Characterizations of Hemirings Based on Probability Spaces

Abstract

The notion of falling fuzzy ℎ-ideals of a hemiring is introduced on the basis of the theory of falling shadows and fuzzy sets. Then the relations between fuzzy ℎ-ideals and falling fuzzy ℎ-ideals are described. In particular, by means of falling fuzzy ℎ-ideals, the charac-terizations of ℎ-hemiregular hemirings are investigated based on independent (prefect positive correlation) probability spaces.

1. Introduction

Starting from a unified treatment of uncertainty by combining probability and fuzzy set theory [1], Goodman [2] put forward the equivalence between a fuzzy set and a class of random sets. Falling shadow representation theory was established based on the collection of Wang and Sanchez [3], which is directly related to the concept of probabilistic fuzzy set membership function. The theory shows the selection methods related to the joint degrees distributions. It provides a reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. Utilizing the theory of falling shadows, in particular, Tan et al. [4, 5] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Yuan and Lee [6] considered a fuzzy subgroup (subring, ideal) as the falling shadow of a cloud of the subgroups (subrings, ideals). Jun and Kang [7] proposed a theoretical approach for BCK algebras.

A semiring plays an important role in studying matrices and determinants. Many aspects of the theory of matrices and determinants over semiring have been studied by Beasley and Pullman [8] and Ghosh [9]. The ideals in semiring are useful for many purposes, but they do not coincide with the usual ring ideals if 𝑆 is ring in general. Their use is thus somewhat limited in terms of obtaining analogues of ring theorems for semirings. In fact, many results in ring apparently have no analogues in hemirings using only ideals. LaTorre [10] investigated ℎ-ideals and 𝑘-ideals in hemirings in an effort to obtain analogues of familiar ring theorems. The fuzzy theory in semirings and hemirings has been discussed by many researchers (see [11–16]). The concept of ℎ-hemiregular hemirings has been introduced by Zhan and Dudek [17] to generalize the regularity in hemirings. Further, some characterizations of ℎ-semisimple and ℎ-intra-hemiregular hemirings were investigated by Yin et al. [18, 19]. It is pointed out that some generalized fuzzy ℎ-ideals of hemirings were investigated by Ma et al.; for example, see [18–25].

Recently, some properties of falling fuzzy ideals of hemirings have also been investigated by Yu and Zhan [26]. As a continuation of our previous investigation of falling fuzzy ideals of hemirings, the present paper is organized as follows. In Section 2, we recall the concepts and properties of hemirings, fuzzy sets, and falling shadows. In Section 3, we introduce the concept of falling fuzzy ℎ-ideals and investigated some related properties. Finally, we investigate characterizations of ℎ-hemiregular hemirings based on independent (prefect positive correlation) probability spaces in Section 4.

2. Preliminaries

A semiring is an algebraic system (𝑆,+,⋅) consisting of a nonempty set 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: 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐,(𝑎+𝑏)𝑐=𝑎𝑐+𝑏𝑐 (1) 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,+) are called a hemiring. For the sake of simplicity, we shall write 𝑎𝑏 for 𝑎⋅𝑏(𝑎,𝑏∈𝑆).

A subset 𝐴 in a hemiring 𝑆 is called a left (right) ideal of 𝑆 if 𝐴 is closed under addition and 𝑆𝐴⊆𝐴 (𝐴𝑆⊆𝐴). Further, 𝐴 is called an ideal of 𝑆 if it is both a left ideal and a right ideal of 𝑆. A left ℎ-ideal of hemiring 𝑆 is defined to be a left ideal 𝐴 of 𝑆, such that, for all 𝑥,𝑧∈𝑆, and, for all 𝑎,𝑏∈𝐴,  𝑥+𝑎+𝑧=𝑏+𝑧→𝑥∈𝐴.

The ℎ-closure 𝐴 of 𝐴 in a hemiring 𝑆 is defined as: 𝐴={𝑥∈𝑆∣𝑥+𝑎₁+𝑧=𝑎₂+𝑧forsome𝑎₁,𝑎₂∈𝐴,𝑧∈𝑆}.

Definition 1 (see [25]). A fuzzy set 𝜇 of a hemiring 𝑆 is called a fuzzy left (right) ℎ-ideal if, for all 𝑥,𝑦,𝑧,𝑎,𝑏∈𝑆, we have(𝐹₁)𝜇(𝑥+𝑦)≥min{𝜇(𝑥),𝜇(𝑦)}; (𝐹₂)𝜇(𝑥𝑦)≥𝜇(𝑦)(𝜇(𝑥𝑦)≥𝜇(𝑥)); (𝐹₃)𝑥+𝑎+𝑧=𝑏+𝑧→𝜇(𝑥)≥min{𝜇(𝑎),𝜇(𝑏)}.
Further, 𝜇 is called a fuzzy ℎ-ideal of 𝑆 if it is both a fuzzy left ℎ-ideal and a fuzzy right ℎ-ideal of 𝑆.

Note that if 𝜇 is a fuzzy ℎ-ideal, then 𝜇(0)≥𝜇(𝑥) for all 𝑥∈𝑆.

For any 𝐴⊆𝑆, we denote the characteristic function of 𝐴 by 𝜒_𝐴.

Theorem 2 (see [17]). A fuzzy set 𝜇 of 𝑆 is a fuzzy ℎ-ideal of 𝑆 if and only if the nonempty subset 𝜇_𝑡 is an ℎ-ideal of 𝑆 for all 𝑡∈[0,1].

It is well known that ideals theory plays a fundamental role in the development of hemirings. Throughout this paper, 𝑆 is a hemiring.

We now display the basic theory on falling shadows. We refer the reader to the papers [2–5] for further information regarding falling shadows. Given a universe of discourse 𝑈, let 𝒫(𝒰) denote the power set of  𝑈. For each 𝑢∈𝑈, let 𝑢̇={𝐸∣𝑢∈𝐸,𝐸⊆𝑈}, (2) and, for each 𝐸∈𝒫(𝑈), let 𝐸̇={𝑢̇∣𝑢∈𝐸}. (3)

An ordered pair (𝒫(𝑈),ℬ) is said to be a hypermeasurable structure on 𝑈 if ℬ is a 𝜎-field in 𝒫 and 𝑈̇⊆ℬ. Given a probability space (Ω,𝒜,𝑃) and a hypermeasurable structure (𝒫(𝑈),ℬ) on 𝑈, a random set on 𝑈 is defined to be a mapping 𝜉∶Ω→𝒫(𝑈), which is 𝒜−ℬ measurable, that is, 𝜉⁻¹(𝐶)={𝜔∣𝜔∈Ω,𝜉(𝜔)∈𝐶}∈𝒜,∀𝐶∈ℬ. (4)

Suppose that 𝜉 is a random set on 𝑈. Let 𝐻(𝑢)=𝑃(𝜔∣𝑢∈𝜉(𝜔)),foreach𝑢∈𝑈. (5)

Then 𝐻 is a kind of fuzzy set in 𝑈. We call 𝐻 a falling shadow of the random set 𝜉, and 𝜉 is called a cloud of 𝐻.

For example, (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚), where 𝒜 is a Borel field on [0,1] and 𝑚 is the usual Lebesgue measure. Let 𝐻 be a fuzzy set in 𝑈 and 𝐻_𝑡={𝑢∈𝑈∣𝐻(𝑢)≥𝑡} be a 𝑡-cut of 𝐻. Then 𝜉∶[0,1]→𝒫(𝑈),𝑡↦𝐻_𝑡 (6) is a random set and 𝜉 is a cloud of  𝐻. We shall call 𝜉 defined above the cut cloud of 𝐻 (see [2]).

3. Falling Fuzzy ℎ-Ideals

In this section, we will introduce the notion of falling fuzzy ℎ-ideals of a hemiring. The relations between fuzzy ℎ-ideals and falling fuzzy ℎ-ideals are provided.

Definition 3 (see [26]). Let (Ω,𝒜,𝒫) be a probability space, and let 𝜉∶Ω→𝒫(𝒮) be a random set. If 𝜉(𝜔) is a left (right) ideal of 𝑆 for any 𝜔∈Ω, then the falling shadow of the random set 𝜉, that is, 𝐻(𝑢)=𝑃(𝜔∣𝑢∈𝜉(𝜔)), is called a falling fuzzy left (right) ideal of 𝑆. Further, 𝐻(𝑢) is called a falling fuzzy ideal of 𝑆 if it is both a falling fuzzy left ideal and a falling fuzzy right ideal of 𝑆.

Let (Ω,𝒜,𝑃) be a probability space and 𝐹(𝑆)={𝑓∣𝑓∶Ω→𝑆}, where 𝑆 is a hemiring.

Define two operations ⊕ and ⊙ on 𝐹(𝑆) by (𝜔)(𝜔)(𝜔),𝜔)(𝜔)(𝜔),𝑓⊕𝑔=𝑓+𝑔𝑓⊙𝑔(=𝑓⋅𝑔 (7) for all 𝜔∈Ω,𝑓,𝑔∈𝐹(𝑆).

Let 𝜃∈𝐹(𝑆) be defined by 𝜃(𝜔)=0, for all 𝜔∈Ω. Then we can check that (𝐹(𝑆),⊕,⊙,𝜃) is a hemiring.

For any subset 𝐴 of 𝑆 and 𝑓∈𝐹(𝑆), let 𝐴_𝑓={𝜔∈Ω∣𝑓(𝜔)∈𝐴}, (𝐹(𝑆,(𝑆)(𝜔),𝜉∶Ω→𝒫))𝜔↦𝑓∈𝐹∣𝑓∈𝐴 (8) and then 𝐴_𝑓∈𝒜.

Definition 4. Let (Ω,𝒜,𝒫) be a probability space, and let 𝜉∶Ω→𝒫(𝒮) be a random set. If 𝜉(𝜔) is an ℎ-ideal of 𝑆 for any 𝜔∈Ω, then the falling shadow of the random set 𝜉, that is, 𝐻(𝑢)=𝑃(𝜔∣𝑢∈𝜉(𝜔)), is called a falling fuzzy ℎ-ideal of 𝑆.

Proposition 5. If 𝐴 is an ℎ-ideal of 𝑆, then 𝜉(𝜔)={𝑓∈𝐹(𝑆)∣𝑓(𝜔)∈𝐴} is an ℎ-ideal of 𝐹(𝑆).

Proof. Assume that 𝐴 is an ℎ-ideal of 𝑆 and 𝜔∈Ω. Let 𝑓,𝑔∈𝐹(𝑆) be such that 𝑓,𝑔∈𝜉(𝜔), and then 𝑓(𝜔),𝑔(𝜔)∈𝐴. Since 𝐴 is an ℎ-ideal of 𝑆, then 𝑓(𝜔)+𝑔(𝜔)∈𝐴. Thus, (𝑓⊕𝑔)(𝜔)=𝑓(𝜔)+𝑔(𝜔)∈𝐴, and so 𝑓⊕𝑔∈𝜉(𝜔). Let 𝑓∈𝜉(𝜔) and 𝑡∈𝐹(𝑆), and then 𝑓(𝜔)∈𝐴. Since 𝐴 is an ℎ-ideal of 𝑆, then (𝑡⊙𝑓)(𝜔)=𝑡(𝜔)⋅𝑓(𝜔)∈𝐴, and so 𝑡⊙𝑓∈𝜉(𝜔), that is, 𝐹(𝑆)⊙𝜉(𝜔)∈𝜉(𝜔). Similarly, we can prove 𝜉(𝜔)⊙𝐹(𝑆)∈𝜉(𝜔).
Let 𝑓,𝑔∈𝜉(𝜔) and 𝑡,ℎ∈𝐹(𝑆), and then 𝑓(𝜔),𝑔(𝜔)∈𝐴. Hence (𝑡⊕𝑓⊕ℎ)(𝜔)=𝑡(𝜔)+𝑓(𝜔)+ℎ(𝜔)=𝑔(𝜔)+ℎ(𝜔)=(𝑔⊕ℎ)(𝜔). Since 𝐴 is an ℎ-ideal of 𝑆, we have 𝑡(𝜔)∈𝐴, that is, 𝑡∈𝜉(𝜔). Hence, 𝜉(𝜔) is an ℎ-ideal of 𝐹(𝑆).

From the above proposition, we know that 𝐻 is a falling fuzzy ideal of 𝐹(𝑆), where 𝐻(𝑓)=𝑃(𝜔∣𝑓(𝜔)∈𝐴}. In fact, since 𝜉⁻¹𝑓=(𝜔=(𝜔)𝜔∈Ω∣𝑓∈𝜉)𝜔∈Ω∣𝑓∈𝐴=𝐴_𝑓∈𝒜, (9) we see that 𝜉 is a random set on 𝐹(𝑆). By Proposition 5, we know that 𝐻 is a falling fuzzy ℎ-ideal of 𝑆.

Example 6. (1) Let 𝑆={0,1,2,3} be a set with an addition operation and a multiplication operation as follows: +01230101232112332223⋅3332012301000020111301110111 (10)
Then (𝑆,+,⋅) is a hemiring [23].
Let (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚) and 𝜉∶[0,1]→𝒫(𝑆) be defined by 𝜉(𝑡)={0}[);{}[);{}[);[].,if𝑡∈0,0.30,1,if𝑡∈0.3,0.50,1,2,if𝑡∈0.5,0.9𝑆,if𝑡∈0.9,1 (11)
Then 𝜉(𝑡) is an ℎ-ideal of 𝑆 for all 𝑡∈[0,1]. Hence 𝐻=𝑃(𝑡∣𝑥∈𝜉(𝑡)) is a falling fuzzy ℎ-ideal of 𝑆.
(2) The set 𝑆={0,1,𝑎,𝑏,𝑐} with the following Cayley tables: +01𝑎𝑏𝑐0101𝑎𝑏𝑐𝑎1𝑏1𝑎1𝑏𝑎1𝑎𝑏𝑎𝑐𝑏𝑎𝑏1𝑏⋅𝑐1𝑎𝑏𝑐01𝑎𝑏𝑐0100000𝑎01𝑎𝑏𝑐𝑏0𝑎𝑎𝑎𝑐𝑐0𝑏𝑎1𝑐0𝑐𝑐𝑐0 (12) Then (𝑆,+,⋅) is a hemiring.
Let (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚) and 𝜉∶[0,1]→𝒫(𝑆) be defined by 𝜉(𝑡)={0}[);{}[);[].,if𝑡∈0,0.30,𝑎,𝑐,if𝑡∈0.3,0.5𝑆,if𝑡∈0.5,1 (13)
Then 𝜉(𝑡) is an ℎ-ideal of 𝑆 for all 𝑡∈[0,1]. Hence 𝐻=𝑃(𝑡∣𝑥∈𝜉(𝑡)) is a falling fuzzy ℎ-ideal of 𝑆.

Theorem 7. Every fuzzy ℎ-ideal of 𝑆 is a falling fuzzy ℎ-ideal of  𝑆.

Proof. Consider the probability space (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚), where 𝒜 is a Bored field on [0,1], and 𝑚 is the usual Lebesgue measure. Let 𝜇 be a fuzzy ℎ-ideal of 𝑆, and then 𝜇_𝑡 is an ℎ-ideal of 𝑆 for all 𝑡∈[0,1]. Let 𝜉∶[0,1]→𝒫(𝑆) be a random set and 𝜉(𝑡)=𝜇_𝑡 for every 𝑡∈[0,1]. Then 𝜇 is a falling fuzzy ℎ-ideal of 𝑆.

Remark 8. The following example shows that the converse of Theorem 7 is not valid.

Example 9. Let 𝑆={0,𝑎,𝑏,𝑐} be a set with an addition operation and a multiplication operation as follows: +0𝑎𝑏𝑐0𝑎0𝑎𝑏𝑐𝑏𝑎0𝑐𝑏𝑐𝑏𝑐0𝑎⋅𝑐𝑏𝑎00𝑎𝑏𝑐0𝑎0000𝑏0000𝑐00𝑐𝑐00𝑐𝑐 (14) Then (𝑆,+,⋅) is a hemiring.

Let (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚) and 𝜉∶[0,1]→𝒫(𝑆) be defined by 𝜉(𝑡)={0}[);{}[);{}[);[].,if𝑡∈0,0.30,𝑎,if𝑡∈0.3,0.50,𝑐,if𝑡∈0.5,0.9𝑆,if𝑡∈0.9,1 (15) Then 𝜉(𝑡) is an ℎ-ideal of 𝑆 for all 𝑡∈[0,1]. Hence 𝐻=𝑃(𝑡∣𝑥∈𝜉(𝑡)) is a falling fuzzy ℎ-ideal of 𝑆, and it is represented as follows: 𝐻(𝑥)=1,if𝑥=0;0.3,if𝑥=𝑎;0.5,if𝑥=𝑐;0.1,if𝑥=𝑏. (16) Then 𝐻_𝑡={0}(];{}(];{}(];[].,if𝑡∈0.5,10,𝑐,if𝑡∈0.3,0.50,𝑎,𝑐,if𝑡∈0.1,0.3𝑆,if𝑡∈0,0.1 (17) If 𝑡∈(0.1,0.3], then 𝐻_𝑡={0,𝑎,𝑐} is not an ℎ-ideal of 𝑆 since 𝑎+𝑐=𝑏⊆{̸0,𝑎,𝑐}. Thus, it follows from Theorem 2 that 𝐻 is not a fuzzy ℎ-ideal of 𝑆.

Let (Ω,𝒜,𝑃) be a probability space and a falling shadow of a random set 𝜉∶Ω→𝒫(𝑆). For any 𝑥∈𝑆, let Ω(𝑥;𝜉)={𝜔∈Ω∣𝑥∈𝜉(𝜔)}. Then Ω(𝑥;𝜉)∈𝒜.

Theorem 10. If a falling shadow 𝐻 of a random set 𝜉∶Ω→𝒫(𝑆) is a falling fuzzy left (right) ℎ-ideal of 𝑆, then, for all 𝑥,𝑦,𝑎,𝑏,𝑧∈𝑆, we have(1)Ω(𝑥;𝜉)∩Ω(𝑦;𝜉)⊆Ω(𝑥+𝑦;𝜉); (2)Ω(𝑦;𝜉)⊆Ω(𝑥𝑦;𝜉)(Ω(𝑥;𝜉)⊆Ω(𝑥𝑦;𝜉));(3)𝑥+𝑎+𝑧=𝑏+𝑧⇒Ω(𝑎;𝜉)∩Ω(𝑏;𝜉)⊆Ω(𝑥;𝜉).

Proof. (1) Let 𝜔∈Ω(𝑥;𝜉)∩Ω(𝑦;𝜉), then 𝑥,𝑦∈𝜉(𝜔). Since 𝜉(𝜔) is a left (right) ℎ-ideal of 𝑆 by Definition 4, then 𝑥+𝑦∈𝜉(𝜔), and so 𝜔∈Ω(𝑥+𝑦;𝜉). This implies that Ω(𝑥;𝜉)∩Ω(𝑦;𝜉)⊆Ω(𝑥+𝑦;𝜉).
(2) Let 𝜔∈Ω(𝑦;𝜉), and then 𝑦∈𝜉(𝜔). Since 𝜉(𝜔) is a left ℎ-ideal of 𝑆 by Definition 4, then 𝑥𝑦∈𝜉(𝜔), and so 𝜔∈Ω(𝑥𝑦;𝜉). This implies that Ω(𝑦;𝜉)⊆Ω(𝑥𝑦;𝜉). Similarly, we can show that Ω(𝑥;𝜉)⊆Ω(𝑥𝑦;𝜉).
(3) Let 𝜔∈Ω(𝑎;𝜉)∩Ω(𝑏;𝜉) and 𝑥,𝑧∈𝑆, and then 𝑎,𝑏∈𝜉(𝜔). Since 𝜉(𝜔) is a left (right) ℎ-ideal of 𝑆 by Definition 4, then 𝑥∈𝜉(𝜔), and so 𝜔∈Ω(𝑥;𝜉). This implies that Ω(𝑎;𝜉)∩Ω(𝑏;𝜉)⊆Ω(𝑥;𝜉). This completes the proof.

Theorem 11. Let 𝐻 be a falling fuzzy ℎ-ideal of 𝑆, and then 𝐻(𝑥𝑦)≥max{𝐻(𝑥),𝐻(𝑦)}.

Proof. Since Ω(𝑥𝑦;𝜉)⊇Ω(𝑥;𝜉)∩Ω(𝑦;𝜉), it follows that 𝐻(𝜔)((𝜔∩(𝜔)((𝜔(𝜔)((𝜔∩(𝜔)𝑃((𝜔(𝜔)𝐻(𝑥),𝐻𝑦.𝑥𝑦=𝑃𝜔∣𝑥𝑦∈𝜉≥𝑃𝜔∣𝑥∈𝜉))𝜔∣𝑦∈𝜉≥𝑃𝜔∣𝑥∈𝜉))+𝑃𝜔∣𝑦∈𝜉−𝑃𝜔∣𝑥∈𝜉))𝜔∣𝑦∈𝜉≥max𝜔∣𝑥∈𝜉)),𝑃𝜔∣𝑦∈𝜉=max (18)

4. Characterizations of ℎ-Hemiregular Hemirings

The concept of ℎ-hemiregularity of a hemiring was first introduced by Zhan and Dudek [17] as a generalization of the concept of regularity of a ring.

Definition 12 (see [25]). A hemiring 𝑆 is said to be ℎ-hemiregular if, for each 𝑎∈𝑆, there exist 𝑥₁,𝑥₂,𝑧∈𝑆 such that 𝑎+𝑎𝑥₁𝑎+𝑧=𝑎𝑥₂𝑎+𝑧.

Lemma 13 (see [17]). If 𝐴 and 𝐵 are, respectively, a right ℎ-ideal and a left ℎ-ideal of a hemiring 𝑆, then 𝐴𝐵⊆𝐴∩𝐵.

Lemma 14 (see [17]). A hemiring 𝑆 is hemiregular if and only if, for any right ℎ-ideal 𝐴 and for any left ℎ-ideal 𝐵, 𝐴𝐵=𝐴∩𝐵.

In the following sections, we divide the results into two parts. In Sections 4.1 and 4.2, we describe the characterizations of ℎ-hemiregular hemirings based on prefect positive correlation and independent probability spaces via falling fuzzy ℎ-ideals, respectively.

4.1. Prefect Positive Correlation Probability Spaces

In this subsection, we describe the characterizations of ℎ-hemiregular hemirings based on prefect positive correlation probability spaces via falling fuzzy ℎ-ideals.

Definition 15. The probability space Ω is called prefect positive correlation if Ω(𝑥;𝜉)⊆Ω(𝑦;𝜉) or Ω(𝑦;𝜉)⊆Ω(𝑥;𝜉) for all 𝑥,𝑦∈𝑆.

Definition 16. Let Ω be a prefect positive correlation probability space and let 𝐻₁ and 𝐻₂ be falling fuzzy ℎ-ideals of 𝑆. Then the 𝑃 product of 𝐻₁ and 𝐻₂ is defined by 𝐻₁⊙_𝑝𝐻₂(𝑥)=sup_{𝑥+∑𝑚𝑖=1𝑎𝑖𝑏𝑖+𝑧=∑𝑛𝑗=1𝑎′𝑗𝑏′𝑗+𝑧}min𝐻₁𝑎_𝑖,𝐻₁𝑎^′_𝑗,𝐻₂𝑏_𝑖,𝐻₂𝑏^′_𝑗∣𝑖=1,…,𝑚,𝑗=1,…𝑛 (19) and (𝐻₁⊙_𝑝𝐻₂)(𝑥)=0 if 𝑥 cannot be expressed as 𝑥+∑^𝑚_𝑖=1𝑎_𝑖𝑏_𝑖+𝑧=∑^𝑛_𝑗=1𝑎^′_𝑗𝑏^′_𝑗+𝑧.

Theorem 17. If Ω is a prefect positive correlation probability space and 𝐻 is a falling fuzzy left (right) ℎ-ideal of 𝑆 for all 𝑥,𝑦,𝑎,𝑏,𝑧∈𝑆, then(1)𝐻(𝑥+𝑦)≥min{𝐻(𝑥),𝐻(𝑦)}; (2)𝐻(𝑥𝑦)≥𝐻(𝑦)(𝐻(𝑥𝑦)≥𝐻(𝑥)); (3)𝑥+𝑎+𝑧=𝑏+𝑧⇒𝐻(𝑥)≥min{𝐻(𝑎),𝐻(𝑏)}.

Proof. (1) By Definition 4, 𝜉(𝜔) is a left ℎ-ideal of 𝑆 for any 𝜔∈Ω. Hence by Theorem 10, Ω(𝑥;𝜉)∩Ω(𝑦;𝜉)⊆Ω(𝑥+𝑦;𝜉).
Thus, we have 𝐻(𝑥+𝑦)=𝑃(𝜔∣𝑥+𝑦∈𝜉(𝜔))≥𝑃({𝜔∣𝑥∈𝜉(𝜔)}∩{𝜔∣𝑦∈𝜉(𝜔)}).
If {𝜔∣𝑥∈𝜉(𝜔)}⊇{𝜔∣𝑦∈𝜉(𝜔)}, then 𝐻(𝑥+𝑦)≥𝐻(𝑥). If {𝜔∣𝑥∈𝜉(𝜔)}⊆{𝜔∣𝑦∈𝜉(𝜔)}, then 𝐻(𝑥+𝑦)≥𝐻(𝑦), and so 𝐻(𝑥+𝑦)≥min{𝐻(𝑥),𝐻(𝑦)}.
(2) By Definition 4, 𝜉(𝜔) is a left ℎ-ideal of 𝑆 for any 𝜔∈Ω. Hence by Theorem 10, Ω(𝑦;𝜉)⊆Ω(𝑥𝑦;𝜉). Thus, we have 𝐻(𝜔)(𝜔)=𝐻𝑦.𝑥𝑦=𝑃𝜔∣𝑥𝑦∈𝜉≥𝑃𝜔∣𝑦∈𝜉 (20)
(3) By Definition 4, 𝜉(𝜔) is a left ℎ-ideal of 𝑆 for any 𝜔∈Ω. Hence by Theorem 10, 𝑥+𝑎+𝑧=𝑏+𝑧⇒Ω(𝑎;𝜉)∩Ω(𝑏;𝜉)⊆Ω(𝑥;𝜉). Thus, we have 𝐻(𝑥)=𝑃(𝜔∣𝑥∈𝜉(𝜔))≥𝑃({𝜔∣𝑎∈𝜉(𝜔)}∩{𝜔∣𝑏∈𝜉(𝜔)}).
If {𝜔∣𝑎∈𝜉(𝜔)}⊇{𝜔∣𝑏∈𝜉(𝜔)}, then 𝐻(𝑥)≥𝐻(𝑎). If {𝜔∣𝑎∈𝜉(𝜔)}⊆{𝜔∣𝑏∈𝜉(𝜔)}, then 𝐻(𝑥)≥𝐻(𝑏), and so 𝐻(𝑥)≥min{𝐻(𝑎),𝐻(𝑏)}.

Proposition 18 (see [5]). If Ω is a prefect positive correlation probability space, then (𝐻₁∩𝐻₂)(𝑥)=min{𝐻₁(𝑥),𝐻₂(𝑥)}.

Proposition 19. If Ω is a prefect positive correlation probability space and 𝐻₁ and 𝐻₂ are two falling fuzzy left (right) ℎ-ideals of 𝑆, then 𝐻₁∩𝐻₂ is a falling fuzzy left (right) ℎ-ideal of 𝑆.

Proof. We only consider the case of left ℎ-ideals, and the proof of right ℎ-ideals is similar:
(i) 𝐻₁∩𝐻₂𝑥+𝑦=min𝐻₁𝑥+𝑦,𝐻₂𝐻𝑥+𝑦≥minmin₁(𝑥),𝐻₁𝑦,min𝐻₂(𝑥),𝐻₂𝑦=minmin𝐻₁(𝑥),𝐻₂(𝑥,𝐻)min₁𝑦,𝐻₂𝑦=min𝐻₁∩𝐻₂(𝑥),𝐻₁∩𝐻₂𝑦, (21)
(ii) 𝐻₁∩𝐻₂𝑥𝑦=min𝐻₁𝑥𝑦,𝐻₂𝐻𝑥𝑦≥min₁𝑦,𝐻₂𝑦=𝐻₁∩𝐻₂𝑦, (22)
(iii) 𝑥+𝑎+𝑧=𝑏+𝑧⟹𝐻₁∩𝐻₂(𝑥)≥minmin𝐻₁(𝑎),𝐻₁(𝑏,𝐻)min₂(𝑎),𝐻₂(𝑏𝐻)=minmin₁(𝑎),𝐻₂(𝑎,𝐻)min₁(𝑏),𝐻₂(𝑏𝐻)=min₁∩𝐻₂(𝑎),𝐻₁∩𝐻₂(𝑏). (23)

Theorem 20. If a falling fuzzy set 𝐻 of 𝑆 is a falling fuzzy left (right) ℎ-ideal of 𝑆, then 𝜒_𝑆⊙_𝑝𝐻⊆𝐻(𝐻⊙_𝑝𝜒_𝑆⊆𝐻).

Proof. We only consider the case of left ℎ-ideals, and the proof of right ℎ-ideals is similar. It is sufficient to show that the condition is satisfied. Let 𝑥∈𝑆. If (𝜒_𝑆⊙_𝑝𝐻)(𝑥)=0, it is clear that (𝜒_𝑆⊙_𝑝𝐻)(𝑥)≤𝐻(𝑥). Otherwise, there exist 𝑎_𝑖,𝑏_𝑖,𝑎^′_𝑗,𝑏^′_𝑗∈𝑆 such that 𝑥+∑^𝑚_𝑖=1𝑎_𝑖𝑏_𝑖+𝑧=∑^𝑛_𝑗=1𝑎^′_𝑗𝑏^′_𝑗+𝑧. Then we have 𝜒_𝑆⊙_𝑝𝐻(𝑥)=sup_{𝑥+∑𝑚𝑖=1𝑎𝑖𝑏𝑖+𝑧=∑𝑛𝑗=1𝑎′𝑗𝑏′𝑗+𝑧}min𝐻𝑏_𝑖,𝐻𝑏^′_𝑗≤sup_{𝑥+∑𝑚𝑖=1𝑎𝑖𝑏𝑖+𝑧=∑𝑛𝑗=1𝑎′𝑗𝑏′𝑗+𝑧}min𝐻𝑎_𝑖𝑏_𝑖,𝐻𝑎^′_𝑗𝑏^′_𝑗≤sup_{𝑥+∑𝑚𝑖=1𝑎𝑖𝑏𝑖+𝑧=∑𝑛𝑗=1𝑎′𝑗𝑏′𝑗+𝑧}min𝐻^𝑚_𝑖=1𝑎_𝑖𝑏_𝑖,𝐻^𝑛_𝑗=1𝑎^′_𝑗𝑏^′_𝑗≤sup_{𝑥+∑𝑚𝑖=1𝑎𝑖𝑏𝑖+𝑧=∑𝑛𝑗=1𝑎′𝑗𝑏′𝑗+𝑧}𝐻(𝑥)=𝐻(𝑥). (24)
This implies that 𝜒_𝑆⊙_𝑝𝐻⊆𝐻.

Theorem 21. Let 𝑆 be a hemiring and 𝐴,𝐵⊆𝑆. Then 𝜒_𝐴⊙_𝑝𝜒_𝐵=𝜒_𝐴𝐵.

Proof. Let 𝑥∈𝑆. If 𝑥∈𝐴𝐵, then 𝜒_𝐴𝐵(𝑥)=1 and 𝑥+∑^𝑚_𝑖=1𝑝_𝑖𝑞_𝑖+𝑧=∑^𝑛_𝑗=1𝑝^′_𝑗𝑞^′_𝑗+𝑧 for some 𝑝_𝑖,𝑝^′_𝑗∈𝐴,𝑞_𝑖,𝑞^′_𝑗∈𝐵 and 𝑧∈𝑆. Thus we have 𝜒_𝐴⊙_𝑝𝜒_𝐵(𝑥)=sup_{𝑥+∑𝑚𝑖=1𝑎𝑖𝑏𝑖+𝑧=∑𝑛𝑗=1𝑎′𝑗𝑏′𝑗+𝑧}min𝜒_𝐴𝑎_𝑖,𝜒_𝐴𝑎^′_𝑗,𝜒_𝐵𝑏_𝑖,𝜒_𝐵𝑏^′_𝑗≥min𝜒_𝐴𝑝_𝑖,𝜒_𝐴𝑝^′_𝑗,𝜒_𝐵𝑞_𝑖,𝜒_𝐵𝑞^′_𝑗=1, (25) and so (𝜒_𝐴⊙_𝑝𝜒_𝐵)(𝑥)=1=𝜒_𝐴𝐵(𝑥).
If 𝑥∉𝐴𝐵, then 𝜒_𝐴𝐵(𝑥). If possible, let (𝜒_𝐴⊙_𝑝𝜒_𝐵)(𝑥)≠0. Then 𝜒_𝐴⊙_𝑝𝜒_𝐵(𝑥)=sup_{𝑥+∑𝑚𝑖=1𝑎𝑖𝑏𝑖+𝑧=∑𝑛𝑗=1𝑎′𝑗𝑏′𝑗+𝑧}min𝜒_𝐴𝑎_𝑖,𝜒_𝐴𝑎^′_𝑗,𝜒_𝐵𝑏_𝑖,𝜒_𝐵𝑏^′_𝑗≠0. (26)
Hence, there exist 𝑝_𝑖,𝑝^′_𝑗,𝑞_𝑖,𝑞^′_𝑗,𝑧∈𝑆 such that 𝑥+∑^𝑚_𝑖=1𝑝_𝑖𝑞_𝑖+𝑧=∑^𝑛_𝑗=1𝑝^′_𝑗𝑞^′_𝑗+𝑧 and min{𝜒_𝐴(𝑝_𝑖),𝜒_𝐴(𝑝^′_𝑗),𝜒_𝐵(𝑞_𝑖),𝜒_𝐵(𝑞^′_𝑗)}≠0, that is, 𝜒_𝐴(𝑝_𝑖)=𝜒_𝐴(𝑝^′_𝑗)=𝜒_𝐵(𝑞_𝑖)=𝜒_𝐵(𝑞^′_𝑗)=1, hence 𝑝_𝑖,𝑝^′_𝑗∈𝐴, 𝑞_𝑖,𝑞^′_𝑗∈𝐵 and 𝑥∈𝐴𝐵, which contradicts with 𝜒_𝐴𝐵(𝑥)=0. Thus we have (𝜒_𝐴⊙_𝑝𝜒_𝐵)(𝑥)=0=𝜒_𝐴𝐵(𝑥).
In any case, we have (𝜒_𝐴⊙_𝑝𝜒_𝐵)(𝑥)=𝜒_𝐴𝐵(𝑥). This completes the proof.

Theorem 22. A hemiring 𝑆 is ℎ-hemiregular if and only if for any falling fuzzy right ℎ-ideal 𝐻₁ and any falling fuzzy left ℎ-ideal 𝐻₂ of 𝑆 we have 𝐻₁⊙_𝑝𝐻₂=𝐻₁∩𝐻₂.

Proof. Let 𝑆 be an ℎ-hemiregular hemiring, 𝐻₁ and 𝐻₂ be any falling fuzzy right ℎ-ideal and any falling fuzzy left ℎ-ideal of 𝑆, respectively. Then by Theorem 21, we have 𝐻₁⊙_𝑝𝐻₂⊆𝐻₁⊙_𝑝𝜒_𝑆⊆𝐻₁ and 𝐻₁⊙_𝑝𝐻₂⊆𝜒_𝑆⊙_𝑝𝐻₂⊆𝐻₂. Thus 𝐻₁⊙_𝑝𝐻₂⊆𝐻₁∩𝐻₂. To show the converse inclusion, let 𝑥 be any element of 𝑆. Since 𝑆 is ℎ-hemiregular, there exist 𝑎,𝑎^′,𝑧∈𝑆 such that 𝑥+𝑥𝑎𝑥+𝑧=𝑥𝑎^′𝑥+𝑧. Then we have 𝐻₁⊙_𝑝𝐻₂(𝑥)=sup_{𝑥+∑𝑚𝑖=1𝑎𝑖𝑏𝑖+𝑧=∑𝑛𝑗=1𝑎′𝑗𝑏′𝑗+𝑧}min𝐻₁𝑎_𝑖,𝐻₁𝑎^′_𝑗,𝐻₂𝑏_𝑖,𝐻₂𝑏^′_𝑗𝐻∣𝑖=1,…,𝑚,𝑗=1,…,𝑛≥min₁(𝑥𝑎),𝐻₁𝑥𝑎^′,𝐻₂(𝑥𝐻)≥min₁(𝑥),𝐻₂(𝑥=𝐻₁)∩𝐻₂(𝑥). (27)
This implies that 𝐻₁⊙_𝑝𝐻₂⊇𝐻₁∩𝐻₂. Therefore, 𝐻₁⊙_𝑝𝐻₂=𝐻₁∩𝐻₂.
Conversely, let 𝜉₁(𝜔) and 𝜉₂(𝜔) be any right ℎ-ideal and any left ℎ-ideal of 𝑆, respectively. Then by Definition 4, 𝐻₁(𝜇) and 𝐻₂(𝜇) are any falling right ℎ-ideal and any falling left fuzzy ℎ-ideal of 𝑆, respectively. The characteristic functions 𝜒_𝜉1(𝜔) and 𝜒_𝜉2(𝜔) are a fuzzy right ℎ-ideal and a fuzzy left ℎ-ideal of 𝑆, respectively. Now, by Theorem 21, we have 𝜒_{𝜉1(𝜔)𝜉2(𝜔)}=𝜒_𝜉1(𝜔)⊙_𝑝𝜒_𝜉2(𝜔)=𝜒_𝜉1(𝜔)∩𝜒_𝜉2(𝜔)=𝜒_{𝜉1(𝜔)∩𝜉2(𝜔)}. (28)
It follows from Theorem 21 that 𝜉₁(𝜔)𝜉₂(𝜔)=𝜉₁(𝜔)∩𝜉₂(𝜔). Thus, we have 𝐻₁𝐻₂𝜇=𝑃𝜔∣𝜇∈𝜉₁(𝜔)𝜉₂(𝜔)=𝑃𝜔∣𝜇∈𝜉₁(𝜔)∩𝜉₂(𝜔)=𝐻₁∩𝐻₂𝜇. (29)
Therefore 𝑆 is ℎ-hemiregular by Lemma 14.

4.2. Independent Probability Spaces

In this subsection, we describe the characterizations of ℎ-hemiregular hemirings based on independent probability spaces via falling fuzzy ℎ-ideals.

Definition 23. The probability space Ω is called independent if Ω(𝑥;𝜉)∩Ω(𝑦;𝜉)=Ω(𝑥;𝜉)Ω(𝑦;𝜉) for all 𝑥,𝑦∈𝑆.

Definition 24. If Ω is an independent probability space and let 𝐻₁ and 𝐻₂ be falling fuzzy ℎ-ideals of 𝑆. Then the 𝐼-product of 𝐻₁ and 𝐻₂ is defined by 𝐻₁⊙_𝑖𝐻₂(𝑥)=𝐻₁𝑎_𝑖𝐻₁𝑎^′_𝑗𝐻₂𝑏_𝑖𝐻₂𝑏^′_𝑗,if𝑥+^𝑚_𝑖=1𝑎_𝑖𝑏_𝑖+𝑧=^𝑛_𝑗=1𝑎^′_𝑗𝑏^′_𝑗+𝑧;0,otherwise. (30)

Theorem 25. If Ω is an independent probability space and 𝐻 is a falling fuzzy left (right) ℎ-ideal of 𝑆 for all 𝑥,𝑦,𝑎,𝑏,𝑧∈𝑆, then(1)𝐻(𝑥+𝑦)≥𝐻(𝑥)𝐻(𝑦); (2)𝐻(𝑥𝑦)≥𝐻(𝑦)(𝐻(𝑥𝑦)≥𝐻(𝑥)); (3)𝑥+𝑎+𝑧=𝑏+𝑧⇒𝐻(𝑥)≥𝐻(𝑎)𝐻(𝑏).

Proof. (1) By Definition 4, 𝜉(𝜔) is a left ℎ-ideal of 𝑆 for any 𝜔∈Ω. Hence by Theorem 10, Ω(𝑥;𝜉)∩Ω(𝑦;𝜉)⊆Ω(𝑥+𝑦;𝜉).
Thus, we have 𝐻(𝜔){(𝜔∩(𝜔((𝜔𝑃(𝜔)=𝐻(𝑥)𝐻𝑦.𝑥+𝑦=𝑃𝜔∣𝑥+𝑦∈𝜉≥𝑃𝜔∣𝑥∈𝜉)}𝜔∣𝑦∈𝜉)=𝑃𝜔∣𝑥∈𝜉))𝜔∣𝑦∈𝜉 (31)
(2) By Definition 4, 𝜉(𝜔) is a left ℎ-ideal of 𝑆 for any 𝜔∈Ω. Hence by Theorem 10, Ω(𝑦;𝜉)⊆Ω(𝑥𝑦;𝜉). Thus, we have 𝐻(𝜔)(𝜔)=𝐻𝑦.𝑥𝑦=𝑃𝜔∣𝑥𝑦∈𝜉≥𝑃𝜔∣𝑦∈𝜉 (32)
(3) By Definition 4, 𝜉(𝜔) is a left ℎ-ideal of 𝑆 for any 𝜔∈Ω. Hence by Theorem 10, 𝑥+𝑎+𝑧=𝑏+𝑧⇒Ω(𝑎;𝜉)∩Ω(𝑏;𝜉)⊆Ω(𝑥;𝜉). Thus, we have 𝐻(𝑥)((𝜔(𝜔∩{(𝜔((𝜔𝑃((𝜔=𝐻(𝑎)𝐻(𝑏).=𝑃𝜔∣𝑥∈𝜉))≥𝑃({𝜔∣𝑎∈𝜉)}𝜔∣𝑏∈𝜉)})=𝑃𝜔∣𝑎∈𝜉))𝜔∣𝑏∈𝜉)) (33)

Proposition 26 (see [5]). If Ω is an independent probability space, then (𝐻₁∩𝐻₂)(𝑥)=𝐻₁(𝑥)𝐻₂(𝑥).

Proposition 27. If Ω is an independent probability space and 𝐻 is a falling fuzzy ℎ-ideal of 𝑆, then 𝐻(𝑥𝑦)≥𝐻(𝑥)+𝐻(𝑦)−𝐻(𝑥)𝐻(𝑦).

Proof. Since Ω(𝑥𝑦;𝜉)⊇Ω(𝑥;𝜉)∩Ω(𝑦;𝜉), it follows that 𝐻(𝜔)((𝜔∩(𝜔)((𝜔(𝜔)((𝜔∩(𝜔)((𝜔(𝜔)((𝜔𝑃(𝜔)=𝐻(𝑥)+𝐻𝑦−𝐻(𝑥)𝐻𝑦.𝑥𝑦=𝑃𝜔∣𝑥𝑦∈𝜉≥𝑃𝜔∣𝑥∈𝜉))𝜔∣𝑦∈𝜉≥𝑃𝜔∣𝑥∈𝜉))+𝑃𝜔∣𝑦∈𝜉−𝑃𝜔∣𝑥∈𝜉))𝜔∣𝑦∈𝜉=𝑃𝜔∣𝑥∈𝜉))+𝑃𝜔∣𝑦∈𝜉−𝑃𝜔∣𝑥∈𝜉))𝜔∣𝑦∈𝜉 (34)

Proposition 28. Let Ω be an independent probability space and 𝐻₁ and 𝐻₂ be two falling fuzzy left (right) ℎ-ideals of 𝑆, and then 𝐻₁∩𝐻₂ is a falling fuzzy left (right) ℎ-ideal of 𝑆.

Proof. We only consider the case of left ℎ-ideals, and the proof of right ℎ-ideals is similar:
(i) 𝐻₁∩𝐻₂𝑥+𝑦=𝐻₁𝑥+𝑦𝐻₂≥𝐻𝑥+𝑦₁(𝑥)𝐻₁𝑦𝐻₂(𝑥)𝐻₂𝑦=𝐻₁∩𝐻₂(𝑥)𝐻₁∩𝐻₂𝑦, (35)
(ii) 𝐻₁∩𝐻₂𝑥𝑦=𝐻₁𝑥𝑦𝐻₂≥𝐻𝑥𝑦₁𝑦𝐻₂𝑦=𝐻₁∩𝐻₂𝑦, (36)
(iii) 𝑥+𝑎+𝑧=𝑏+𝑧⟹𝐻₁∩𝐻₂(𝑥)≥𝐻₁(𝑎)𝐻₁(𝑏)𝐻₂(𝑎)𝐻₂(𝑏)=𝐻₁∩𝐻₂(𝑎)𝐻₁∩𝐻₂(𝑏). (37)

Theorem 29. If a falling fuzzy set 𝐻 of 𝑆 is a falling fuzzy left (right) ℎ-ideal of 𝑆, then 𝜒_𝑆⊙_𝑖𝐻⊆𝐻(𝐻⊙_𝑖𝜒_𝑆⊆𝐻).

Proof. We only consider the case of left ℎ-ideals, and the proof of right ℎ-ideals is similar. It is sufficient to show that the condition is satisfied. Let 𝑥∈𝑆. If (𝜒_𝑆⊙_𝑖𝐻)(𝑥)=0, it is clear that (𝜒_𝑆⊙_𝑖𝐻)(𝑥)≤𝐻(𝑥). Otherwise, there exist 𝑎_𝑖,𝑏_𝑖,𝑎^′_𝑗,𝑏^′_𝑗∈𝑆 such that 𝑥+∑^𝑚_𝑖=1𝑎_𝑖𝑏_𝑖+𝑧=∑^𝑛_𝑗=1𝑎^′_𝑗𝑏^′_𝑗+𝑧. Then we have 𝜒_𝑆⊙_𝑖𝐻(𝑥)=𝐻𝑏_𝑖𝐻𝑏^′_𝑗≤𝐻𝑎_𝑖𝑏_𝑖𝐻𝑎^′_𝑗𝑏^′_𝑗≤𝐻^𝑚_𝑖=1𝑎_𝑖𝑏_𝑖𝐻^𝑛_𝑗=1𝑎^′_𝑗𝑏^′_𝑗≤𝐻(𝑥). (38)
This implies that 𝜒_𝑆⊙_𝑖𝐻⊆𝐻.

Theorem 30. Let 𝑆 be a hemiring and 𝐴,𝐵⊆𝑆, and then 𝜒_𝐴⊙_𝑖𝜒_𝐵=𝜒_𝐴𝐵.

Proof. Let 𝑥∈𝑆. If 𝑥∈𝐴𝐵, then 𝜒_𝐴𝐵(𝑥)=1 and 𝑥+∑^𝑚_𝑖=1𝑝_𝑖𝑞_𝑖+𝑧=∑^𝑛_𝑗=1𝑝^′_𝑗𝑞^′_𝑗+𝑧 for some 𝑝_𝑖,𝑝^′_𝑗∈𝐴, 𝑞_𝑖,𝑞^′_𝑗∈𝐵 and 𝑧∈𝑆. Thus we have 𝜒_𝐴⊙_𝑖𝜒_𝐵(𝑥)=𝜒_𝐴𝑎_𝑖𝜒_𝐴𝑎^′_𝑗𝜒_𝐵𝑏_𝑖𝜒_𝐵𝑏^′_𝑗≥𝜒_𝐴𝑝_𝑖𝜒_𝐴𝑝^′_𝑗𝜒_𝐵𝑞_𝑖𝜒_𝐵𝑞^′_𝑗=1, (39) and so (𝜒_𝐴⊙_𝑖𝜒_𝐵)(𝑥)=1=𝜒_𝐴𝐵(𝑥).
If 𝑥∉𝐴𝐵, then 𝜒_𝐴𝐵(𝑥)=0. If possible, let (𝜒_𝐴⊙_𝑖𝜒_𝐵)(𝑥)≠0. Then 𝜒_𝐴⊙_𝑖𝜒_𝐵(𝑥)=𝜒_𝐴𝑎_𝑖𝜒_𝐴𝑎^′_𝑗𝜒_𝐵𝑏_𝑖𝜒_𝐵𝑏^′_𝑗≠0. (40)
Hence, there exist 𝑝_𝑖,𝑝^′_𝑗,𝑞_𝑖,𝑞^′_𝑗,𝑧∈𝑆 such that 𝑥+∑^𝑚_𝑖=1𝑝_𝑖𝑞_𝑖+𝑧=∑^𝑛_𝑗=1𝑝^′_𝑗𝑞^′_𝑗+𝑧 and 𝜒_𝐴(𝑝_𝑖)𝜒_𝐴(𝑝^′_𝑗)𝜒_𝐵(𝑞_𝑖)𝜒_𝐵(𝑞^′_𝑗)≠0, that is, 𝜒_𝐴(𝑝_𝑖)=𝜒_𝐴(𝑝^′_𝑗)=𝜒_𝐵(𝑞_𝑖)=𝜒_𝐵(𝑞^′_𝑗)=1; hence, 𝑝_𝑖,𝑝^′_𝑗∈𝐴, 𝑞_𝑖,𝑞^′_𝑗∈𝐵 and 𝑥∈𝐴𝐵, which contradicts with 𝜒_𝐴𝐵(𝑥)=0. Thus we have (𝜒_𝐴⊙_𝑖𝜒_𝐵)(𝑥)=0=𝜒_𝐴𝐵(𝑥).
In any case, we have (𝜒_𝐴⊙_𝑖𝜒_𝐵)(𝑥)=𝜒_𝐴𝐵(𝑥). This completes the proof.

Theorem 31. A hemiring 𝑆 is ℎ-hemiregular if and only if for any falling fuzzy right ℎ-ideal 𝐻₁ and any falling fuzzy left ℎ-ideal 𝐻₂ of 𝑆, 𝐻₁⊙_𝑖𝐻₂=𝐻₁∩𝐻₂.

Proof. (1) Let 𝑆 be an ℎ-hemiregular hemiring, 𝐻₁ any falling fuzzy right ℎ-ideal, and 𝐻₂ any falling fuzzy left ℎ-ideal of 𝑆, respectively. Then by Theorem 29, we have 𝐻₁⊙_𝑖𝐻₂⊆𝐻₁⊙_𝑖𝜒_𝑆⊆𝐻₁ and 𝐻₁⊙_𝑖𝐻₂⊆𝜒_𝑆⊙_𝑖𝐻₂⊆𝐻₂. Thus 𝐻₁⊙_𝑖𝐻₂⊆𝐻₁∩𝐻₂. To show the converse inclusion, let 𝑥 be any element of 𝑆. Since 𝑆 is ℎ-hemiregular, there exist 𝑎,𝑎^′,𝑧∈𝑆 such that 𝑥+𝑥𝑎𝑥+𝑧=𝑥𝑎^′𝑥+𝑧. Then we have 𝐻₁⊙_𝑖𝐻₂(𝑥)=𝐻₁𝑎_𝑖𝐻₁𝑎^′_𝑗𝐻₂𝑏_𝑖𝐻₂𝑏^′_𝑗≥𝐻₁(𝑥𝑎𝑥)𝐻₁𝑥𝑎^′𝑥≥𝐻₁(𝑎𝑥)𝐻₂𝑎^′𝑥≥𝐻₁(𝑥)𝐻₂(𝑥)=𝐻₁∩𝐻₂(𝑥). (41)
This implies that 𝐻₁⊙_𝑖𝐻₂⊇𝐻₁∩𝐻₂. Therefore, we have 𝐻₁⊙_𝑖𝐻₂=𝐻₁∩𝐻₂.
Conversely, let 𝜉₁(𝜔) and 𝜉₂(𝜔) be any right ℎ-ideal and any left ℎ-ideal of 𝑆, respectively. Then by Definition 4, 𝐻₁(𝜇) and 𝐻₂(𝜇) are a falling right ℎ-ideal and a falling left fuzzy ℎ-ideal of 𝑆, respectively. The characteristic functions 𝜒_𝜉1(𝜔) and 𝜒_𝜉2(𝜔) of 𝜉₁(𝜔) and 𝜉₂(𝜔) are a fuzzy right ℎ-ideal and a fuzzy left ℎ-ideal of 𝑆, respectively. Now, by Theorem 30, we have 𝜒_{𝜉1(𝜔)𝜉2(𝜔)}=𝜒_𝜉1(𝜔)⊙_𝑖𝜒_𝜉2(𝜔)=𝜒_𝜉1(𝜔)∩𝜒_𝜉2(𝜔)=𝜒_{𝜉1(𝜔)∩𝜉2(𝜔)}. (42)
It follows from Theorem 30 that 𝜉₁(𝜔)𝜉₂(𝜔)=𝜉₁(𝜔)∩𝜉₂(𝜔). So (𝐻₁𝐻₂)(𝜇)=𝑃(𝜔∣𝜇∈𝜉₁(𝜔)𝜉₂(𝜔))=𝑃(𝜔∣𝜇∈𝜉₁(𝜔)∩𝜉₂(𝜔))=(𝐻₁∩𝐻₂)(𝜇).
Therefore, 𝑆 is ℎ-hemiregular by Lemma 14.

5. Conclusions

In this paper, we introduce the notion of falling fuzzy ℎ-ideals of a hemiring. Then we investigate some characteristics of ℎ-hemiregular by means of falling fuzzy ℎ-ideals based on independent (prefect positive correlation) probability spaces. In future work, one can consider ℎ-hemiregular using falling fuzzy ℎ-𝑏𝑖-ideals and falling fuzzy ℎ-quasi-ideals. One also can apply fuzzy inference relations to hemirings. Further, one can investigate this theory to information sciences and intelligent and fuzzy systems.

Acknowledgments

The authors are extremely grateful to the referees and Editor: Professor Qiankun Song for giving them many valuable comments and helpful suggestions which helped them improve the presentation of this paper. 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), and Natural Science Foundation of Education Committee of Hubei Province (D20131903).