hm�hl.(2)) (3) This is straightforward by Lemma 3.10.(3)) (1) Assume that (3) holds. Let l be any fuzzy h-quasi-ideal of S. Then since vS is a fuzzy h-ideal of S, we have

l ¼ l \ vS � l�hvS�hl:

Therefore S is h-hemiregular by Theorem 4.6. h


Corollary 4.8. Let S be a hemiring. Then the following conditions are equivalent.

(1) S is h-hemiregular.(2) B \ A ¼ BAB for every h-bi-ideal B and every h-ideal A of S.(3) Q \ A ¼ QAQ for every h-quasi-ideal Q and every h-ideal A of S.

Proof. (1)) (2) Assume that (1) holds. Let B and A be any h-bi-ideal and any h-ideal of S, respectively. Then by Lemma 3.7,the characteristic functions vB and vA of B and A are a fuzzy h-bi-ideal and a fuzzy h-ideal of S, respectively. Thus, by Theorem4.7 and Lemma 2.9, we have

vB\A ¼ vB \ vA � vB�hvA�hvB ¼ vBAB:

Then it follows from Lemma 2.9 that B \ A � BAB. On the other hand, since B and A is an h-bi-ideal and an h-ideal of S, respec-tively, we have BAB � BSB � B ¼ B and BAB � A ¼ A. Thus BAB � B \ A and so BAB ¼ B \ A.

(2)) (3) This is straightforward.(3)) (1) Assume that (3) holds. Let Q be any h-quasi-ideal of S. Then since S itself is an h-ideal of S, we have

Q ¼ Q \ S ¼ QSQ : Therefore S is h-hemiregular by Lemma 4.5. h


(1) S is h-hemiregular.(2) l \ m � l�hm for every fuzzy h-bi-ideal l and every fuzzy left h-ideal m of S.(3) l \ m � l�hm for every fuzzy h-quasi-ideal l and every fuzzy left h-ideal m of S.(4) l \ m � l�hm for every fuzzy right h-ideal l and every fuzzy h-bi-ideal m of S.(5) l \ m � l�hm for every fuzzy right h-ideal l and every fuzzy h-quasi-ideal m of S.(6) l \ m \ x � l�hm�hx for every fuzzy right h-ideal l, every fuzzy h-bi-ideal m and every fuzzy left h-ideal x of S.(7) l \ m \ x � l�hm�hx for every fuzzy right h-ideal l, every fuzzy h-quasi-ideal m and every fuzzy left h-ideal x of S.

Proof. (1)) (2) Assume that (1) holds. Let l and m be any fuzzy h-bi-ideal and any fuzzy left h-ideal of S, respectively.Now let x be any element of S. Since S is h-hemiregular, there exist a; a0; z 2 S such that xþ xaxþ z ¼ xa0xþ z. Then wehave


i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞ; mðbiÞ; mðb0jÞgÞP minflðxÞ; mðaxÞ; mða0xÞgP minflðxÞ; mðxÞg ¼ ðl \ mÞðxÞ:

This implies that l \ m � l�hm.(2)) (1) Assume that (2) holds. Let l and m be any fuzzy right h-ideal and any fuzzy left h-ideal of S, respectively. Then, it

is easy to see that l is a fuzzy h-bi-ideal of S. By the assumption, we have

l \ m � l�hm � l�hvS \ vS�hm � l \ m:

Hence l \ m ¼ l�hm and so S is h-hemiregular by Theorem 4.3.Similarly, we can show that ð1Þ () ð3Þ, ð1Þ () ð4Þ, ð1Þ () ð5Þ.(1)) (6) Assume that (1) holds. Let l, m and x be any fuzzy right h-ideal, any fuzzy h-bi-ideal and any fuzzy left h-ideal of

S, respectively. Now let x be any element of S. Since S is h-hemiregular, there exist a; a0; z 2 S such that xþ xaxþ z ¼ xa0xþ z.Then we have

ðl�hm�hxÞðxÞ ¼ supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminfðl�hmÞðaiÞ; ðl�hmÞða0jÞ;xðbiÞ;xðb0jÞgÞP minfðl�hmÞðxÞ;xðaxÞ;xða0xÞg

¼ min supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞ; mðbiÞ; mðb0jÞgÞ;xðaxÞ;xða0xÞ

8<:

9=;

P minfminflðxaÞ; lðxa0Þ; mðxÞg;xðaxÞ;xða0xÞgP minflðxÞ; mðxÞ;xðxÞg ¼ ðl \ m \ xÞðxÞ:

This implies that l \ m \ x � l�hm�hx.(6)) (7) This is straightforward by Lemma 3.10.(7)) (1) Assume that (7) holds. Let l and m be any fuzzy right h-ideal and any fuzzy left h-ideal of S, respectively. Since vS

is a fuzzy h-quasi-ideal of S, by the assumption, we have

l \ m ¼ l \ vS \ m � l�hvS�hm � l�hm � l�hvS \ vS�hm � l \ m:

Hence l \ m ¼ l�hm and so S is h-hemiregular by Theorem 4.3. This completes the proof. h


Corollary 4.10. Let S be a hemiring. Then the following conditions are equivalent.

(1) S is h-hemiregular.(2) B \ L � BL for every h-bi-ideal B and every left h-ideal L of S.(3) Q \ L � QL for every h-quasi-ideal Q and every left h-ideal L of S.(4) R \ B � RB for every right h-ideal R and every h-bi-ideal B of S.(5) R \ Q � RQ for every right h-ideal R and every h-quasi-ideal Q of S.(6) R \ B \ L � RBL for every right h-ideal R, every h-bi-ideal B and every left h-ideal L of S.(7) R \ Q \ L � RQL for every right h-ideal R, every h-quasi-ideal Q and every left h-ideal L of S.

Proof. The proof is analogous to that of Corollary 4.8 h.

Definition 4.11. A subset A in a hemiring S is called idempotent if A ¼ A2. A fuzzy subset l in a hemiring S is called idempotentif l ¼ l�hl.

Example 4.12. Consider Example 2.2. Let A ¼ f0; ag. Evidently A2 ¼ f0g and A2 ¼ f0; ag ¼ A. Define a fuzzy subset l in S bylð0Þ ¼ 0:5; lðaÞ ¼ 0:5; lðbÞ ¼ 0. Then l ¼ l�hl.

Lemma 4.13. A hemiring S is h-hemiregular if and only if the right and left h-ideals of S are idempotent and for any right h-ideal Rand any left h-ideal L of S, the set RL is an h-quasi-ideal of S.

Proof. Assume that S is h-hemiregular. Let R and L be any right h-ideal and any left h-ideal of S, respectively. Then we haveR2 � R ¼ R. Let x 2 R. Since S is h-hemiregular, there exist a; a0; z 2 S such that xþ xaxþ z ¼ xa0xþ z. Since R is a right h-idealof S, we have xa; xa0 2 R and so xax; xa0x 2 R2. Hence x 2 R2, that is, R � R2. Thus R ¼ R2 and so R is idempotent. Similarly, wemay show that L is idempotent. Since S is h-hemiregular, we have R \ L ¼ RL by Lemma 4.2. Then it follows from the proof ofLemma 4.5 that RL ¼ R \ L is an h-quasi-ideal of S.

Conversely, assume that the given conditions hold. Let Q be any h-quasi-ideal of S. Then it is easy to see that Q þ SQ is aleft h-ideal of S. Thus by the assumption, we have

Q � Q þ SQ ¼ ðQ þ SQÞ ðQ þ SQÞ � ðQ þ SQÞ ðQ þ SQÞ ¼ ðQ þ SQÞðQ þ SQÞ ¼ Q2 þ QSQ þ SQ2 þ SQSQ

� SQ þ SQ þ SQ þ SQ ¼ SQ ;

that is, Q � SQ . Similarly, we can show that Q � QS. Thus Q � SQ \ QS � Q and so Q ¼ SQ \ QS.On the other hand, it is clear that QS and SQ are a right h-ideal and a left h-ideal of S, respectively. Then, by the

assumption, we know ðQSÞ2 ¼ QS, ðSQÞ2 ¼ SQ and that the set ðQSÞðSQÞ is an h-quasi-ideal of S. Thus we have

Q � QSQ � QSSQ ¼ ðQSÞ ðSQÞ � ðQSÞ ðSQÞ ¼ ððQSÞ ðSQÞÞS \ SððQSÞ ðSQÞÞ ¼ ððQSÞðSQÞÞS \ SððQSÞ ðSQÞÞ

¼ ðQSÞðSQÞS \ SðQSÞðSQÞ ¼ QðSSÞQS \ SQðSSÞQ ¼ Q ðSSÞðQSÞ \ ðSQÞ ðSSÞQ ¼ Q S ðQSÞ \ ðSQÞS Q

¼ ðQSÞðQSÞ \ ðSQÞðSQÞ ðsince S itself is an h-ideal of S and so SS ¼ S ¼ SÞ ¼ ðQSÞ2 \ ðSQÞ2 ¼ QS \ SQ ¼ Q ;

that is, Q ¼ QSQ and so S is h-hemiregular. h

Theorem 4.14. A hemiring S is h-hemiregular if and only if the fuzzy right and fuzzy left h-ideals of S are idempotent and for anyfuzzy right h-ideal l and any fuzzy left h-ideal m of S, the set l�hm is a fuzzy h-quasi-ideal of S.

Proof. Assume that S is h-hemiregular. Let l be any fuzzy right h-ideal of S. Then l�hl � l�hvS � l. Moreover, since S is h-hemiregular, it follows from the proof of Theorem 4.3 that l � l�hl and so l ¼ l�hl, hence l is idempotent. In a similar waywe may prove that the fuzzy left h-ideals of S are idempotent. Now let l and m be any fuzzy right h-ideal and any fuzzy left h-ideal of S, respectively. Using Theorem 4.3, we have l�hm ¼ l \ m and it follows from Lemma 3.9 that l�hm is a fuzzy h-quasi-ideal of S.

Conversely, assume that the given conditions hold. Let R be any right h-ideal of S. Then by Lemma 3.7, the characteristicfunction vR of R is a fuzzy right h-ideal of S. By the assumption and Lemma 2.9, we know vR ¼ vR�hvR ¼ v

R2 . Then, by Lemma2.9, we have R ¼ R2, that is, R is idempotent. In a similar way we may prove that the left h-ideals of S are idempotent. Now letR be a right h-ideal and L a left h-ideal of S. Then using the assumption and Lemma 2.9 again, vRL ¼ vR�hvL is a fuzzy h-quasi-ideal of S, and it follows from Lemma 3.7 that RL is an h-quasi-ideal of S. Therefore S is h-hemiregular by Lemma 4.13. h

5. h-Intra-hemiregular hemirings

In this section, we introduce the concept of h-intra-hemiregularity of a hemiring, and investigate the characterizations ofh-intra-hemiregular hemirings and hemirings that are both h-hemiregular and h-intra-hemiregular.


We first give the concept of h-intra-hemiregularity of a hemiring as follows.

Definition 5.1. A hemiring S is said to be h-intra-hemiregular if for each x 2 S, there exist ai; a0i; bj; b0j; z 2 S such that

xþPm

i¼1aix2a0i þ z ¼Pn

j¼1bjx2b0j þ z. Equivalent definitions: (1) x 2 Sx2S 8x 2 S, (2) A � SA2S 8A � S.

It is not difficult to observe that in the case of rings the h-intra-hemiregularity coincides with the classical intra-regularityof rings.

Example 5.2. (1) The set N0 of all non-negative integers with usual addition and multiplication is a hemiring, but it isneither h-hemiregular nor h-intra-hemiregular. Indeed, the element 2 2 N0 cannot be expressed as 2þ 2a2þ z ¼ 2a02þ z or2þ

Pmi¼1ai2

2a0i þ z0 ¼Pn

j¼1bj22b0j þ z0 for all a; a0; ai; a0i; bj; b

0j; z; z

0 2 N0:

(2) Let S ¼ f0; a; b; cg be a set with a addition operation (+) and a multiplication operation (�) as follows:

Then S is a hemiring that is both h-hemiregular and h-intra-hemiregular.


(1) S is h-intra-hemiregular.(2) L \ R � LR for every left h-ideal L and every right h-ideal R of S.

Proof. (1)) (2) Assume that (1) holds. Let L and R be any left h-ideal and any right h-ideal of S, respectively. Since S is h-intra-hemiregular, we have

L \ R � SðL \ RÞ2S ¼ ðSðL \ RÞÞððL \ RÞSÞ � ðSLÞðRSÞ � LR:

(2)) (1) Assume that (2) holds. Let a 2 S. Then it is easy to see that SxþMx and xSþ Nx, where M ¼ f0;1;2; . . .g andN ¼ f0;1;2; . . .g, are the principal left h-ideal and principal right h-ideal of S generated by x, respectively. By the assumption,we have

x ¼ x � 0þ 1 � x 2 SxþMx \ xSþ Nx � ðSxþMxÞðxSþ NxÞ ¼ ðSxþMxÞðxSþ NxÞ

¼ ðSxÞðxSÞ þ ðSxÞðNxÞ þ ðMxÞðxSÞ þ ðMxÞðNxÞ:

Thus we have

xþXm

i¼1

aix2a0i þ z ¼Xn

j¼1

bjx2b0j þ z

for some ai; a0i; bj; b0j; z 2 S: This implies that S is h-intra-hemiregular. h


(1) S is h-intra-hemiregular.(2) l \ m � l�hm for every fuzzy left h-ideal l and every fuzzy right h-ideal m of S.

Proof. (1)) (2) Assume that (1) holds. Let l and m be any fuzzy left h-ideal and any fuzzy right h-ideal of S, respectively.Now let x 2 S. Since S is h-intra-hemiregular, there exist ai; a0i; bj; b

0j; z 2 S such that

xþXm

i¼1

aix2a0i þ z ¼Xn

j¼1

bjx2b0j þ z;

that is,

xþXm

i¼1

ðaixÞðxa0iÞ þ z ¼Xn

j¼1

ðbjxÞðxb0jÞ þ z:

Then we have


i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞ; mðbiÞ; mðb0jÞgÞP minflðaixÞ; lðbjxÞ; mðxa0iÞ; mðxb0jÞgP minflðxÞ; mðxÞg

¼ ðl \ mÞðxÞ:

This implies that l \ m � l�hm.


(2)) (1) Assume that (2) holds. Let L and R be any left h-ideal and any right h-ideal of S, respectively. Then by Lemma 3.7,the characteristic functions vL and vR of L and R are a fuzzy left h-ideal and a fuzzy right h-ideal of S, respectively. Now, by theassumption and Lemma 2.9, we have

vL\R ¼ vL \ vR � vL�hvR ¼ vLR:

Then it follows from Lemma 2.9 that L \ R � LR: Therefore S is h-intra-regular by Lemma 5.3. h


(1) S is h-intra-hemiregular.(2) lðxÞ ¼ lðx2Þ for all fuzzy h-ideal l of S and for all x 2 S.

Proof. (1)) (2) Assume that (1) holds. Let l be any fuzzy h-ideal of S and x any element of S. Since S is h-intra-hemiregular,there exist ai; a0i; bj; b

0j; z 2 S such that

xþXm

i¼1

aix2a0i þ z ¼Xn

j¼1

bjx2b0j þ z;

then we have

lðxÞP min lXm

i¼1

aix2a0i

!; l

Xn

j¼1

bjx2b0j

!( )P minflðaix2a0iÞ; lðbjx2b0jÞgP lðx2ÞP lðxÞ;

this implies that lðxÞ ¼ lðx2Þ.(2)) (1) Assume that (2) holds. Let x be any element of S. Then it is easy to see that Mx2 þ Sx2 þ x2SþSx2S, where

M ¼ f0;1;2; . . .g, is the principal h-ideal of S generated by x2. Now, by Lemma 3.7, the characteristic function vMx2þSx2þx2SþSx2S

of Mx2 þ Sx2 þ x2Sþ Sx2S is a fuzzy h-ideal of S. Since

x2 ¼ 0 � x2 þ 1 � x2 2 Mx2 þ Sx2 þ x2Sþ Sx2S;

we have

vMx2þSx2þx2SþSx2S

ðxÞ ¼ vMx2þSx2þx2SþSx2S

ðx2Þ ¼ 1;

and so x 2 Mx2 þ Sx2 þ x2Sþ Sx2S. Thus we have

xþXm

i¼1

aix2a0i þ z ¼Xn

j¼1

bjx2b0j þ z

for some ai; a0i; bj; b0j; z 2 S: Therefore S is h-intra-hemiregular. h


(1) S is both h-hemiregular and h-intra-hemiregular.(2) B ¼ B2 for every h-bi-ideal B of S.(3) Q ¼ Q2 for every h-quasi-ideal Q of S.

Proof. (1)) (2) Assume that (1) holds. Let B be any h-bi-ideal of S and x any element of B. Then B2 � B ¼ B: Since S is bothh-hemiregular and h-intra-hemiregular, there exist some elements a1; a2; pi; p0i; qj; q0j; z1 and z2 of S such that

xþ xa1xþ z1 ¼ xa2xþ z1 ð1Þ

and

xþXm

i¼1

pix2p0i þ z2 ¼

Xn

j¼1

qjx2q0j þ z2: ð2Þ

Then by (1) we have

xa1xþ xa1xa1xþ z1a1x ¼ xa2xa1xþ z1a1x ð3Þ
and
xa2xþ xa1xa2xþ z1a2x ¼ xa2xa2xþ z1a2x: ð4Þ

Combing (1), (3) and (4), we have

xþ xa1xþ z1 þ xa1xa1xþ xa1xa2xþ z1a1xþ z1a2x ¼ xa2xþ z1 þ xa1xa1xþ xa1xa2xþ z1a1xþ z1a2x;


and so

xþ xa2xa1xþ xa1xa2xþ z3 ¼ xa1xa1xþ xa2xa2xþ z3; ð5Þ

where z3 ¼ z1 þ z1a1xþ z1a2x. By (2), we have

xa2xa1xþXm

i¼1

ðxa2pixÞðxp0ia1xÞ þ xa2z2a1x ¼Xn

j¼1

ðxa2qjxÞðxq0ja1xÞ þ xa2z2a1x ð6Þ

xa1xa2xþXm

i¼1


j¼1


xa1xa1xþXm

i¼1


j¼1


xa2xa2xþXm

i¼1


j¼1

ðxa2qjxÞðxq0ja2xÞ þ xa2z2a2x: ð9Þ

Combing (5) with (6)–(9), we have

xþXn

j¼1

ðxa2qjxÞðxq0ja1xÞ þXn

j¼1

ðxa1qjxÞðxq0ja2xÞ þXm

i¼1

ðxa1pixÞðxp0ia1xÞ þXm

i¼1

ðxa2pixÞðxp0ia2xÞ þ z

¼Xm

i¼1


i¼1

ðxa1pixÞðxp0ia2xÞ þXn

j¼1


j¼1

ðxa2qjxÞðxq0ja2xÞ þ z:

where z4 ¼ z3 þ xa2z2a1xþ xa1z2a2xþ xa1z2a1xþ xa2z2a2x. Since B is an h-bi-ideal of S, we have

Xn

j¼1


j¼1


i¼1


i¼1

ðxa2pixÞðxp0ia2xÞ 2 B2

and

Xm

i¼1


i¼1


j¼1


j¼1

ðxa2qjxÞðxq0ja2xÞ 2 B2:

Thus we have x 2 B2 and so B � B2. Therefore B ¼ B2.(2)) (3) This is straightforward.(3)) (1) Assume that (3) holds. Let L and R be any left h-ideal and any right h-ideal of S, respectively. Then it is clear that

L \ R is an h-quasi-ideal of S, by the assumption, we have

L \ R ¼ ðL \ RÞðL \ RÞ � RL � R \ L ¼ R \ L and L \ R ¼ ðL \ RÞðL \ RÞ � LR:

Therefore S is both h-hemiregular and h-intra-hemiregular. h


(1) S is both h-hemiregular and h-intra-hemiregular.(2) l ¼ l�hl for every fuzzy h-bi-ideal l of S.(3) l ¼ l�hl for every fuzzy h-quasi-ideal l of S.

Proof. (1)) (2) Assume that (1) holds. Let l be any fuzzy h-bi-ideal of S and x any element of S. Then it is easy to see thatl�hl � l. Since S is both h-hemiregular and h-intra-hemiregular, there exist some elements a1; a2; pi; p

0i; qj; q

0j and z of S such

that

xþXn

j¼1


j¼1


i¼1


i¼1


¼Xm

i¼1


i¼1


j¼1


j¼1


Thus we have

ðl�hlÞðxÞ ¼ supxþPm

i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞ; lðbiÞ; lðb0jÞgÞ

P minflðxa2qjxÞ; lðxq0ja1xÞ; lðxa1qjxÞ; lðxq0ja2xÞ; lðxa1pixÞ; lðxp0ia1xÞ; lðxa2pixÞ; lðxp0ia2xÞgP lðxÞ:


This implies that l � l�hl. Therefore l ¼ l�hl.(2)) (3) This is straightforward by Lemma 3.10.(3)) (1) Assume that (3) holds. Let Q be any h-quasi-ideal of S. Then by Lemma 3.7, the characteristic function vQ of Q is a

fuzzy h-quasi-ideal of S. Now, by the assumption and Lemma 2.9, we have

vQ ¼ vQ�hvQ ¼ vQ2 :

Then it follows from Lemma 2.9 that Q ¼ Q 2. Hence S is both h-hemiregular and h-intra-hemiregular by Lemma 5.6. h


(1) S is both h-hemiregular and h-intra-hemiregular.(2) l \ m � l�hm for all fuzzy h-bi-ideals l and m of S.(3) l \ m � l�hm for every fuzzy h-bi-ideal l and every fuzzy h-quasi-ideal m of S.(4) l \ m � l�hm for every fuzzy h-quasi-ideal l and every fuzzy h-bi-ideal m of S.(5) l \ m � l�hm for all fuzzy h-quasi-ideals l and m of S.

Proof. Assume that (1) holds. Let l and m be any two fuzzy h-bi-ideals of S and x any element of S. Since S is both h-hemi-regular and h-intra-hemiregular, there exist some elements a1; a2; pi; p

0i; qj; q

0j and z of S such that

xþXn

j¼1


j¼1


i¼1


i¼1


¼Xm

i¼1


i¼1


j¼1


j¼1


Thus we have


i¼1aibiþz¼

Pn

j¼1a0

jb0jþz

ðminflðaiÞ; lða0jÞ; mðbiÞ; mðb0jÞgÞ

P minflðxa2qjxÞ; lðxa1qjxÞ; lðxa1pixÞ; lðxa2pixÞ; mðxq0ja1xÞ; mðxq0ja2xÞ; mðxp0ia1xÞ; mðxp0ia2xÞgP minflðxÞ; mðxÞg¼ ðl \ mÞðxÞ;

this implies that l \ m � l�hm. Hence (1) implies (2). It is clear that ð2Þ ) ð3Þ ) ð5Þ and ð2Þ ) ð4Þ ) ð5Þ by Lemma 3.10. Let Qbe any quasi-ideal of S. Then by Lemma 3.7, the characteristic function vQ of Q is a fuzzy quasi-ideal of S. Now, by theassumption and Lemma 2.9, we have

vQ ¼ vQ \ vQ � vQ�hvQ ¼ vQ 2 :

Then it follows from Lemma 2.9 that Q � Q2. Since the converse inclusion always holds, we have Q ¼ Q2 and so (5) implies(1) by Lemma 5.6. This completes the proof. h

6. Conclusions

Since Zadeh proposed the notion of fuzzy sets, his ideas have been applied to various fields. In the paper, we applied theseideas to hemirings. We introduced the concepts of fuzzy h-bi-ideals and fuzzy h-quasi-ideals of a hemiring, and gave some oftheir properties. We provided the notion of h-intra-hemiregularity of a hemiring as a generalization of the notion of intra-reg-ularity of a ring. We also investigated the characterizations of h-hemiregular hemirings, h-intra-hemiregular hemirings andhemirings that are both h-hemiregular and h-intra-hemiregular in terms of fuzzy left, fuzzy right h-ideals, fuzzy h-bi-idealsand fuzzy h-quasi-ideals. Based on these results, we will apply intuitionistic or interval-valued fuzzy sets to other h-ideals ofhemirings.

Acknowledgements

We express our warmest thanks to Professor Witold Pedrycz, Editor-in-Chief, for editing, communicating the paper, andhis useful suggestions. We also express our warmest thanks to the referees for their interest in our work and their value timeto read the manuscript very carefully and their valuable comments for improving the paper. This research was supported byNational Natural Science Foundation of China (60774049) and Major State Basic Research Development Program of China(2002CB312200).


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The characterizations of h-hemiregular hemirings and h-intra-hemiregular hemirings