Hybrid structures applied to hemirings

Abstract

Using the notions of a soft set and a fuzzy set, Jun’s introduced a new type of notion called a hybrid structure, which is a generalization of soft set and fuzzy set. In this paper, we apply hybrid structure in hemirings and introduce the concept of hybrid h-ideals of hemirings and the related properties are investigated. Some operational properties of hybrid structures are first investigated. The concepts of a hybrid h-sum and a hybrid h-product are discussed and the related properties are investigated. The notions of hybrid h-ideals, hybrid h-bi-ideals and hybrid h-quasi-ideals are introduced and several properties are provided. The notion of a hybrid level set is introduced and the basic properties of hybrid h-ideals, hybrid h-bi-ideals and hybrid h-quasi-ideals are studied in detail. The characterizations of h-hemiregular hemirings are discussed and several important results of a h-hemiregular hemirings are provided.

Keywords

Hybrid structure hybrid structures operations

1. Introduction

In order to model vagueness and uncertainty, Molodstov [20], introduced soft set theory and it has received much attention since its inception. Soft set theory emphasizes a balanced coverage of both theory and practice. Nowadays, it has promoted a breadth of the discipline of Informations sciences with intelligent systems, approximate reasoning, expert and decision support systems, self-adaptation and self-organizational systems, information and knowledge, modeling and computing with words. Soft set theory has been regarded as a new mathematical tool for dealing with uncertainties and it has seen a wide-ranging applications in the mean of algebraic structures such as groups [1], semirings [6], hemirings [15, 17], and so on e.g., (see [2 –5]).

Ideals of hemirings and semirings play a central role in the structure theory and are useful for many purposes. However, in general, they do not coincide with the usual ring ideals. Many results in rings apparently have no analogues in hemirings using only ideals. In (1985) Henriksen [7], defined a more restricted class of ideals in semirings, called k-ideals, with the property that if the semiring R is the ring, then a complex in R is a k-ideal if and only if it is a ring ideal. Another more restricted, but very important, class of ideals in hemirings, called now h-ideals, has been given and investigated by Iizuka in (1959) (see [8]) and LaTorre in (1965) (see [9]).

Soft sets are a set-valued function, and have been studied in several other fields of mathematics, including uncertainty formalisms. In this paper, we intend to continue the Jun’s work in Jun et al. [12], and Jun, Khan and Anis [13], and introduced the notion of a hybrid structure in hemirings. The concepts of hybrid h-ideals, hybrid h-bi-ideals and hybrid h-quasi-ideals are introduced and some basic properties of these notions are provided. The notions of hybrid h-sum and hybrid h-product are introduced and the related properties are studied. The characterizations of h-hemiregular hemirings are investigated.

The rest of the paper is organized as follows. In Section 2, we review some fundamental concepts and preliminaries of hemirings previously used in algebraic structures. In Section 3, the basic operations of soft sets are given and some basic properties are studied. In Section 4, some operational properties of hybrid structure are introduced and some basic results are obtained. In Section 5, the concept of a hybrid left (right) h-ideal is discussed and some results are studied. In Section 6, the notions of a hybrid h-bi-ideal and h-quasi-ideal are defined and several important characterizations of regular hemirings are discussed.

2. Preliminaries

Using the notions of a fuzzy set and a soft set, Jun et al. in [12], introduced a new notion called a hybrid structure and several properties have been investigated. Applications of hybrid structures in BCK/BCI-algebras and linear spaces have been discussed in detail. In [13], hybrid structures have been applied to semigroups and notions of hybrid subsemigroups and hybrid left (resp., right) ideals in semigroups are introduced, and several properties are investigated.

A semiring is an algebraic system (R, + , ·) consisting of a nonempty set R together with two binary operations on R called addition and multiplication (denoted in the usual manner) such that (R, +) and (R, ·) are semigroups and the following distributive laws $\begin{matrix} a \cdot (b + c) = a \cdot b + a \cdot c and \\ (a + b) \cdot c = a \cdot c + b \cdot c \end{matrix}$

are satisfied for all a, b, c ∈ R.

By zero of a semiring (R, + , ·) we mean an element 0 ∈ R such that 0 · x = x · 0 =0 for all x ∈ R. A semiring with zero and a commutative semigroup (R, +) is called a hemiring. For simplicity we will use ab instead of a · b (a, b ∈ R).

A subset A in a hemiring R is called a left (resp. right) ideal of R if A is closed under addition and R A ⊆ A (resp. AR ⊆ A ). A subset A of R is called an ideal if it is both a left and a right ideal of R . A subset A in a hemiring R is called a bi-ideal if A is closed under addition and multiplication satisfying ARA ⊆ A.

A left ideal A of R is called a left h-ideal of R if x, z ∈ R, a, b ∈ A and x + a + z = b + z implies x ∈ A. Right h-ideals, h-ideals and h-bi-ideals are defined similarly.

The h-closure $\bar{A}$ of A in a hemiring R is defined as $\begin{matrix} A = {x \in R | x + a + z = a + z for some \\ a \in A, z \in R} . \end{matrix}$

A subset A in a hemiring R is called a quasi-ideal of R if A is closed under addition and RA ∩ AR ⊆ A. A quasi-ideal A of R is called an h-quasi-ideal of R if $\bar{RA} \cap \bar{AR} \subseteq A$ and x + a + z = b + z implies x ∈ A for all x, z ∈ R and a, b ∈ A. If A, B and C are subsets of a hemiring R, then $A \subseteq \bar{A}$ , $\bar{AB} = \bar{\bar{A} \bar{B}}$ , $\bar{AB} \subseteq A \cap B,$ and $\bar{ABC} = \bar{\bar{A} \bar{B} \bar{C}}$ . A subset A in a hemiring R is called an h-idempotent if $A = \bar{A^{2}}$ .

3. Soft sets

In what follows, we take E = R as the set of parameters, which is a hemiring, unless otherwise specified.

From now on, U is an initial universe set, E is a set of parameters, $P (U)$ is the power set of U and A, B, C . . . , ⊆ E.

Definition 3.1. (see [20]). A soft set f_A over U is defined as $f_{A} : E ⟶ P (U) such that f_{A} (x) = \emptyset if x \notin A .$ Hence f_A is also called an approximation function.

A soft set f_A over U can be represented by the set of ordered pairs $f_{A} = {(x, f_{A} (x)) | x \in E, f_{A} (x) \in P (U)} .$

It is clear that a soft set is a parameterized family of subsets of U. Note that the set of all soft sets over U will be denoted by S (U).

Definition 3.2. (see [20]). Let f_A, f_B ∈ S (U) . Then f_A is called a soft subset of f_B, denoted by $f_{A} \tilde{\subseteq} f_{B}$ if f_A (x) ⊆ f_B (x) for all x ∈ E.

Definition 3.3. (see [20]). Two soft sets f_A and f_B are said to be equal soft sets if $f_{A} \tilde{\subseteq} f_{B}$ and $f_{B} \tilde{\subseteq} f_{A}$ and is denoted by $f_{A} \tilde{=} f_{B} .$

Definition 3.4. (see [20]). Let f_A, f_B ∈ S (U) . Then the soft union of f_A and f_B, denoted by $f_{A} \tilde{\cup} f_{B} = f_{A \cup B},$ is defined by $(f_{A} \tilde{\cup} f_{B}) (x) = f_{A} (x) \cup f_{B} (x)$ for all x ∈ E.

Definition 3.5. (see [20]). Let f_A, f_B ∈ S (U) . Then the soft intersection of f_A and f_B, denoted by $f_{A} \tilde{\cap} f_{B} = f_{A \cap B},$ is defined by $(f_{A} \tilde{\cap} f_{B}) (x) = f_{A} (x) \cap f_{B} (x)$ for all x ∈ E.

4. Basic operations of hybrid structures

In what follows, let I be the unit interval, X a set of parameters and $P (U)$ denote the power set of an initial universe set U.

Definition 4.1. (cf. [12]). A hybrid structure in X over U is defined to be a mapping $\begin{matrix} {\tilde{f}}_{λ} : = (\tilde{f}, λ) : X \to (U) \times I x \mapsto \\ (\tilde{f} (x), λ (x)) \end{matrix}$

where $\tilde{f} : X ⟶ P (U)$ and λ : X ⟶ I are mappings.

For two hybrid structures ${\tilde{f}}_{λ}$ and ${\tilde{g}}_{γ}$ in X over U, we define ${\tilde{f}}_{λ} ⪡ {\tilde{g}}_{γ} \Leftrightarrow \tilde{f} \tilde{\subseteq} \tilde{g} and λ \geq γ,$ where $\tilde{f} \tilde{\subseteq} \tilde{g}$ means that $\tilde{f} (x) \subseteq \tilde{g} (x)$ and λ ≥ γ means that λ (x) ≥ γ (x) for all x ∈ X.

We also define the intersection and union of two hybrid structures in a hemiring R over U as follows: let ${\tilde{f}}_{λ}$ and ${\tilde{g}}_{γ}$ be two hybrid structures in a hemiring R over U. The intersection of ${\tilde{f}}_{λ}$ and ${\tilde{g}}_{γ},$ denoted by ${\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ},$ is a hybrid structure ${\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} = {〈 x, (\tilde{f} \tilde{\cap} \tilde{g}) (x), \lor (λ (x), γ (x)) 〉 : x \in R}$

where $(\tilde{f} \tilde{\cap} \tilde{g}) (x) = \tilde{f} (x) \cap \tilde{g} (x)$ and ⋁ (λ (x) , γ (x)) = (λ (x) ∨ γ (x)) for all x ∈ R .

The union of ${\tilde{f}}_{λ}$ and ${\tilde{g}}_{γ},$ denoted by ${\tilde{f}}_{λ} ⋓ {\tilde{g}}_{γ},$ is a hybrid structure ${\tilde{f}}_{λ} ⋓ {\tilde{g}}_{γ} = {〈 x, (\tilde{f} \cup \tilde{g}) (x), \land (λ (x), γ (x)) 〉 : x \in R}$

where $(\tilde{f} \tilde{\cup} \tilde{g}) (x) = \tilde{f} (x) \cup \tilde{g} (x)$ and ⋀ (λ (x) , γ (x)) = (λ (x) ∧ γ (x)) for all x ∈ R .

For a nonempty subset A of X and $ɛ, δ \in P (U)$ with $ε ⊉ δ$ and s, t ∈ [0, 1] with t < s, consider a hybrid structure

$χ_{A (s, t)}^{(ɛ, δ)} ({\tilde{f}}_{λ}) : = (χ_{A ({\tilde{f}}_{λ})}^{(ɛ, δ)}, χ_{A (λ)}^{(s, t)}),$ where

$χ_{A ({\tilde{f}}_{λ})}^{(ɛ, δ)} : X ⟶ P (U),$ x $⟼ {\begin{matrix} ɛ if x \in A, \\ δ otherwise, \end{matrix}$ and

$χ_{A (λ)}^{(s, t)} : X ⟶ I,$ x $⟼ {\begin{matrix} t if x \in A, \\ s otherwise, \end{matrix}$

which is called the $\frac{(ɛ, δ)}{(s, t)}$ -characteristic hybrid structure in X over U. The hybrid structure $χ_{X (s, t)}^{(ɛ, δ)} ({\tilde{f}}_{λ}) : = (χ_{X ({\tilde{f}}_{λ})}^{(ɛ, δ)}, χ_{X (λ)}^{(s, t)}),$ is called the $\frac{(ɛ, δ)}{(s, t)}$ -identity hybrid structure in X over U. The $\frac{(ɛ, δ)}{(s, t)}$ -characteristic (resp., identity) hybrid structure in X over U with ɛ = U, δ = ∅ , t = 0 and s = 1 is called the characteristic (resp., identity) hybrid structure in X over U, and is denoted by $χ_{A} ({\tilde{f}}_{λ}) : = (χ_{A} (\tilde{f}), χ_{A} (λ))$ $(resp ., χ_{A} ({\tilde{f}}_{λ}) : = (χ_{\tilde{f}}, χ_{λ})) .$

In the following, we introduce the hybrid h-sum and hybrid h-product of two hybrid structures in a hemiring R over U.

Definition 4.2. Let ${\tilde{f}}_{λ}$ and ${\tilde{g}}_{γ}$ be two hybrid structures in a hemiring R over U. Then the hybrid h-sum of ${\tilde{f}}_{λ}$ and ${\tilde{g}}_{γ}$ is a hybrid structure ${\tilde{f}}_{λ} \oplus_{h} {\tilde{g}}_{γ} = {〈 x, (\tilde{f} {\tilde{+}}_{h} \tilde{g}) (x), (λ +_{h} γ) (x) 〉 : x \in R}$ which is briefly denoted by ${\tilde{f}}_{λ} \oplus_{h} {\tilde{g}}_{γ} = 〈 \tilde{f} {\tilde{+}}_{h} \tilde{g}, λ +_{h} γ 〉$

and is defined by $(\tilde{f} {\tilde{+}}_{h} \tilde{g}) (x)$ $= ⋃_{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (b_{1}) \\ \cap \tilde{g} (a_{2}) \cap \tilde{g} (b_{2}) \end{matrix}},$ and (λ + _hγ) (x) $= \underset{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \lor λ (b_{1}) \\ \lor γ (a_{2}) \lor γ (b_{2}) \end{matrix}},$

for all x, z, a₁, b₁, a₂, b₂ ∈ R .

Definition 4.3. Let ${\tilde{f}}_{λ}$ and ${\tilde{g}}_{γ}$ be two hybrid structures in a hemiring R over U. Then the hybrid h-product of ${\tilde{f}}_{λ}$ and ${\tilde{g}}_{γ}$ is a hybrid structure ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} = {〈 x, (\tilde{f} °_{h} \tilde{g}) (x), (λ °_{h} γ) (x) 〉 : x \in R}$ which is briefly denoted by ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ}$ $= 〈 \tilde{f} {\tilde{\circ}}_{h} \tilde{g}, λ \circ_{h} γ 〉$ where $\tilde{f} {\tilde{\circ}}_{h} \tilde{g}$ and λ ∘ _hγ respectively, are defined by $\begin{matrix} (\tilde{f} {\tilde{\circ}}_{h} \tilde{g}) (x) \\ = {\begin{matrix} ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \\ \cap \tilde{f} (b_{1}) \\ \cap \tilde{g} (a_{2}) \cap \tilde{g} (b_{2}) \end{matrix}} \\ if x is expressible as \\ x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z, \\ \emptyset \\ if x is not expressible as \\ x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z, \end{matrix} \end{matrix}$

and (λ ∘ _hγ) (x) $= {\begin{matrix} \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \\ \lor λ (b_{1}) \\ \lor γ (a_{2}) \lor γ (b_{2}) \end{matrix}} \\ if x is expressible as \\ x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z, \\ 1 if x is not expressible as \\ x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z, \end{matrix}$ for all x, z, a₁, b₁, a₂, b₂ ∈ R .

We denote by H (R), the set of all hybrid structures in a hemiring R over U.

Lemma 4.4. Let ${\tilde{f}}_{λ}, {\tilde{g}}_{γ}, {\tilde{h}}_{ζ} \in H (R) .$ Then we have

(1) ${\tilde{f}}_{λ} \oplus_{h} ({\tilde{g}}_{γ} ⋓ {\tilde{h}}_{ζ}) = ({\tilde{f}}_{λ} \oplus_{h} {\tilde{g}}_{γ}) ⋓ ({\tilde{f}}_{λ} \oplus_{h} {\tilde{h}}_{ζ}),$

(2) ${\tilde{f}}_{λ} \oplus_{h} ({\tilde{g}}_{γ} ⋒ {\tilde{h}}_{ζ}) = ({\tilde{f}}_{λ} \oplus_{h} {\tilde{g}}_{γ}) ⋒ ({\tilde{f}}_{λ} \oplus_{h} {\tilde{h}}_{ζ}) .$

Proof. Assume that x is any element in R. Let x be expressed as x + (a₁ + b₁) + z = (a₂ + b₂) + z . Then $\begin{matrix} (\tilde{f} {\tilde{+}}_{h} (\tilde{g} \tilde{\cup} \tilde{h})) (x) \\ = ⋃_{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (b_{1}) \cap \\ (\tilde{g} \tilde{\cup} \tilde{h}) (a_{2}) \\ \cap (\tilde{g} \tilde{\cup} \tilde{h}) (b_{2}) \end{matrix}} \\ = ⋃_{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} [\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (b_{1}) \\ \cap {\tilde{g} (a_{2}) \cup \tilde{h} (a_{2})} \\ \cap {\tilde{g} (b_{2}) \cup \tilde{h} (b_{2})} \end{matrix}] \end{matrix}$ $\begin{matrix} = ⋃_{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} [\begin{matrix} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{g} (a_{2}) \\ \cap \tilde{f} (b_{1}) \cap \tilde{g} (b_{2}) \end{matrix}} \\ \cup {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{h} (a_{2}) \\ \cap \tilde{f} (b_{1}) \cap \tilde{h} (b_{2}) \end{matrix}} \end{matrix}] \\ = ⋃_{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{g} (a_{2}) \\ \cap \tilde{f} (b_{1}) \cap \tilde{g} (b_{2}) \end{matrix}} \\ \cup ⋃_{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{h} (a_{2}) \\ \cap \tilde{f} (b_{1}) \cap \tilde{h} (b_{2}) \end{matrix}} \\ = (\tilde{f} {\tilde{+}}_{h} \tilde{g}) (x) \cup (\tilde{f} {\tilde{+}}_{h} \tilde{h}) (x) \\ = ((\tilde{f} {\tilde{+}}_{h} \tilde{g}) \tilde{\cup} (\tilde{f} {\tilde{+}}_{h} \tilde{h})) (x) \end{matrix}$

and $\begin{matrix} (λ +_{h} (γ \land ξ)) (x) \\ = \underset{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \lor λ (b_{1}) \\ \lor (γ \land ξ) (a_{2}) \\ \lor (γ \land ξ) (b_{2}) \end{matrix}} \\ = \underset{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} [\begin{matrix} (a_{1}) \lor λ (b_{1}) \\ \lor {γ (a_{2}) \land ξ (a_{2})} \\ \lor {γ (b_{2}) \land ξ (b_{2})} \end{matrix}] \\ = \underset{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} [\begin{matrix} {\begin{matrix} λ (a_{1}) \lor γ (a_{2}) \\ \lor λ (b_{1}) \lor γ (b_{2}) \end{matrix}}, \\ {\begin{matrix} λ (a_{1}) \lor ξ (a_{2}) \\ \lor λ (b_{1}) \lor ξ (b_{2}) \end{matrix}} \end{matrix}] \\ = \underset{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} \\ {λ (a_{1}) \lor γ (a_{2}) \lor λ (b_{1}) \lor γ (b_{2})}, \\ \underset{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} \\ {λ (a_{1}) \lor ξ (a_{2}) \lor λ (b_{1}) \lor ξ (b_{2})} \\ = ⋁ (λ +_{h} γ) (x), ⋁ (λ +_{h} ξ) (x) \\ = ⋁ ((λ +_{h} γ), (λ +_{h} ξ)) (x) . \end{matrix}$

Hence (1) holds. (2) Similar to part (1). ■

Lemma 4.5. Let ${\tilde{f}}_{λ}, {\tilde{g}}_{γ}, {\tilde{h}}_{ζ} \in H (R) .$ Then we have

(1) ${\tilde{f}}_{λ} ⋓ ({\tilde{g}}_{γ} ⋒ {\tilde{h}}_{ζ}) = ({\tilde{f}}_{λ} ⋓ {\tilde{g}}_{γ}) ⋒ ({\tilde{f}}_{λ} ⋓ {\tilde{h}}_{ζ}),$

(2) ${\tilde{f}}_{λ} ⋒ ({\tilde{g}}_{γ} ⋓ {\tilde{h}}_{ζ}) = ({\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ}) ⋓ ({\tilde{f}}_{λ} ⋒ {\tilde{h}}_{ζ}),$

(3) ${\tilde{f}}_{λ} \otimes_{h} ({\tilde{g}}_{γ} ⋒ {\tilde{h}}_{ζ}) = ({\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ}) ⋒ ({\tilde{f}}_{λ} \otimes_{h} {\tilde{h}}_{ζ}),$

(4) ${\tilde{f}}_{λ} \otimes_{h} ({\tilde{g}}_{γ} ⋓ {\tilde{h}}_{ζ}) = ({\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ}) ⋓ ({\tilde{f}}_{λ} \otimes_{h} {\tilde{h}}_{ζ}) .$

Proof. The proofs of (1) and (2) are straightforward.

(3) Let x be any element of R. If x is not expressed as x + a₁b₁ + z = a₂b₂ + z, then $(\tilde{f} {\tilde{\circ}}_{h} (\tilde{g} \tilde{\cap} \tilde{h})) (x) = \emptyset = ((\tilde{f} {\tilde{\circ}}_{h} \tilde{g}) \tilde{\cap} (\tilde{f} {\tilde{\circ}}_{h} \tilde{h})) (x)$ and $(λ \circ_{h} (γ \lor ζ)) (x) = 1 = ((λ \circ_{h} γ) \lor (λ \circ_{h} ζ)) (x) .$

Therefore, ${\tilde{f}}_{λ} \otimes_{h} ({\tilde{g}}_{γ} ⋒ {\tilde{h}}_{ζ}) = ({\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ}) ⋒ ({\tilde{f}}_{λ} \otimes_{h} {\tilde{h}}_{ζ}) .$ Assume that x is expressed as x + a₁b₁ + z = a₂b₂ + z, then $\begin{matrix} (\tilde{f} {\tilde{\circ}}_{h} (\tilde{g} \tilde{\cap} \tilde{h})) (x) \\ = ⋃_{x + a_{1} b_{1} + z = a_{2} b_{2} + z} [\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (a_{2}) \\ \cap {(\tilde{g} \tilde{\cap} \tilde{h}) (b_{1})} \\ \cap {(\tilde{g} \tilde{\cap} \tilde{h}) (b_{2})} \end{matrix}] \\ = ⋃_{x + a_{1} b_{1} + z = a_{2} b_{2} + z} [\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (a_{2}) \\ \cap {\tilde{g} (b_{1}) \cap \tilde{h} (b_{1})} \\ \cap {\tilde{g} (b_{2}) \cap \tilde{h} (b_{2})} \end{matrix}] \\ = ⋃_{x + a_{1} b_{1} + z = a_{2} b_{2} + z} [\begin{matrix} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (a_{2}) \\ \cap \tilde{g} (b_{1}) \cap \tilde{g} (b_{2}) \end{matrix}} \\ \cap {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (a_{2}) \\ \cap \tilde{h} (b_{1}) \cap \tilde{h} (b_{2}) \end{matrix}} \end{matrix}] \\ = ⋃_{x + a_{1} b_{1} + z = a_{2} b_{2} + z} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (a_{2}) \\ \cap \tilde{g} (b_{1}) \cap \tilde{g} (b_{2}) \end{matrix}} \cap \\ ⋃_{x + a_{1} b_{1} + z = a_{2} b_{2} + z} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (a_{2}) \\ \cap \tilde{h} (b_{1}) \cap \tilde{h} (b_{2}) \end{matrix}} \\ = (\tilde{f} {\tilde{\circ}}_{h} \tilde{g}) (x) \cap (\tilde{f} {\tilde{\circ}}_{h} \tilde{h}) (x) \\ = ((\tilde{f} {\tilde{\circ}}_{h} \tilde{g}) \tilde{\cap} (\tilde{f} {\tilde{\circ}}_{h} \tilde{h})) (x) \end{matrix}$ and $\begin{matrix} (λ \circ_{h} (γ \lor ζ)) (x) \\ = \underset{x + a_{1} b_{1} + z = a_{2} b_{2} + z}{⋀} [\begin{matrix} λ (a_{1}) \lor λ (a_{2}) \\ \lor {(γ \lor ζ) (b_{1})} \\ \lor {(γ \lor ζ) (b_{2})} \end{matrix}] \\ = \underset{x + a_{1} b_{1} + z = a_{2} b_{2} + z}{⋀} [\begin{matrix} λ (a_{1}) \lor λ (a_{2}) \\ \lor {(γ (b_{1}) \lor ζ (b_{1}))} \\ \lor {(γ (b_{2}) \lor ζ (b_{2}))} \end{matrix}] \\ = \underset{x + a_{1} b_{1} + z = a_{2} b_{2} + z}{⋀} [\begin{matrix} {\begin{matrix} λ (a_{1}) \lor λ (a_{2}) \\ \lor γ (b_{1}) \lor γ (b_{2}) \end{matrix}} \\ \lor {\begin{matrix} λ (a_{1}) \lor λ (a_{2}) \\ \lor ζ (b_{1}) \lor ζ (b_{2}) \end{matrix}} \end{matrix}] \\ = \underset{x + a_{1} b_{1} + z = a_{2} b_{2} + z}{⋀} {\begin{matrix} λ (a_{1}) \lor λ (a_{2}) \\ \lor γ (b_{1}) \lor γ (b_{2}) \end{matrix}} \lor \\ \underset{x + a_{1} b_{1} + z = a_{2} b_{2} + z}{⋀} {\begin{matrix} λ (a_{1}) \lor λ (a_{2}) \\ \lor ζ (b_{1}) \lor ζ (b_{2}) \end{matrix}} \\ = (λ \circ_{h} γ) (x) \lor (λ \circ_{h} ζ) (x) \\ = ((λ \circ_{h} γ) \lor (λ \circ_{h} ζ)) (x) . \end{matrix}$

Hence (3) holds.

(4) Similar to part (3). ■

Proposition 4.6 For any nonempty subsets A and B of a hemiring R, we have

(1) $A \subseteq B \Leftrightarrow χ_{A} ({\tilde{f}}_{λ}) ⪡ χ_{B} ({\tilde{f}}_{λ}),$ i.e., $A \subseteq B \Leftrightarrow χ_{A} (\tilde{f}) \tilde{\subseteq} χ_{B} (\tilde{f})$ and χ_A (λ) ≽ χ_B (λ) ,

(2) $χ_{A} ({\tilde{f}}_{λ}) ⋒ χ_{B} ({\tilde{f}}_{λ}) = χ_{A \cap B} ({\tilde{f}}_{λ}),$ i.e., $χ_{A} (\tilde{f})$ $\tilde{\cap} χ_{B} (\tilde{f}) = χ_{A \cap B} (\tilde{f})$ and χ_A (λ) ∨ χ_B (λ) = χ_A∪B (λ) ,

(3) $χ_{A} ({\tilde{f}}_{λ}) \oplus_{h} χ_{B} ({\tilde{f}}_{λ}) = χ_{\bar{A + B}} ({\tilde{f}}_{λ}),$ i.e., $χ_{A} (\tilde{f})$ ${\tilde{+}}_{h} χ_{B} (\tilde{f}) = χ_{\bar{A + B}} (\tilde{f})$ and $χ_{A} (λ) +_{h} χ_{B} (λ) = χ_{\bar{A + B}} (λ),$

(4) $χ_{A} ({\tilde{f}}_{λ}) \otimes_{h} χ_{B} ({\tilde{f}}_{λ}) = χ_{\bar{AB}} ({\tilde{f}}_{λ}),$ i.e., $χ_{A} (\tilde{f})$ ${\tilde{\circ}}_{h} χ_{B} (\tilde{f}) = χ_{\bar{AB}} (\tilde{f})$ and $χ_{A} (λ) \circ_{h} χ_{B} (λ) = χ_{\bar{AB}} (λ) .$

Proof The proofs of (1) and (2) are straightforward.

(3) Let A and B be subsets of R and x ∈ R. If $x \in \bar{A + B},$ then there exist a₁, a₂ ∈ A and b₁, b₂ ∈ B such that x + (a₁ + a₂) + z = (b₁ + b₂) + z for some z ∈ R. Thus

$\begin{matrix} (χ_{A} (\tilde{f}) {\tilde{+}}_{h} χ_{B} (\tilde{f})) (x) \\ = ⋃_{x + (a_{1} + a_{2}) + z = (b_{1} + b_{2}) + z} {\begin{matrix} χ_{A} (\tilde{f}) (a_{1}) \\ \cap χ_{A} (\tilde{f}) (a_{2}) \\ \cap χ_{B} (\tilde{f}) (b_{1}) \\ \cap χ_{B} (\tilde{f}) (b_{2}) \end{matrix}} \\ = U \\ = χ_{\bar{A + B}} (\tilde{f}) (x) \end{matrix}$ and $\begin{matrix} (χ_{A} (λ) +_{h} χ_{B} (λ)) (x) \\ = \underset{x + (a_{1} + a_{2}) + z = (b_{1} + b_{2}) + z}{⋀} {\begin{matrix} χ_{A} (λ) (a_{1}) \\ \lor χ_{A} (λ) (a_{2}) \\ \lor χ_{B} (λ) (b_{1}) \\ \lor χ_{B} (λ) (b_{2}) \end{matrix}} \\ = 0 = χ_{\bar{A + B}} (λ) (x) . \end{matrix}$

If $x \notin \bar{A + B}$ , then there do not exist a₁, a₂ ∈ A and b₁, b₂ ∈ B such that x + (a₁ + a₂) + z = (b₁ + b₂) + z for some z ∈ R . Thus $(χ_{A} (\tilde{f}) {\tilde{+}}_{h} χ_{B} (\tilde{f})) (x) = \emptyset = χ_{\bar{A + B}} (\tilde{f}) (x)$ and $(χ_{A} (λ) +_{h} χ_{B} (λ)) (x) = 1 = χ_{\bar{A + B}} (λ) (x) .$ Therefore, $χ_{A} ({\tilde{f}}_{λ}) \oplus_{h} χ_{B} ({\tilde{f}}_{λ}) = χ_{\bar{A + B}} ({\tilde{f}}_{λ}) .$

(4) If $x \in \bar{AB}$ , then there exist a₁, a₂ ∈ A and b₁, b₂ ∈ B such that x + (a₁ . a₂) + z = (b₁ . b₂) + z for some z ∈ R . Thus

$\begin{matrix} (χ_{A} (\tilde{f}) {\tilde{\circ}}_{h} χ_{B} (\tilde{f})) (x) \\ = ⋃_{x + (a_{1} a_{2}) + z = (b_{1} b_{2}) + z} {\begin{matrix} χ_{A} (\tilde{f}) (a_{1}) \\ \cap χ_{A} (\tilde{f}) (a_{2}) \\ \cap χ_{B} (\tilde{f}) (b_{1}) \\ \cap χ_{B} (\tilde{f}) (b_{2}) \end{matrix}} \\ = U \\ = χ_{\bar{AB}} (\tilde{f}) (x) \end{matrix}$

and

$\begin{matrix} (χ_{A} (λ) \circ_{h} χ_{B} (λ)) (x) \\ = \underset{x + (a_{1} a_{2}) + z = (b_{1} b_{2}) + z}{⋀} {\begin{matrix} χ_{A} (λ) (a_{1}) \\ \lor χ_{A} (λ) (a_{2}) \\ \lor χ_{B} (λ) (b_{1}) \\ \lor χ_{B} (λ) (b_{2}) \end{matrix}} \\ = 0 \\ = χ_{\bar{AB}} (λ) (x) . \end{matrix}$

If $x \notin \bar{AB}$ , then there do not exist a₁, a₂ ∈ A and b₁, b₂ ∈ B such that x + (a₁ . a₂) + z = (b₁ . b₂) + z for some z ∈ R . Thus $(χ_{A} (\tilde{f}) {\tilde{\circ}}_{h} χ_{B} (\tilde{f})) (x) = \emptyset = χ_{\bar{AB}} (\tilde{f}) (x)$ and $(χ_{A} (λ) \circ_{h} χ_{B} (λ)) (x) = 1 = χ_{\bar{AB}} (λ) (x) .$ Therefore, $χ_{A} ({\tilde{f}}_{λ}) \otimes_{h} χ_{B} ({\tilde{f}}_{λ}) = χ_{\bar{AB}} ({\tilde{f}}_{λ}) .$ ■

5. Hybrid left (right) h-ideals

Definition 5.1. A hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U is called a hybrid left ideal of R over U if for all a, b ∈ R, we have

(H1a) $\tilde{f} (a + b) \supseteq \tilde{f} (a) \cap \tilde{f} (b),$

(H1b) λ (a + b) ⪯ λ (a) ∨ λ (b) ,

(H2a) $\tilde{f} (ab) \tilde{\supseteq} \tilde{f} (b),$

(H2b) λ (ab) ⪯ λ (b) .

A hybrid right ideal of a hemiring R over U is defined analogously.

A hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U is called a hybrid ideal of R over U if it is both a hybrid left and a hybrid right ideal of a hemiring R over U .

Definition 5.2. A hybrid left ideal ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U is called a hybrid left h-ideal of R over U if for all a, b, z ∈ R, we have

(H3a) $x + a + z = b + z ⟶ \tilde{f} (x) \supseteq \tilde{f} (a) \cap \tilde{f} (b),$

(H3b) x + a + z = b + z ⟶ λ (x) ⪯ λ (a) ∨ λ (b) .

A hybrid right h-ideal in a hemiring R over U is defined analogously.

A hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U is called a hybrid h-ideal of R over U if it is both a hybrid left and a hybrid right h-ideal of a hemiring R over U .

Note that a hybrid left h-ideal ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ of a semiring R with zero over U also satisfies the inequalities $\tilde{f} (0) \supseteq \tilde{f} (x), λ (0) \leq λ (x)$ for all x ∈ R.

Example 5.3. Let R = {0, 1, 2, 3} be a set with addition and multiplication operations as follows:

Define a hybrid structure

{\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉

in R over

U = ℤ

as follows given in Table 1:

Table 1
Tabular representation of hybrid structure ${\tilde{f}}_{λ}$

R	$\tilde{f} (x)$	λ (x)
0	$ℤ$	0.9
1	$4 ℕ$	0.5
2	$2 ℤ$	0.3
3	$8 ℕ$	0.1

Then ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid h-ideal of R over U.

Let ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ be a hybrid structure in set X over U, t ∈ [0, 1] and δ ⊆ U . The set $U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t) : = {x \in X | \tilde{f} (x) \supseteq δ, λ (x) \leq t}$

is called the hybrid level set of ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉 .$

Lemma 5.4. For a hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U, the following statements are equivalent:

(1) ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid left h-ideal of R over U.

(2) The nonempty hybrid level set of ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a left h-ideal of R.

Proof. Assume that ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid left h-ideal of R and t ∈ [0, 1] and δ ⊆ U be such that $U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t) \neq \emptyset .$ Let $a, b \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t)$ , then $\tilde{f} (a) \supseteq δ, λ (a) \leq t$ and $\tilde{f} (a) \supseteq δ, λ (a) \leq t$ . Since $\tilde{f} (a + b) \supseteq \tilde{f} (a) \cap \tilde{f} (b)$ and λ (a + b) ≤ λ (a) ∨ λ (b), so we have $\tilde{f} (a + b) \supseteq δ$ and λ (a + b) ≤ t and $a + b \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t) .$ For r ∈ R, we have $\tilde{f} (ra) \supseteq \tilde{f} (a) \supseteq δ$ and λ (ra) ≤ λ (a) ≤ t . It follows that $\tilde{f} (ra) \supseteq δ$ and λ (ra) ≤ t and $ra \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t) .$ Now let x + a + z = b + z for some $a, b \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t)$ and x, z ∈ R. Then $\tilde{f} (a) \supseteq δ, λ (a) \leq t$ and $\tilde{f} (b) \supseteq δ, λ (b) \leq t .$ Since $\tilde{f} (x) \supseteq \tilde{f} (a) \cap \tilde{f} (b)$ and λ (x) ≤ λ (a) ∨ λ (b), so we have $x \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t) .$ Conversely, suppose that $U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t) \neq \emptyset$ for all t ∈ [0, 1] and δ ⊆ U is a left h-ideal of R. If there exist a, b ∈ R such that $\tilde{f} (a + b) \subset \tilde{f} (a) \cap \tilde{f} (b) = δ$ and λ (a + b) > λ (a) ∨ λ (b) = t, for some t ∈ [0, 1] and δ ⊆ U . Then $a, b \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t)$ but $a + b \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t),$ a contradiction. Thus $\tilde{f} (a + b) \supseteq \tilde{f} (a) \cap \tilde{f} (b)$ and λ (a + b) ≤ λ (a) ∨ λ (b), for all a, b ∈ R . For r, a ∈ R, if $\tilde{f} (ra) \subset \tilde{f} (a) = δ$ and λ (ra) > λ (a) = t, then $a \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t)$ but $ra \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t),$ a contradiction. Hence, $\tilde{f} (ra) \supseteq \tilde{f} (a) = δ$ and λ (ra) ≤ λ (a) for all a, r ∈ R . Finally, if there exist a, b, x, z ∈ R, x + a + z = b + z such that $\tilde{f} (x) \subset \tilde{f} (a) \cap \tilde{f} (b) = δ$ and λ (x) > λ (a) ∨ λ (b) = t, then $a, b \in U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t)$ but $x \notin U_{(\tilde{f}, λ)}^{(δ, t)} (\tilde{f}; δ, t),$ a contradiction. Therefore, $\tilde{f} (x) \supseteq \tilde{f} (a) \cap \tilde{f} (b)$ and λ (x) ≤ λ (a) ∨ λ (b) for all x, a, b, z ∈ R. Consequently, ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid left h-ideal of R over U. ■

Similarly, we have the following lemma.

Lemma 5.5 For a hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U, the following statements are equivalent:

(1) ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid right h-ideal of R over U.

(2) The nonempty hybrid level set of ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a right h-ideal of R.

Lemma 5.6 A hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U is a hybrid left h-ideal of R over U if and only if:

(1) $\tilde{f} (x + y) \supseteq \tilde{f} (x) \cap \tilde{f} (y)$ and λ (x + y) ≤ λ (x) ∨ λ (y), for all x, y ∈ R .

(2) $χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ},$

(3) $x + a + z = b + z ⟶ \tilde{f} (x) \supseteq \tilde{f} (a) \cap \tilde{f} (b)$ and λ (x) ≤ λ (a) ∨ λ (b) for all x, a, b, z ∈ R.

Proof. The proof will follow if we show that condition (2) is equivalent to conditions (H2a) and (H2b) of Definition 13. Assume that ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid left h-ideal of R over U . Let x ∈ R. If $(χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ}) (x) = 〈 \emptyset, 1 〉$ , then it is clear that $χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} .$ Otherwise, there exist x, z, a₁, a₂, b₁, b₂ ∈ R such that x + (a₁b₁) + z = (a₂b₂) + z. Since ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid left h-ideal of R over U, we have

$\begin{matrix} (χ_{R} (\tilde{f}) {\tilde{\circ}}_{h} \tilde{f}) (x) \\ = ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\begin{matrix} χ_{R} (\tilde{f}) (a_{1}) \cap χ_{R} (\tilde{f}) (a_{2}) \\ \cap \tilde{f} (b_{1}) \cap \tilde{f} (b_{2}) \end{matrix}} \\ \subseteq ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {U \cap \tilde{f} (a_{1} b_{1}) \cap \tilde{f} (a_{2} b_{2})} \\ = ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\tilde{f} (a_{1} b_{1}) \cap \tilde{f} (a_{2} b_{2})} \\ \subseteq ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} \tilde{f} (x) \\ = \tilde{f} (x) \end{matrix}$

and $\begin{matrix} (χ_{R} (λ) \circ_{h} λ) (x) \\ = \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {\begin{matrix} χ_{R} (λ) (a_{1}) \\ \lor χ_{R} (λ) (a_{2}) \\ \lor λ (b_{1}) \lor λ (b_{2}) \end{matrix}} \\ \geq \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {1 \lor λ (a_{1} b_{1}) \lor λ (a_{2} b_{2})} \\ = \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {λ (a_{1} b_{1}) \lor λ (a_{2} b_{2})} \\ \geq \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} λ (x) = λ (x) . \end{matrix}$

Therefore, $χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} .$

Conversely, suppose that $χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} .$ For any elements x, z of R, let x + (a₁b₁) + z = (a₂b₂) + z.

Then $\begin{matrix} \tilde{f} (xy) \\ \supseteq (χ_{R} (\tilde{f}) {\tilde{\circ}}_{h} \tilde{f}) (xy) \\ = ⋃_{xy + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\begin{matrix} χ_{R} (\tilde{f}) (a_{1}) \\ \cap χ_{R} (\tilde{f}) (a_{2}) \\ \cap \tilde{f} (b_{1}) \cap \tilde{f} (b_{2}) \end{matrix}} \\ \supseteq {U \cap U \cap \tilde{f} (b_{1}) \cap \tilde{f} (b_{2})} \\ = {\tilde{f} (b_{1}) \cap \tilde{f} (b_{2})} \\ = \tilde{f} (b_{1}) \\ (since xy + 0 y + 0 = xy + 0) \end{matrix}$

and $\begin{matrix} λ (xy) \\ \leq (χ_{R} (λ) \circ_{h} λ) (xy) \\ = \underset{xy + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {\begin{matrix} χ_{R} (λ) (a_{1}) \\ \lor χ_{R} (λ) (a_{2}) \\ \lor λ (b_{1}) \lor λ (b_{2}) \end{matrix}} \\ \leq {1 \lor 1 \lor λ (b_{1}) \lor λ (b_{2})} \\ = {λ (b_{1}) \lor λ (b_{2})} \\ = λ (b_{1}) . \end{matrix}$

Hence ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid left h-ideal of R over U . ■

Similarly, we can prove the following:

Lemma 5.7. A hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U is a hybrid right h-ideal of R over U if and only if:

(1) $\tilde{f} (x + y) \supseteq \tilde{f} (x) \cap \tilde{f} (y)$ and λ (x + y) ≤ λ (x) ∨ λ (y), for all a, b ∈ R .

(2) ${\tilde{f}}_{λ} \otimes_{h} χ_{R} ({\tilde{f}}_{λ}) ⪡ {\tilde{f}}_{λ},$

(3) $x + a + z = b + z ⟶ \tilde{f} (x) \supseteq \tilde{f} (a) \cap \tilde{f} (b)$ and λ (x) ≤ λ (a) ∨ λ (b) for all x, a, b, z ∈ R.

6. Hybrid h-bi-ideals and h-quasi-ideals

Definition 6.1 A hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U is called a hybrid h-bi-ideal of R over U if for all x, y, z, a, b ∈ R, we have

(H4a) $\tilde{f} (x + y) \supseteq \tilde{f} (x) \cap \tilde{f} (y),$

(H4b) λ (x + y) ≤ λ (x) ∨ λ (y) ,

(H5a) $\tilde{f} (xy) \tilde{\supseteq} \tilde{f} (x) \cap \tilde{f} (y),$

(H5b) λ (xy) ≤ λ (x) ∨ λ (y) ,

(H6a) $\tilde{f} (xyz) \tilde{\supseteq} \tilde{f} (x) \cap \tilde{f} (z),$

(H6b) λ (xyz) ≤ λ (x) ∨ λ (z) ,

(H7a) $x + a + z = b + z ⟶ \tilde{f} (x) \supseteq \tilde{f} (a) \cap \tilde{f} (b),$

(H7b) x + a + z = b + z ⟶ λ (x) ≤ λ (a) ∨ λ (b) .

Example 6.2 Let $U = 〈 a, b 〉 | a^{2} = b^{2} = e, a b = b a} = {e, a, b, b a},$ Dihedral group, be the universal set. Consider the hemiring $R = ℤ_{4} = {0, 1, 2, 3}$ be the set of non-negative integers module 4, as the set of paramenters.

Define a hybrid structure

{\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉

over U by

R	$\tilde{f}$		λ
0	{e, a, b}	0.8
1	{a}	0.5
2	{e, a}	0.4
3	{a}	0.1

Then, one can easily check that ${\tilde{f}}_{λ}$ is a hybrid h-bi-ideal of R over U.

Theorem 6.3 A hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U is a hybrid h-bi-ideal of R over U if and only if

(1) ${\tilde{f}}_{λ} \oplus_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ},$

(2) ${\tilde{f}}_{λ} \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ},$

(3) ${\tilde{f}}_{λ} \otimes_{h} χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} .$

Proof (1) Assume that ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid h-bi-ideal of R over U . Let x ∈ R, then

$\begin{matrix} (\tilde{f} {\tilde{+}}_{h} \tilde{f}) (x) \\ = ⋃_{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (a_{2}) \\ \cap \tilde{f} (b_{1}) \cap \tilde{f} (b_{2}) \end{matrix}} \\ \subseteq ⋃_{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1} + a_{2}) \\ \cap \tilde{f} (b_{1} + b_{2}) \end{matrix}} (using (H 4 a)) \\ \subseteq ⋃_{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} \tilde{f} (x) \\ = \tilde{f} (x) (using (H 7 a)) \end{matrix}$

and $\begin{matrix} (λ +_{h} λ) (x) \\ = \underset{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \lor λ (a_{2}) \\ \lor λ (b_{1}) \lor λ (b_{2}) \end{matrix}} \\ \geq \underset{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1} + a_{2}) \\ \lor λ (b_{1} + b_{2}) \end{matrix}} (using (H 4 b)) \\ \geq \underset{x + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} λ (x) \\ = λ (x) (using (H 7 b)) \end{matrix}$

Thus ${\tilde{f}}_{λ} \oplus_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} .$

Conversely, assume that condition (1) holds. First we show that $\tilde{f} (0) \supseteq \tilde{f} (x)$ and λ (0) ≤ λ (x) for all x ∈ R. Let x ∈ R, $\begin{matrix} \tilde{f} (0) \\ \supseteq \tilde{f} (x {\tilde{+}}_{h} y) (0) \\ = ⋃_{0 + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (a_{2}) \\ \cap \tilde{f} (b_{1}) \cap \tilde{f} (b_{2}) \end{matrix}} \\ \supseteq {\tilde{f} (x) \cap \tilde{f} (x) \cap \tilde{f} (x) \cap \tilde{f} (x)} \\ = \tilde{f} (x) because 0 + x + x + z = x + x + z \end{matrix}$

and $\begin{matrix} λ (0) \\ \leq λ (x +_{h} y) (0) \\ = \underset{0 + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \lor λ (a_{2}) \\ \lor λ (b_{1}) \lor λ (b_{2}) \end{matrix}} \\ \leq {λ (x) \lor λ (x) \lor λ (x) \lor λ (x)} \\ = λ (x) . \end{matrix}$

Thus $\tilde{f} (0) \supseteq \tilde{f} (x)$ and λ (0) ≤ λ (x) for all x ∈ R.

Now $\begin{matrix} \tilde{f} (x + y) \\ \supseteq \tilde{f} (x {\tilde{+}}_{h} y) (x + y) \end{matrix}$ $\begin{matrix} = ⋃_{x + y + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \\ \cap \tilde{f} (a_{2}) \\ \cap \tilde{f} (b_{1}) \\ \cap \tilde{f} (b_{2}) \end{matrix}} \\ \supseteq {\begin{matrix} \tilde{f} (0) \\ \cap \tilde{f} (0) \\ \cap \tilde{f} (x) \\ \cap \tilde{f} (y) \end{matrix}} \\ because x + y + 0 + 0 + z \\ = x + y + z \\ = \tilde{f} (x) \cap \tilde{f} (y) \end{matrix}$

and $\begin{matrix} λ (x + y) \\ \leq λ (x +_{h} y) (x + y) \\ = \underset{x + y + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \\ \lor λ (a_{2}) \\ \lor λ (b_{1}) \\ \lor λ (b_{2}) \end{matrix}} \\ \leq {\begin{matrix} λ (0) \lor λ (0) \\ \lor λ (x) \lor λ (y) \end{matrix}} \\ = λ (x) \lor λ (y) . \end{matrix}$

Again $\begin{matrix} \tilde{f} (x + y) \\ \supseteq \tilde{f} (x {\tilde{+}}_{h} y) (x + y) \\ = ⋃_{x + y + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \\ \cap \tilde{f} (a_{2}) \\ \cap \tilde{f} (b_{1}) \\ \cap \tilde{f} (b_{2}) \end{matrix}} . \end{matrix}$

and

$\begin{matrix} λ (x + y) \\ \leq λ (x +_{h} y) (x + y) \end{matrix}$ $\begin{matrix} = \underset{x + y + (a_{1} + b_{1}) + z = (a_{2} + b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \\ \lor λ (a_{2}) \\ \lor λ (b_{1}) \\ \lor λ (b_{2}) \end{matrix}} . \end{matrix}$

If x + a + z = b + z then x + a + 0 + z = b + 0 + z and we have

$\tilde{f} (x) \supseteq \tilde{f} (a) \cap \tilde{f} (0) \cap \tilde{f} (b) \cap \tilde{f} (0) = \tilde{f} (a) \cap \tilde{f} (b)$

and

λ (x) ≤ λ (a) ∨ λ (0) ∨ λ (b) ∨ λ (0) = λ (a) ∨ λ (b) .

(2) Let x ∈ R. If $(\tilde{f} {\tilde{\circ}}_{h} \tilde{f}) (x) = \emptyset$ and (λ ∘ _hλ) (x) = 1 . Then $\tilde{f} {\tilde{\circ}}_{h} \tilde{f} \tilde{\subseteq} \tilde{f}$ and λ ∘ _hλ ≥ λ . Thus ${\tilde{f}}_{λ} \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} .$ Let a₁, a₂, b₁, b₂ ∈ R. Then we have $\begin{matrix} (\tilde{f} {\tilde{\circ}}_{h} \tilde{f}) (x) \\ = ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \cap \tilde{f} (a_{2}) \\ \cap \tilde{f} (b_{1}) \cap \tilde{f} (b_{2}) \end{matrix}} \\ \subseteq ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1} b_{1}) \\ \cap \tilde{f} (a_{2} b_{2}) \end{matrix}} (using (H 5 a)) \\ = ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} \tilde{f} (x) \\ = \tilde{f} (x) \end{matrix}$

and $\begin{matrix} (λ \circ_{h} λ) (x) \\ = \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \lor λ (a_{2}) \\ \lor λ (b_{1}) \lor λ (b_{2}) \end{matrix}} \\ \geq \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1} b_{1}) \\ \lor λ (a_{2} b_{2}) \end{matrix}} (using (H 5 b)) \\ = \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} λ (x) \\ = λ (x) . \end{matrix}$

Therefore, ${\tilde{f}}_{λ} \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} .$

Conversely, assume that condition (2) holds. The remaining proof is similar to the converse of part (1).

(3) The proof follows from part (1) and (2).■

Definition 6.4. A hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U is called a hybrid h-quasi-ideal of R over U if for all x, y, z, a, b ∈ R, we have

(H8a) $\tilde{f} (x + y) \supseteq \tilde{f} (x) \cap \tilde{f} (y),$

(H8b) λ (x + y) ≤ λ (x) ∨ λ (y) ,

(H9) ${\tilde{f}}_{λ} \otimes_{h} χ_{R} ({\tilde{f}}_{λ}) ⋒ χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ},$

(H10a) $x + a + z = b + z ⟶ \tilde{f} (x) \supseteq \tilde{f} (a) \cap \tilde{f} (b),$

(H10b) x + a + z = b + z ⟶ λ (x) ≤ λ (a) ∨ λ (b) .

Note that a hybrid h-bi-ideal and hybrid h-quasi-ideal ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U satisfy the inequalities $\tilde{f} (0) \supseteq \tilde{f} (x)$ and λ (0) ≤ λ (x) for all x ∈ R.

Example 6.5. Assume that

U = ℤ^{+}

be the set of positive integers, is the universal set and

R = ℤ_{6} = {0, 1, 2, 3, 4, 5}

non-negative positive integers module 6, is the set of parameters. Define a hybrid structure

{\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉

in R over U by

R	$\tilde{f}$	λ
0	$ℤ^{+}$	0.5
1	{6n\|n∈ $ℤ^{+}$ }	0.8
2	{2n\|n∈ $ℤ^{+}$ }	0.6
3	{3n\|n∈ $ℤ^{+}$ }	0.5
4	{2n\|n∈ $ℤ^{+}$ }	0.6
5	{6n\|n∈ $ℤ^{+}$ }	0.5

Then ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid h-quasi-ideal of R over U.

Lemma 6.6. For a hybrid structure ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ in a hemiring R over U, the following are equivalent:

(1) ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid h-bi-ideal or hybrid h-quasi-ideal of R over U,

(2) The nonempty cubic level set of ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is an h-bi-ideal or h-quasi-ideal of R .

Proof. Proof follows from Lemma 15. ■

Proposition 6.7. Let A be a nonempty subset of a hemiring R. Then the hybrid characteristic mapping $χ_{A} ({\tilde{f}}_{λ}) = 〈 χ_{A} (\tilde{f}), χ_{A} (λ) 〉$ of A is a hybrid h-ideal (resp. hybrid h-bi-ideal and h-quasi-ideal) of R over U if and only if A is an h-ideal (resp. h-bi-ideal and h-quasi-ideal) of R.

Lemma 6.8. Let ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ and ${\tilde{g}}_{γ} = 〈 \tilde{g}, γ 〉$ be a hybrid right h-ideal and hybrid left h-ideal of a hemiring R over U, respectively. Then ${\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} = 〈 \tilde{f} \tilde{\cap} \tilde{g}, λ \lor γ 〉$ is a hybrid h-quasi-ideal of R over U .

Proof. Let x, y be any elements of R. Then $\begin{matrix} (\tilde{f} \tilde{\cap} \tilde{g}) (x + y) \\ = \tilde{f} (x + y) \cap \tilde{g} (x + y) \\ \supseteq {\tilde{f} (x) \cap \tilde{f} (y)} \\ \cap {\tilde{f} (y) \cap \tilde{g} (y)} \\ = {\tilde{f} (x) \cap \tilde{g} (x)} \\ \cap {\tilde{f} (y) \cap \tilde{g} (y)} \\ = (\tilde{f} \tilde{\cap} \tilde{g}) (x) \\ \cap (\tilde{f} \tilde{\cap} \tilde{g}) (y) \end{matrix}$

and $\begin{matrix} (λ \lor γ) (x + y) \\ = λ (x + y) \lor γ (x + y) \\ \leq {λ (x) \lor λ (y)} \\ \lor {γ (y) \lor γ (y)} \\ = {λ (x) \lor γ (x)} \\ \lor {λ (y) \lor γ (y)} \\ = (λ \lor γ) (x) \\ \lor (λ \lor γ) (y) . \end{matrix}$

Let a, b, x, z ∈ R such that x + a + z = b + z. Then $\begin{matrix} (\tilde{f} \tilde{\cap} \tilde{g}) (x) & = \tilde{f} (x) \cap \tilde{g} (x) \\ \supseteq {\tilde{f} (a) \cap \tilde{f} (b)} \cap {\tilde{g} (a) \cap \tilde{g} (b)} \\ = {\tilde{f} (a) \cap \tilde{g} (a)} \cap {\tilde{f} (b) \cap \tilde{g} (b)} \\ = (\tilde{f} \tilde{\cap} \tilde{g}) (a) \cap (\tilde{f} \tilde{\cap} \tilde{g}) (b) \end{matrix}$

and $\begin{matrix} (λ \lor γ) (x) & = λ (x) \lor γ (x) \\ \leq {λ (a) \lor λ (b)} \lor {γ (a) \lor γ (b)} \end{matrix}$ $\begin{matrix} = {λ (a) \lor γ (a)} \lor {λ (b) \lor γ (b)} \\ = (λ \lor γ) (a) \lor (λ \lor γ) (b) . \end{matrix}$

Also, $\begin{matrix} ({\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ}) \otimes_{h} χ_{R} ({\tilde{f}}_{λ}) \\ ⋒ χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} ({\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ}) \\ ⪡ {\tilde{f}}_{λ} \otimes_{h} χ_{R} ({\tilde{f}}_{λ}) \\ ⋒ χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{g}}_{γ} \\ ⪡ {\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} . \end{matrix}$ ■

Lemma 6.9. Any hybrid h-quasi-ideal of a hemiring R over U is a hybrid h-bi-ideal of R over U.

Proof. Let ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ be hybrid h-quasi-ideal of a hemiring R over U . It is sufficient to show that $\tilde{f} (xyz) \supseteq \tilde{f} (x) \cap \tilde{f} (z), λ (xyz) \leq λ (x) \lor λ (z)$

and $\tilde{f} (xy) \supseteq \tilde{f} (x) \cap \tilde{f} (y), λ (xy) \leq λ (x) \lor λ (y) .$

Since ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ is a hybrid h-quasi-ideal of R over U, we have $\begin{matrix} \tilde{f} (xyz) \\ \supseteq (\tilde{f} {\tilde{\circ}}_{h} χ_{R} (\tilde{f})) \\ \tilde{\cap} (χ_{R} (\tilde{f}) {\tilde{\circ}}_{h} \tilde{f}) (xyz) \\ = (\tilde{f} {\tilde{\circ}}_{h} χ_{R} (\tilde{f})) (xyz) \\ \cap (χ_{R} (\tilde{f}) {\tilde{\circ}}_{h} \tilde{f}) (xyz) \\ = ⋃_{xyz + (a_{1} b_{1}) + z^{'} = (a_{2} b_{2}) + z^{'}} {\begin{matrix} \tilde{f} (a_{1}) \\ \cap \tilde{f} (a_{2}) \end{matrix}} \\ \cap ⋃_{xyz + (a_{1} b_{1}) + z^{'} = (a_{2} b_{2}) + z^{'}} {\begin{matrix} \tilde{f} (b_{1}) \\ \cap \tilde{f} (b_{2}) \end{matrix}} \\ \supseteq {\tilde{f} (0) \cap \tilde{f} (x)} \end{matrix}$ $\begin{matrix} \cap {\tilde{f} (0) \cap \tilde{f} (z)} \\ (since xyz + 00 + 0 \\ = x (yz) + 0 \\ and xyz + 00 + 0 \\ = (xy) z + 0) \\ = \tilde{f} (x) \cap \tilde{f} (z) \end{matrix}$

and $\begin{matrix} λ (xyz) \\ \leq (λ \circ_{h} χ_{R} (λ)) \\ \lor (χ_{R} (λ) \circ_{h} λ) (xyz) \\ = (λ \circ_{h} χ_{R} (λ)) (xyz) \\ \lor (χ_{R} (λ) \circ_{h} λ) (xyz) \\ = \underset{xyz + (a_{1} b_{1}) + z^{'} = (a_{2} b_{2}) + z^{'}}{⋀} {\begin{matrix} λ (a_{1}) \\ \lor λ (a_{2}) \end{matrix}} \\ \lor \underset{xyz + (a_{1} b_{1}) + z^{'} = (a_{2} b_{2}) + z^{'}}{⋀} {\begin{matrix} λ (b_{1}) \\ \lor λ (b_{2}) \end{matrix}} \\ \leq {λ (0) \lor λ (x)} \\ \lor {λ (0) \lor λ (z)} \\ = λ (x) \lor λ (z) . \end{matrix}$

Similarly, we can show that $\tilde{f} (xy) \supseteq \tilde{f} (x) \cap \tilde{f} (y), λ (xy) \leq λ (x) \lor λ (y)$ for all x, y ∈ R. ■

The converse of the above lemma is not true in general, as shown in the following example.

Example 6.10. We denote by $ℕ_{0}$ the set of all non-negative integers and let R be the set of all 2 × 2 $(\begin{matrix} a_{11} a_{12} \\ a_{21} a_{22} \end{matrix}) (a_{ij} \in ℕ_{0}) .$ Then R is a hemiring with respect to the usual addition and multiplication of matrices. Let $Q = {(\begin{matrix} a 0 \\ 00 \end{matrix}) | a \in ℕ_{0}}$ be the set of all 2 × 2 matrices is an h-quasi-ideal of R and not a left (right) h-ideal of R, and consequently is not an h-bi-ideal of R. Then by Proposition 25, the characteristic hybrid mapping $χ_{Q} ({\tilde{f}}_{λ}) = 〈 χ_{Q} (\tilde{f}), χ_{Q} (λ) 〉$ of Q is a hybrid h-quasi-ideal, but not a hybrid left (right) h-ideal of R and hence is not a hybrid h-bi-ideal of R.

7. h-Hemiregular hemirings

In literature the concept of h-hemiregularity was given by Yin et al. in (2008) (see [23]) as a generalization of the concept of regular rings. In this section, we try to study this concept and characterize h-hemiregular hemirings by using the notion of hybrid h-ideals. We recall the following definition from Yin and Li [23].

Definition 7.1 (cf. [23]) A hemiring R is said to be h-hemiregular if for each x ∈ R, there exist such that $x + x a x + z = x a' x + z$ .

Lemma 7.2 (cf. [23])) A hemiring R is h-hemiregular if and only if for any right h-ideal A and any left h-ideal B of R, we have $\bar{AB} = A \cap B$ .

Theorem 7.3 A hemiring R is h-hemiregular if and only if for any hybrid right h-ideal ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ and any hybrid left h-ideal ${\tilde{g}}_{γ} = 〈 \tilde{g}, γ 〉$ of R over U, we have ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} = {\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} .$

Proof Assume that R is an h-hemiregular hemiring, ${\tilde{f}}_{λ} = 〈 \tilde{f}, λ 〉$ a hybrid right h-ideal and ${\tilde{g}}_{γ} = 〈 \tilde{g}, γ 〉$ a hybrid left h-ideal of R over U, respectively. Then by Lemma 17 and 18, we have ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} ⪡ {\tilde{f}}_{λ} \otimes_{h} χ_{R} ({\bar{f}}_{λ}) ⪡ {\tilde{f}}_{λ}$ and ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} ⪡ χ_{R} ({\bar{f}}_{λ}) \otimes_{h} {\tilde{g}}_{γ} ⪡ {\tilde{g}}_{γ} .$ It follows that ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} ⪡ {\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} .$ For the converse inclusion, let x be any element of R. Since R is h-hemiregular, there exist $a, a', z \in R$ such that $x + x a x + z = x a' x + z$ . Then we have $\begin{matrix} (\tilde{f} {\tilde{\circ}}_{h} \tilde{g}) (x) \\ = ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b 2) + z} {\begin{matrix} \tilde{f} (a_{1}) \\ \cap \tilde{f} (a_{2}) \\ \cap \tilde{g} (b_{1}) \\ \cap \tilde{g} (b_{2}) \end{matrix}} \\ \supseteq {\begin{matrix} \tilde{f} (xa) \cap \tilde{f} ({xa}^{'}) \\ \cap \tilde{g} (x) \end{matrix}} \\ \supseteq {\tilde{f} (x) \cap \tilde{g} (x)} \\ = (\tilde{f} \tilde{\cap} \tilde{g}) (x) \end{matrix}$

and $\begin{matrix} (λ \circ_{h} γ) (x) \\ = \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \\ \lor λ (a_{2}) \\ \lor γ (b_{1}) \\ \lor γ (b_{2}) \end{matrix}} \\ \leq {\begin{matrix} λ (xa) \lor λ ({xa}^{'}) \\ \lor γ (x) \end{matrix}} \\ \leq {λ (x) \lor γ (x)} \\ = (λ \lor γ) (x) . \end{matrix}$

Thus ${\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} ⪡ {\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ}$ and ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} = {\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} .$

Conversely, suppose that A and B be any right and left h-ideals of R, respectively. Then by Proposition 25, the characteristic functions $χ_{A} ({\tilde{f}}_{λ})$ and $χ_{B} ({\tilde{f}}_{λ})$ are hybrid right and hybrid left h-ideal of R over U, respectively. By hypothesis, we have $χ_{A} ({\tilde{f}}_{λ}) \otimes_{h} χ_{B} ({\tilde{f}}_{λ}) = χ_{A} ({\tilde{f}}_{λ}) ⋒ χ_{B} ({\tilde{f}}_{λ}) .$

By Proposition 11, $χ_{A} ({\tilde{f}}_{λ}) \otimes_{h} χ_{B} ({\tilde{f}}_{λ}) = χ_{\bar{AB}} ({\tilde{f}}_{λ})$ and $χ_{A} ({\tilde{f}}_{λ}) ⋒ χ_{B} ({\tilde{f}}_{λ}) = χ_{A \cap B} ({\tilde{f}}_{λ}) .$ Thus $χ_{\bar{AB}} ({\tilde{f}}_{λ}) = χ_{A \cap B} ({\tilde{f}}_{λ})$ and $\bar{AB} = A \cap B .$ Therefore, R h-hemiregular (Lemma 24). ■

Lemma 7.4. (cf. [23]) Let R be a hemiring. Then the following conditions are equivalent.

(1) R is h-hemiregular,

(2) $B = \bar{BRB}$ for every h-bi-ideal B of R.

(3) $Q = \bar{QRQ}$ for every h-quasi-ideal Q of R.

Theorem 7.5. Let R be a hemiring. Then the following conditions are equivalent.

(1) R is h-hemiregular,

(2) ${\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} \otimes_{h} χ_{A} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ}$ for every hybrid h-bi-ideal ${\tilde{f}}_{λ}$ of R over U.

(3) ${\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} \otimes_{h} χ_{A} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ}$ for every hybrid h-quasi-ideal ${\tilde{f}}_{λ}$ of R over U.

Proof. (1) ⇒ (2) Suppose that R is h-hemiregular and x be any element of R. Let ${\tilde{f}}_{λ}$ be any hybrid h-bi-ideal of R over U . Since R is h-hemiregular, there exist $a, a', z \in R$ such that $x + x a x + z = x a' x + z$ . Then we have $\begin{matrix} (\tilde{f} {\tilde{\circ}}_{h} χ_{A} (\tilde{f}) \circ_{h} \tilde{f}) (x) \\ = ⋃_{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\begin{matrix} (\tilde{f} \circ_{h} χ_{A} (\tilde{f})) (a_{1}) \\ \lor (\tilde{f} \circ_{h} χ_{A} (\tilde{f})) (a_{2}) \\ \lor \tilde{f} (b_{1}) \\ \lor \tilde{f} (b_{1}) \end{matrix}} \\ \supseteq {\begin{matrix} (\tilde{f} {\tilde{\circ}}_{h} χ_{A} (\tilde{f})) (xa) \\ \cap (\tilde{f} {\tilde{\circ}}_{h} χ_{A} (\tilde{f})) ({xa}^{'}) \\ \cap \tilde{f} (x) \end{matrix}} \\ = {\begin{matrix} ⋃_{xa + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\begin{matrix} \tilde{f} (a_{1}) \\ \cap \tilde{f} (a_{2}) \end{matrix}} \\ \cap ⋃_{{xa}^{'} + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\begin{matrix} \tilde{f} (b_{1}) \\ \cap \tilde{f} (b_{2}) \end{matrix}} \\ \cap \tilde{f} (x) \end{matrix}} \\ \supseteq {\begin{matrix} {\tilde{f} (xax) \cap \tilde{f} ({xa}^{'} x)} \\ \cap {\tilde{f} (xax) \cap \tilde{f} ({xa}^{'} x)} \\ \cap \tilde{f} (x) \end{matrix}} \\ (\begin{matrix} since xa + xaxa + za \\ = {xaxa}^{'} + za and \\ x a^{' + x a x a' + z a'} \\ = xa' xa' + za' \end{matrix}) \\ \supseteq {\tilde{f} (x) \cap \tilde{f} (x) \cap \tilde{f} (x)} \\ = \tilde{f} (x) \end{matrix}$

and $\begin{array}{l} (λ °_{h} χ_{A} (λ) °_{h} % λ) (x) \\ \underset{= x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{\land} {\begin{matrix} (λ °_{h} χ_{A} (λ)) (a_{1}) \\ \lor (λ °_{h} χ_{A} (λ)) (a_{2}) \\ \lor λ (b_{1}) \\ \lor λ (b_{1}) \end{matrix}} \\ \leq {\begin{matrix} (λ °_{h} χ_{A} (λ)) (x a) \\ \lor (λ °_{h} χ_{A} (λ)) (x a') \\ \lor λ (x) \end{matrix}} \\ = {\begin{matrix} \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{\land} {\begin{matrix} λ (a_{1}) \\ \lor λ (a_{2}) \end{matrix}} \\ \cap \underset{x a' + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{\land} {\begin{matrix} λ (b_{1}) \\ \lor λ (b_{2}) \end{matrix}} \\ \lor λ (x) \end{matrix}} \\ \leq {\begin{matrix} {λ (x a x) \lor λ (x a' x)} \\ \lor {λ (x a x) \lor λ (x a' x)} \\ \lor λ (x) \end{matrix}} \\ \supseteq {λ (x) \cap λ (x) \cap λ (x)} \\ = λ (x) \end{array}$

(2)⇒(3) Since every hybrid h-quasi-ideal is a hybrid h-bi-ideal of R. The implication is straightforward (Lemma 27).

(3)⇒(1) Assume that (3) holds. Let Q be any h-hquasi-ideal of R over U. Then by Proposition 25, the characteristic function $χ_{Q} ({\tilde{f}}_{λ})$ is a hybrid h-quasi-ideal of R over U. By assumption, we have $χ_{Q} ({\tilde{f}}_{λ}) ⪡ χ_{Q} ({\tilde{f}}_{λ}) \otimes_{h} χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} χ_{Q} ({\tilde{f}}_{λ}) .$

By Proposition 11, we have $χ_{Q} ({\tilde{f}}_{λ}) \otimes_{h} χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} χ_{Q} ({\tilde{f}}_{λ}) = χ_{\bar{QRQ}} ({\tilde{f}}_{λ})$ and so $χ_{Q} ({\tilde{f}}_{λ}) ⪡ χ_{\bar{QRQ}} ({\tilde{f}}_{λ}) .$ Therefore, $Q \subseteq \bar{QRQ}$ by Proposition 11. For the reverse inclusion, since Q is a h-quasi-ideal of R, we have $\bar{QRQ} \subseteq Q$ and $Q = \bar{QRQ}$ . Consequently, R is h-hemiregular. ■

Theorem 7.6. Let R be a hemiring. Then the following conditions are equivalent:

(1) R is h-hemiregular.

(2) ${\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} ⪡ {\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} \otimes_{h} {\tilde{f}}_{λ}$ for every hybrid h-bi-ideal ${\tilde{f}}_{λ}$ and every hybrid h-ideal ${\tilde{g}}_{γ}$ of R over U.

(3) ${\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} ⪡ {\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} \otimes_{h} {\tilde{f}}_{λ}$ for every hybrid h-quasi-ideal ${\tilde{f}}_{λ}$ and every hybrid h-ideal ${\tilde{g}}_{γ}$ of R over U.

Proof. (1)⇒(2) Assume that (1) holds. Let ${\tilde{f}}_{λ}$ be a hybrid h-bi-ideal and ${\tilde{g}}_{γ}$ be a hybrid h-ideal of R over U, respectively. Let x be any element of R, then since R is h-hemiregular, there exist $a, a', z \in R$ such that $x + x a x + z = x a' x + z$ . Then we have

\begin{array}{l} (\tilde{f} {\tilde{°}}_{h} {\tilde{g}}_{h} \tilde{f}) (x) \\ = \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{\cup} {\begin{matrix} ({\tilde{f}}_{\tilde{o}}_{h} \tilde{g}) (a_{1}) \\ \cap ({\tilde{f}}_{\tilde{o}} \tilde{g}) (a_{2}) \\ \cap \tilde{f} (b_{1}) \\ \cap \tilde{f} (b_{1}) \end{matrix}} \\ \supseteq {\begin{matrix} (\tilde{f} {\tilde{°}}_{h} \tilde{g}) (x a) \\ \cap (\tilde{f} {\tilde{°}}_{h} \tilde{g}) (x a') \\ \cap \tilde{f} (x) \end{matrix}} \\ = {\begin{matrix} \underset{x s + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{\cup} {\begin{matrix} (\tilde{f}) (a_{1}) \\ \cap (\tilde{f}) (a_{2}) \\ \cap \tilde{g} (b_{1}) \\ \cap \tilde{g} (b_{2}) \end{matrix}} \\ \cap \underset{x a' + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{\cup} {\begin{matrix} (\tilde{f}) (a_{1}) \\ \cap (\tilde{f}) (a_{2}) \\ \cap \tilde{g} (b_{1}) \\ \cap \tilde{g} (b_{2}) \end{matrix}} \\ \cap (\tilde{f}) (x) \end{matrix}} \\ \supseteq {\begin{matrix} {\begin{matrix} \tilde{f} (x) \cap \tilde{g} (a x a) \\ \cap \tilde{g} (a x a') \end{matrix}} \\ \cap {\begin{matrix} \tilde{f} (x) \cap \tilde{g} (a x a') \\ \cap \tilde{g} (a' z a') \end{matrix}} \\ \cap (\tilde{f}) (x) \end{matrix}} \\ (\begin{matrix} since x a + x a x a + z a \\ = x a' x a + z a and \\ x a' + x a x a' + z a' \\ = x a' x a + z a' \end{matrix}) \\ \supseteq {\tilde{f} (x) \cap \tilde{g} (x)} \\ \cap {\tilde{f} (x) \cap \tilde{g} (x)} \\ = \tilde{f} (x) \cap \tilde{g} (x) \\ = (\tilde{f} \tilde{\cap} \tilde{g}) (x) \end{array}

and $\begin{matrix} (λ \circ_{h} \tilde{γ} \circ_{h} \tilde{λ}) (x) \\ = \underset{x + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {\begin{matrix} (λ \circ_{h} γ) (a_{1}) \\ \lor (λ \circ_{h} γ) (a_{2}) \\ \lor λ (b_{1}) \\ \lor λ (b_{1}) \end{matrix}} \\ \supseteq {\begin{matrix} (λ \circ_{h} γ) (xa) \\ \lor (λ \circ_{h} γ) ({xa}^{'}) \lor λ (x) \end{matrix}} \\ = {\begin{matrix} \underset{xa + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z}{⋀} {\begin{matrix} λ (a_{1}) \\ \lor λ (a_{2}) \\ \lor γ (b_{1}) \\ \lor γ (b_{2}) \end{matrix}} \\ \lor ⋀_{{xa}^{'} + (a_{1} b_{1}) + z = (a_{2} b_{2}) + z} {\begin{matrix} λ (a_{1}) \\ \lor λ (a_{2}) \\ \lor γ (b_{1}) \\ \lor γ (b_{2}) \end{matrix}} \\ \lor λ (x) \end{matrix}} \\ \leq {\begin{matrix} {\begin{matrix} λ (x) \lor γ (axa) \\ \lor γ (a x a^{'}) \end{matrix}} \\ \lor {\begin{matrix} λ (x) \lor γ ({axa}^{'}) \\ \lor γ (a^{'} {xa}^{'}) \end{matrix}} \\ \lor \tilde{f} (x) \end{matrix}} \\ \leq {λ (x) \lor γ (x)} \\ \lor {λ (x) \lor γ (x)} \\ = λ (x) \lor γ (x) \\ = (λ \lor γ) (x) . \end{matrix}$

Hence, ${\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} ⪡ {\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} \otimes_{h} {\tilde{f}}_{λ}$

(2)⇒(3) This is straightforward.

(3) ⇒ (1) Assume that (3) is true. Let ${\tilde{f}}_{λ}$ be any hybrid h-quasi-ideal of R over U. Since $χ_{R} ({\tilde{f}}_{λ})$ hybrid h-ideal of R over U, we have ${\tilde{f}}_{λ} = {\tilde{f}}_{λ} ⋒ χ_{R} ({\tilde{f}}_{λ}) ⪡ {\tilde{f}}_{λ} \otimes_{h} χ_{R} ({\tilde{f}}_{λ}) \otimes_{h} {\tilde{f}}_{λ} .$

Therefore, R is h-hemiregular by Lemma 32. ■

Lemma 7.7 [23] A hemiring R is h-hemiregular if and only if the right and left h-ideals of R are idempotent and for any right h-ideal A and any left h-ideal B of R, the set $\bar{AB}$ is an h-quasi-ideal of R.

Theorem 7.8. A hemiring R is h-hemiregular if and only if the hybrid right h-ideal and the hybrid left h-ideal of R are idempotent and for any hybrid right h-ideal ${\tilde{f}}_{λ}$ and any hybrid left h-ideal ${\tilde{g}}_{γ}$ of R over U the hybrid product ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ}$ is a hybrid h-quasi-ideal of R.

Proof. Assume that R is h-hemiregular. Let ${\tilde{f}}_{λ}$ be any hybrid right h-ideal of R over U. Then ${\tilde{f}}_{λ} \otimes_{h} {\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} \otimes_{h} χ_{R} ({\tilde{f}}_{λ}) ⪡ {\tilde{f}}_{λ} .$ Since R is h-hemiregular so by Theorem 31, ${\tilde{f}}_{λ} ⪡ {\tilde{f}}_{λ} \otimes_{h} {\tilde{f}}_{λ}$ and hence ${\tilde{f}}_{λ} \otimes_{h} {\tilde{f}}_{λ} = {\tilde{f}}_{λ} .$ Hence, ${\tilde{f}}_{λ}$ is idempotent. In a similar way we can show that the hybrid left h-ideal of R over U is idempotent. Now let ${\tilde{f}}_{λ}$ and ${\tilde{g}}_{γ}$ be any hybrid right and left h-ideals of R over U, respectively. Using Theorem 31, we have that ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ} = {\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ} .$ Since ${\tilde{f}}_{λ} ⋒ {\tilde{g}}_{γ}$ is a hybrid h-quasi-ideal of R over U (Lemma 26). It follows that ${\tilde{f}}_{λ} \otimes_{h} {\tilde{g}}_{γ}$ is a hybrid h-quasi-ideal of R over U. Conversely, assume that the given conditions hold. Let A be any right h-ideal of R. Then by Proposition 25, the hybrid characteristic function $χ_{A} ({\tilde{f}}_{λ})$ of A is a hybrid right h-ideal of R over U. By hypothesis and Proposition 11, we have $χ_{A} ({\tilde{f}}_{λ}) \otimes_{h} χ_{A} ({\tilde{f}}_{λ}) = χ_{\bar{AA}} ({\tilde{f}}_{λ}) = χ_{\bar{A^{2}}} ({\tilde{f}}_{λ}) .$ Using Proposition 11 again, we have that A = A². Thus A is idempotent. In a similar way, we can prove that the left h-ideals of R are idempotent. Now let A be a right h-ideal and B a left h-ideal of R. By using the assumption and Proposition 11, we have $χ_{\bar{AB}} ({\tilde{f}}_{λ}) = χ_{A} ({\tilde{f}}_{λ}) \otimes_{h} χ_{B} ({\tilde{f}}_{λ})$ is a hybrid h-quasi-ideal of R over U, hence it follows from Proposition 11, that $\bar{AB}$ is an h-quasi-ideal of R. Thus R is h-hemiregular (Lemma 31). This completes the proof. ■

8. Concluding remarks

In this paper, we applied hybrid structures in hemirings, which is an extension of soft sets and fuzzy sets. We introduced the concept of hybrid h-ideals of hemirings and the related properties are investigated. Some operational properties of hybrid structures are investigated. The concepts of a hybrid h-sum and a hybrid h-product are discussed and the related properties are investigated.

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Hybrid structures applied to hemirings

Abstract

Keywords

1. Introduction

2. Preliminaries

3. Soft sets

4. Basic operations of hybrid structures

5. Hybrid left (right) h-ideals

Table 1 Tabular representation of hybrid structure f ˜ λ

7. h-Hemiregular hemirings

8. Concluding remarks

References

Table 1
Tabular representation of hybrid structure ${\tilde{f}}_{λ}$