On hybrid k -ideals in semirings

Abstract

Many uncertainties arise in real-world problems, making them impossible to solve using conventional approaches. Researchers all over the world have developed new mathematical theories like fuzzy set theory and rough set theory to better understand the uncertainties that occur in various fields. Soft set theory, which was recently introduced, offers a novel approach to real-world problem solving by removing the need to set the membership function. This is helpful in resolving a variety of issues, and much progress is being made these days. Recently, Jun introduced the concept of a hybrid structure, which blends the concepts of a fuzzy set as well as a soft set. In this paper, we define the hybrid k-sum and hybrid k-product of k-ideals of semiring and investigate their properties. We illustrate with an example that the hybrid sum and hybrid product of two k-ideals are not always hybrid ideals. We also describe semiring regularity constraints in terms of hybrid k-ideal structures.

Keywords

Semiring hybrid structure ideal hybrid product hybrid ideals

1 Introduction

There are several universal algebraic concepts that can be used to generalise an associative ring. Some of them, like near-rings and different kinds of semirings, have turned out to be very useful. Semirings can be used to analyse matrices and determinants. Beasley, Pullman, and Ghosh, among others, are researching various aspects of matrices theory and determinants over semirings. While ideals in semirings are useful for a number of reasons, they often do not coincide with the normal ring ideals if the semiring is a ring, so their usage in attempting to obtain semiring precursors of ring theorems is somewhat restricted. In fact, when only ideals are used, it looks like most ring results don’t have semiring antecedents.

In his groundbreaking paper [29], Zadeh first presented the concepts of fuzzy subset f of a non-empty set A as a function from A into the unit interval I = [0, 1]. This definition was extended by Rosenfeld [26] to the theory of groupoids and groups. Furthermore, several generalizations and branches of fuzzy sets have been presented in the literature(e.g., [4 , 28]).

Fuzzy set theory has been applied in a wide range of contexts, including robot design and computer simulation, as well as engineering and water resource planning. Numerous problems in these fields include data sets with complexities that are treated using a number of current theories like (intuitionistic) fuzzy set theory, rough set theory, the theory of interval mathematics, ambiguous sets, and the theory of probability. Molodtsov [17] established soft set theory as a modern mathematical strategy and successfully applied it in a couple of areas by many researchers(e.g., [15, 16]). Besides that, several analysts applied fuzzy set theory and related principles to a wide range of algebraic structures(e.g., [1 , 30]).

Jun, Song, and Muhiuddin [11] presented and investigated the properties of a hybrid subalgebra, a hybrid linear space, and a hybrid field in BCK/BCI-algebras. Anis et al. [2] presented the concepts of hybrid substructures in semigroups and acquired a number of properties.

Elavarasan and Jun presented the concepts of hybrid k-ideals, hybrid k-closure in semirings, and explored some of their features in [8]. G. Muhiuddin et al. utilised the hybrid structure in semimodules over semirings and looked into the representations of hybrid subsemimodules and hybrid ideals by employing hybrid products in [18]. In [19], G. Muhiuddin et al. established the concepts of hybrid ideals and k-hybrid ideals in a ternary semiring. They also proved different things about k-hybrid ideals and gave some descriptions of hybrid intersection in terms of these k-hybrid ideals. In [5] and [6], Elavarasan and Jun adapted the hybrid structure to semigroups and defined some similar requirements for a semigroup to be regular and intra-regular, in terms of hybrid ideals and hybrid bi-ideals. Additionally, they used hybrid ideals and hybrid bi-ideals to characterise the left and right simple semigroups as well as the completely regular semigroups. The concept of hybrid structures has been applied to various algebraic systems, providing numerous related results (e.g., [7 , 23–25]).

In this paper, we look at the hybrid setting of k-ideals in semirings. Using a set of semiring left k-ideals, we construct hybrid left k-ideals of semiring $ℝ$ . We discuss hybrid k-sum and k-product of semiring hybrid k-ideals, as well as k-regular semirings in terms of hybrid left(right) k-ideals.

2 Preliminaries

This section includes a few definitions and observations that will help us interpret our main findings.

A semiring is a set $ℝ (\neq φ)$ that has two binary operations, “+" and “ · ", and satisfies the below axioms:

(i) $(ℝ, +)$ is a commutative semigroup,

(ii) $(ℝ, \cdot)$ is a semigroup,

(iii) (w + u) · s = w · s + u · s;

s · (w + u) = s · w + s · u for any $w, u, s \in ℝ$ .

A zero element of $ℝ$ is a “0 ": $0 . x = x . 0 = 0 \forall x \in ℝ .$ Unless otherwise stated, $ℝ$ is a semiring with zero element and for a set Q, its power set is $P (Q)$ .

Definition 2.1. Let $L \in P (ℝ) .$

(i) L is described as a left (resp., right) ideal of $ℝ$ if (L, +) is closed and $ℝ L \subseteq L (resp ., L ℝ \subseteq L) .$

If L is both a left and a right ideal of $ℝ$ , it is referred to as an ideal of $ℝ$ .

(ii) L is described as a left (resp., right) k-ideal of $ℝ$ if L is a left (resp., right) ideal of $ℝ$ and for s∈ L ; $e \in ℝ,$ if e + s ∈ L, then e ∈ L . L is described as a k- ideal of $ℝ$ if L is both a left as well as a right k- ideal of $ℝ$ .

Definition 2.2. [11] For an unit interval I = [0, 1] and initial universal set $𝕩$ , a hybrid structure in $ℝ$ over $𝕩$ is a mapping ${\tilde{l}}_{β} : = (\tilde{l}, β) : ℝ \to P (𝕩) \times I, c \mapsto (\tilde{l} (c), β (c))$

where $\tilde{l} : ℝ \to P (𝕩)$ and $β : ℝ \to I$ are mappings.

A relation ⪡ defined on the gathering of all hybrid structures, denoted by $H (ℝ),$ in $ℝ$ over $𝕩$ as follows: $(\forall {\tilde{j}}_{β}, {\tilde{h}}_{γ} \in H (ℝ)) ({\tilde{j}}_{β} ⪡ {\tilde{h}}_{γ} \Leftrightarrow \tilde{j} \tilde{\subseteq} \tilde{h}, β ⪰ γ)$ where $\tilde{j} \tilde{\subseteq} \tilde{h}$ represents $\tilde{j} (k) \subseteq \tilde{h} (k)$ and β ⪰ γ represents $β (k) \geq γ (k) \forall k \in ℝ .$ Then the set $(H (ℝ), ⪡)$ is partially ordered.

Definition 2.3. [8] For ${\tilde{g}}_{β} \in H (ℝ)$ . ${\tilde{g}}_{β}$ is described as a hybrid left (resp., right) ideal of $ℝ$ if ${\tilde{g}}_{β}$ fulfils the below criteria:

(i) $(\begin{matrix} \tilde{g} (s_{1} + c) \supseteq \tilde{g} (s_{1}) \cap \tilde{g} (c) \\ β (s_{1} + c) \leq β (s_{1}) \lor β (c) \end{matrix}),$

(ii) $(\begin{matrix} \tilde{g} (s_{1} c) \supseteq \tilde{g} (c) \\ β (s_{1} c) \leq β (c) \end{matrix})$ $\forall s_{1}, c \in ℝ .$

Definition 2.4. [8] A hybrid left (resp., right) ideal ${\tilde{g}}_{β}$ in $ℝ$ is described as hybrid left (resp., right) k-ideal of $ℝ$ if it fulfils the below criteria:

(i) $\tilde{g} (s_{1}) \supseteq \tilde{g} (s_{1} + c) \cap \tilde{g} (c),$

(ii) β (s₁) ≤ β (s₁ + c) ∨ β (c) $\forall c, s_{1} \in ℝ .$

If $(ℝ, +)$ is non-commutative, the case is reduced to

(i) $\tilde{g} (s) \supseteq ⋂ {\tilde{g} (s + w) \cup \tilde{g} (w + s), \tilde{g} (w)},$

(ii) β (s) ≤ ⋁ {β (s + w) ∧ β (w + s) , β (w)} $\forall w, s \in ℝ .$

Note 2.5.The assertions mentioned below are equivalent for a hybrid left (resp., right) k-ideal ${\tilde{g}}_{μ}$ of $ℝ$ over $𝕩,$

(i) $(\forall s, c \in ℝ)$ $(\begin{matrix} \tilde{g} (s) \supseteq \tilde{g} (s + c) \cap \tilde{g} (c) \\ μ (s) \leq μ (s + c) \lor μ (c) \end{matrix}),$

(ii) $\forall s, c, s_{1} \in ℝ,$ $s + c = s_{1} \Rightarrow (\begin{matrix} \tilde{g} (s) \supseteq \tilde{g} (s_{1}) \cap \tilde{g} (c) \\ μ (s) \leq μ (s_{1}) \lor μ (c) \end{matrix}) .$

Definition 2.6. For ${\tilde{h}}_{β} \in H (ℝ)$ and $φ \neq Q \subseteq ℝ$ , $χ_{Q} ({\tilde{h}}_{β})$ is the characteristic hybrid structure in $ℝ$ over $𝕩,$ which is described as: $\begin{matrix} χ_{Q} ({\tilde{h}}_{β}) = (χ_{Q} (\tilde{h}), χ_{Q} (β)) : ℝ ⟶ P (𝕩) \times I, \\ s_{1} \mapsto (χ_{Q} (\tilde{h}) (s_{1}), χ_{Q} (β) (s_{1})), \end{matrix}$

where $\begin{matrix} χ_{Q} (\tilde{h}) & : ℝ \to P (𝕩), s_{1} \mapsto {\begin{matrix} 𝕩 & if s_{1} \in Q \\ φ & otherwise, \end{matrix} \\ χ_{Q} (β) & : ℝ \to I, s_{1} \mapsto {\begin{matrix} 0 & if s_{1} \in Q \\ 1 & otherwise . \end{matrix} \end{matrix}$

Definition 2.7. Let ${\tilde{l}}_{ν}, {\tilde{h}}_{β} \in H (ℝ) .$ Then

(i) ${\tilde{l}}_{ν} ⊙ {\tilde{h}}_{β}$ is the hybrid product of ${\tilde{l}}_{ν}$ and ${\tilde{h}}_{β}$ , which is described as ${\tilde{l}}_{ν} ⊙ {\tilde{h}}_{β} = (\tilde{l} \tilde{\circ} \tilde{h}, ν \tilde{\circ} β)$ ,

where $\begin{matrix} (\tilde{l} \tilde{\circ} \tilde{h}) (t) = {\begin{matrix} ⋃_{t = sq} {\tilde{l} (s) \cap \tilde{h} (q)} & if t = sq \\ φ & otherwise \end{matrix} \end{matrix}$ and $\begin{matrix} (ν \tilde{\circ} β) (t) = {\begin{matrix} ⋀_{t = sq} {ν (s) \lor β (q)} & if t = sq \\ 1 & otherwise \end{matrix} \end{matrix}$ for $s, q, t \in ℝ .$

(ii) ${\tilde{l}}_{ν} ⊙_{k} {\tilde{h}}_{β}$ is the hybrid k-product of ${\tilde{l}}_{ν}$ and ${\tilde{h}}_{β}$ , which is described as ${\tilde{l}}_{ν} ⊙_{k} {\tilde{h}}_{β} = (\tilde{l} {\tilde{\circ}}_{k} \tilde{h}, ν {\tilde{\circ}}_{k} β)$ , where, for $t \in ℝ,$ if t + a₁b₁ = a₂b₂ for some $t, a_{1}, b_{1}, a_{2}, b_{2} \in ℝ,$ then $(\tilde{l} {\tilde{\circ}}_{k} \tilde{h}) (t) = ⋃_{t} {\tilde{l} (a_{1}) \cap \tilde{l} (a_{2}) \cap \tilde{h} (b_{1}) \cap \tilde{h} (b_{2})}$ and $(ν {\tilde{\circ}}_{k} β) (t) = ⋀_{t} {ν (a_{1}) \lor ν (a_{2}) \lor β (b_{1}) \lor β (b_{2})} .$ Otherwise, $(\tilde{l} {\tilde{\circ}}_{k} \tilde{h}) (t) = φ$ and $(ν {\tilde{\circ}}_{k} β) (t) = 1 .$

Definition 2.8. Let ${\tilde{f}}_{ν}, {\tilde{j}}_{β} \in H (ℝ) .$ Then

(i) the hybrid sum ${\tilde{f}}_{ν} \oplus {\tilde{j}}_{β}$ is described as ${\tilde{f}}_{ν} \oplus {\tilde{j}}_{β} = (\tilde{f} \tilde{+} \tilde{j}, ν \tilde{+} β)$ , where, for $t \in ℝ$ , if t = q₂ + s₂ for some $q_{2}, s_{2} \in ℝ,$ then $(\tilde{f} \tilde{+} \tilde{j}) (t) = ⋃_{t} {\tilde{f} (q_{2}) \cap \tilde{j} (s_{2})}$ and $(ν \tilde{+} β) (t) = ⋀_{t} {ν (q_{2}) \lor β (s_{2})} .$ Otherwise, $(\tilde{f} \tilde{+} \tilde{j}) (t) = φ$ and $(ν \tilde{+} β) (t) = 1 .$

(ii) the hybrid k-sum ${\tilde{f}}_{ν} \oplus_{k} {\tilde{j}}_{β}$ is described as ${\tilde{f}}_{ν} \oplus_{k} {\tilde{j}}_{β} = (\tilde{f} {\tilde{+}}_{k} \tilde{j}, ν {\tilde{+}}_{k} β)$ ,

where, for $t \in ℝ$ , if t + (q₁ + s₁) = q₂ + s₂ for some $t, q_{1}, s_{1}, q_{2}, s_{2} \in ℝ,$ then $(\tilde{f} {\tilde{+}}_{k} \tilde{j}) (t) = ⋃_{t} {⋂ {\tilde{f} (q_{j}), \tilde{j} (s_{j}), j = 1, 2}}$ and $(ν {\tilde{+}}_{k} β) (t) = ⋀_{t} {⋁ {ν (q_{j}), β (s_{j}), j = 1, 2}}$

Otherwise, $(\tilde{f} {\tilde{+}}_{k} \tilde{j}) (t) = φ$ and $(ν {\tilde{+}}_{k} β) (t) = 1$ .

Definition 2.9. [11] For ${\tilde{l}}_{ν}, {\tilde{h}}_{β} \in H (ℝ),$ ${\tilde{l}}_{ν} ⋒ {\tilde{h}}_{β}$ is the hybrid intersection of ${\tilde{l}}_{ν}$ and ${\tilde{h}}_{β},$ which is described as below: ${\tilde{l}}_{ν} ⋒ {\tilde{h}}_{β} : ℝ \to P (𝕩) \times I, s \mapsto ((\tilde{l} \tilde{\cap} \tilde{h}) (s), (ν \lor β) (s)),$ where $\tilde{l} \tilde{\cap} \tilde{h} : ℝ ⟶ P (𝕩), s \mapsto \tilde{l} (s) \cap \tilde{h} (s)$ and $ν \lor β : ℝ \to I, s \mapsto ν (s) \lor β (s)$ for $s \in ℝ .$

Definition 2.10. For any $R \in P (ℝ),$ the k-closure $\bar{R}$ of R is defined by $\bar{R} = {u \in ℝ : u + q_{1} = q_{2}$ for some q₁, q₂ ∈ R} .

Definition 2.11. [10] A semiring $ℝ$ is described as a k-regular if for each $t \in ℝ, \exists a_{1}, a_{2} \in ℝ$ such that t + ta₁t = ta₂t .

3 Hybrid k-ideals in semirings

Proposition 3.1. [8] For ${\tilde{g}}_{β} \in H (ℝ)$ and $φ \neq Q \subseteq ℝ,$ the statements mentioned below are equivalent:

(i) $χ_{Q} ({\tilde{g}}_{β})$ is a hybrid left (resp., right) ideal in $ℝ,$

(ii) Q is a left (resp., right) ideal of $ℝ .$

Proposition 3.2. Let $Q (\neq φ) \subseteq ℝ$ and ${\tilde{g}}_{β} \in H (ℝ) .$ Then assertions mentioned below are equivalent:

(i) Q is a left (resp., right) k-ideal of $ℝ,$

(ii) $χ_{Q} ({\tilde{g}}_{β})$ is a hybrid left (resp., right) k-ideal of $ℝ .$

Proof. For a left k-ideal Q of $ℝ,$ by Proposition 3.1, $χ_{Q} ({\tilde{g}}_{β})$ of $ℝ$ is a hybrid left ideal. Let $e, s, q \in ℝ$ : e + q = s .

If q, s ∈ Q, then e ∈ Q . Thus $χ_{Q} (\tilde{g}) (e) = 𝕩 = χ_{Q} (\tilde{g}) (q) \cap χ_{Q} (\tilde{g}) (s)$ and χ_Q (β) (e) =0 = χ_Q (β) (q) ∨ χ_Q (β) (s). Otherwise q ∉ Q or s ∉ Q . Then $χ_{Q} (\tilde{g}) (q) \cap χ_{Q} (\tilde{g}) (s) = φ \subseteq χ_{Q} (\tilde{g}) (e)$ and χ_Q (β) (q) ∨ χ_Q (β) (s) =1 ≥ χ_Q (β) (e).

Therefore $χ_{Q} ({\tilde{g}}_{β})$ is hybrid left k-ideal in $ℝ$ .

Conversely, assume $χ_{Q} ({\tilde{g}}_{β})$ is a hybrid left k-ideal of $ℝ .$ By Proposition 3.1, Q of $ℝ$ is a left ideal. Let $e \in ℝ$ and q, s ∈ Q : e + q = s . Then $χ_{Q} (\tilde{g}) (e) \supseteq χ_{Q} (\tilde{g}) (q) \cap χ_{Q} (\tilde{g}) (s) = 𝕩$ and χ_Q (β) (e) ≤ χ_Q (β) (q) ∨ χ_Q (β) (s) =0 which imply e ∈ Q . So Q of $ℝ$ is a left k-ideal.

Definition 3.3. For ${\tilde{l}}_{β} \in H (ℝ), Q \in P (𝕩)$ and ∈ ∈ I, the set ${\tilde{l}}_{β} [Q, \in] : = {a_{1} \in ℝ ∣ \tilde{l} (a_{1}) \supseteq Q, β (a_{1}) \leq \in}$ is described as the [Q, ∈] -hybrid cut of ${\tilde{l}}_{β} .$

It is simple to prove that for any hybrid left(resp., right) ideal ${\tilde{l}}_{β}$ in $ℝ$ , ${\tilde{l}}_{β} [Q, \in]$ of $ℝ$ is a left (resp., right) ideal.

Proposition 3.4. For ${\tilde{l}}_{λ} \in H (ℝ),$ the assertions mentioned below are equivalent:

(i) for any $Q \in P (𝕩)$ and t₁ ∈ I such that ${\tilde{l}}_{λ} [Q, t_{1}] \neq φ,$ ${\tilde{l}}_{λ} [Q, t_{1}]$ is a left (resp., right) k-ideal of $ℝ,$

(ii) ${\tilde{l}}_{λ}$ in $ℝ$ is a hybrid left (resp., right) k-ideal.

Proof. Assume that for any $Q \in P (𝕩)$ and t₁ ∈ I, ${\tilde{l}}_{λ} [Q, t_{1}] \neq φ$ and ${\tilde{l}}_{λ} [Q, t_{1}]$ is a left k-ideal of $ℝ .$ For $c, x \in ℝ,$ let $\tilde{l} (c) = Q_{1}, \tilde{l} (c + x) = Q_{2}$ and λ (c) = t₂, λ (c + x) = t₃ for some $Q_{1}, Q_{2} \in P (𝕩); t_{2}, t_{3} \in I .$

If we take Q : = Q₁ ∩ Q₂ and t₁ : = t₂ ∨ t₃, then $\tilde{l} (c) \supseteq Q;$ λ (c) ≤ t₁ and $\tilde{l} (c + x) \supseteq Q;$ λ (c + x) ≤ t₁ which imply $c, c + x \in {\tilde{l}}_{λ} [Q, t_{1}] .$ Since ${\tilde{l}}_{λ} [Q, t_{1}]$ is left k-ideal, we have $x \in {\tilde{l}}_{λ} [Q, t_{1}] .$ Now $\tilde{l} (x) \supseteq Q = Q_{1} \cap Q_{2} = \tilde{l} (c) \cap \tilde{l} (c + x)$ and λ (x) ≤ t₁ = λ (c) ∨ λ (c + x) . So ${\tilde{l}}_{λ}$ is hybrid left k-ideal.

On the other hand, assume that ${\tilde{l}}_{λ}$ is a hybrid left k-ideal of $ℝ .$ Let $c \in {\tilde{l}}_{λ} [Q, t_{1}]$ and $x \in ℝ$ with $c + x \in {\tilde{l}}_{λ} [Q, t_{1}] .$ Then $\tilde{l} (c) \supseteq Q, λ (c) \leq t_{1}$ and $\tilde{l} (c + x) \supseteq Q, λ (c + x) \leq t_{1} .$ So $\tilde{l} (x) \supseteq \tilde{l} (c + x) \cap \tilde{l} (c) \supseteq Q$ and λ (x) ≤ λ (c + x) ∨ λ (c) ≤ t₁ . Thus $x \in {\tilde{l}}_{λ} [Q, t_{1}]$ and hence ${\tilde{l}}_{λ} [Q, t_{1}]$ of $ℝ$ is a left k-ideal.

Proposition 3.5. For each left (resp., right) k-ideal K of $ℝ,$ ∃ a hybrid left (resp., right)k-ideal ${\tilde{l}}_{λ}$ of $ℝ : {\tilde{l}}_{λ} [α, t_{1}] = K$ for some $α \in P (𝕩) ∖ {φ}$ and t₁ ∈ I ∖ {1} .

Proof. Let K be a k-ideal of $ℝ$ and for $α \in P (𝕩) ∖ {φ}$ and t₁ ∈ I ∖ {1} , define a hybrid structure ${\tilde{l}}_{λ}$ of $ℝ$ by $\tilde{l} (a) : = {\begin{matrix} α & if a \in K, \\ {φ} & otherwise \end{matrix} and λ (a) : = {\begin{matrix} t_{1} & if a \in K, \\ 1 & otherwise . \end{matrix}$ Then $\tilde{l} [α, t_{1}] = K .$ For a given $β \in P (𝕩)$ and t₂ ∈ I, we have ${\tilde{l}}_{λ} [β, t_{1}] = {\begin{matrix} {\tilde{l}}_{λ} [φ, 1] (= ℝ) & if β = φ and t_{2} = 1 \\ {\tilde{l}}_{λ} [α, t] (= K) & if α \subset β and t_{1} \geq t_{2} \\ φ & otherwise . \end{matrix}$

Since $ℝ$ and K are itself left k-ideals of $ℝ,$ we have ${\tilde{l}}_{λ} [β, t_{1}]$ of $ℝ$ is left k-ideal. By Proposition 3.4, ${\tilde{l}}_{λ}$ in $ℝ$ is a hybrid left k-ideal.

Lemma 3.6. Let ${\tilde{f}}_{ν} \in H (ℝ)$ and $Q, S \subseteq ℝ .$ Then $χ_{Q} ({\tilde{f}}_{ν}) ⊙_{k} χ_{S} ({\tilde{f}}_{ν}) = χ_{\bar{QS}} ({\tilde{f}}_{ν}) .$

Proof. Let $t \in ℝ .$ If $t \in \bar{QS},$ then $χ_{\bar{QS}} (\tilde{f}) (t) = 𝕩; χ_{\bar{QS}} (ν) (t) = 0$ and t + p₁q₁ = p₂q₂ for some p₁, p₂ ∈ Q and q₁, q₂ ∈ S . Now

$(χ_{Q} (\tilde{f}) {\tilde{\circ}}_{k} χ_{S} (\tilde{f})) (t) = ⋃_{t + p_{1} q_{1} = p_{2} q_{2}} {⋂ {χ_{Q} (\tilde{f}) (p_{m}), χ_{S} (\tilde{f}) (q_{m}), m = 1, 2}} \supseteq ⋂ {χ_{Q} (\tilde{f}) (p_{m}), χ_{S} (\tilde{f}) (q_{m}), m = 1, 2} = 𝕩,$

$(χ_{Q} (ν) {\tilde{\circ}}_{k} χ_{S} (ν)) (t) = ⋀_{t + p_{1} q_{1} = p_{2} q_{2}} {⋁ {χ_{Q} (ν) (p_{i}), χ_{S} (ν) (q_{i}), i = 1, 2}} \leq ⋁ {χ_{Q} (ν) (p_{i}), χ_{S} (ν) (q_{i}), i = 1, 2} = 0 .$

So $χ_{Q} ({\tilde{f}}_{ν}) ⊙_{k} χ_{S} ({\tilde{f}}_{ν}) = χ_{\bar{QS}} ({\tilde{f}}_{ν})$ .

If $t \notin \bar{QS},$ then $χ_{\bar{QS}} (\tilde{f}) (t) = φ$ and $χ_{\bar{QS}} (ν) (t) = 1 .$

If $χ_{Q} (\tilde{f}) {\tilde{\circ}}_{k} χ_{S} (\tilde{f})) (t) \neq φ$ , then

$(χ_{Q} (\tilde{f}) {\tilde{\circ}}_{k} χ_{S} (\tilde{f})) (t) = ⋃_{t + p_{1} q_{1} = p_{2} q_{2}} {⋂ {χ_{Q} (\tilde{f}) (p_{i}), χ_{S} (\tilde{f}) (q_{i}), i = 1, 2}} \neq φ$ .

Hence $\exists p_{1}, p_{2}, q_{1}, q_{2} \in ℝ : t + p_{1} q_{1} = p_{2} q_{2}$ and $⋂ {χ_{Q} (\tilde{f}) (p_{i}), χ_{S} (\tilde{f}) (q_{i}), i = 1, 2} \neq φ$ which imply $χ_{Q} (\tilde{f}) (p_{1}) = χ_{Q} (\tilde{f}) (p_{2}) = χ_{S} (\tilde{f}) (q_{1}) = χ_{S} (\tilde{f}) (q_{2}) = 𝕩 .$ Thus p₁, p₂ ∈ Q and q₁, q₂ ∈ S, and hence $t \in \bar{QS},$ a contradiction.

So $(χ_{Q} (\tilde{f}) {\tilde{\circ}}_{k} χ_{S} (\tilde{f})) (t) = φ = χ_{\bar{QS}} (\tilde{f}) (t) .$

Similar way, if $(χ_{Q} (ν) {\tilde{\circ}}_{k} χ_{S} (ν)) (t) \neq 1,$ then

$(χ_{Q} (ν) {\tilde{\circ}}_{k} χ_{S} (ν)) (t) = ⋀_{t + p_{i} q_{i} = p_{j} q_{j}} {⋁ {χ_{Q} (ν) (p_{i}), χ_{S} (ν) (q_{i}), i = 1, 2}} \neq 1$ .

Hence $\exists p_{1}, p_{2}, q_{1}, q_{2} \in ℝ : t + p_{1} q_{1} = p_{2} q_{2}$ and ⋁ {χ_Q (ν) (p_i) , χ_S (ν) (q_i) , i = 1, 2} ≠1 which imply χ_Q (ν) (p₁) = χ_Q (ν) (p₂) = χ_S (ν) (q₁) = χ_S (ν) (q₂) =0 . Thus p₁, p₂ ∈ Q and q₁, q₂ ∈ S, and hence $t \in \bar{QS},$ a contradiction. So $χ_{\bar{QS}} (ν) (t) = 1 = (χ_{Q} (ν) {\tilde{\circ}}_{k} χ_{S} (ν)) (t) .$ Therefore $χ_{Q} ({\tilde{f}}_{ν}) ⊙_{k} χ_{S} ({\tilde{f}}_{ν}) = χ_{\bar{QS}} ({\tilde{f}}_{ν}) .$

Proposition 3.7. For ${\tilde{l}}_{β}, {\tilde{h}}_{γ} \in H (ℝ),$ the assertions mentioned below are equivalent:

(i) ${\tilde{l}}_{β} \oplus {\tilde{l}}_{β} ⪡ {\tilde{l}}_{β}$ and $χ_{ℝ} ({\tilde{h}}_{γ}) ⊙ {\tilde{l}}_{β} ⪡ {\tilde{l}}_{β}$ (resp., ${\tilde{l}}_{β} ⊙ χ_{ℝ} ({\tilde{h}}_{γ}) ⪡ {\tilde{l}}_{β}$ ),

(ii) ${\tilde{l}}_{β}$ is a hybrid left (resp., right) ideal of $ℝ .$

Proof. (i) ⇒ (ii) Let $q, s \in ℝ .$ Then $\begin{matrix} \tilde{l} (q + s) & \supseteq (\tilde{l} \tilde{+} \tilde{l}) (q + s) \\ = ⋃_{q + s = q_{1} + s_{1}} {\tilde{l} (q_{1}) \cap \tilde{l} (s_{1})} \\ \supseteq \tilde{l} (q) \cap \tilde{l} (s), \\ β (q + s) & \leq (β \tilde{+} β) (q + s) \\ = ⋀_{q + s = q_{1} + s_{1}} {β (q_{1}) \lor β (s_{1})} \\ \leq β (q) \lor β (s) . \end{matrix}$ Also, $\begin{matrix} \tilde{l} (qs) & \supseteq ((χ_{ℝ} (\tilde{h}) \tilde{○} \tilde{l}) (qs) \\ = ⋃_{qs = q_{1} s_{1}} {χ_{ℝ} (\tilde{h}) (q_{1}) \cap \tilde{l} (s_{1})} \\ \supseteq χ_{ℝ} (\tilde{h}) (q) \cap \tilde{l} (s) = \tilde{l} (s), \\ β (qs) & \leq (χ_{ℝ} (γ) \tilde{○} β) (qs) \\ = ⋀_{qs = q_{1} s_{1}} {χ_{ℝ} (γ) (q_{1}) \lor β (s_{1})} \\ \leq χ_{ℝ} (γ) (q) \lor β (s) = β (s) . \end{matrix}$

Hence ${\tilde{l}}_{β}$ is hybrid left ideal.

(ii) ⇒ (i) Consider the hybrid left ideal ${\tilde{l}}_{β}$ of $ℝ$ and ${\tilde{h}}_{γ} \in H (ℝ) .$ Then by Theorem 2.11 of [8], ${\tilde{l}}_{β} \oplus {\tilde{l}}_{β} ⪡ {\tilde{l}}_{β}$ . Let $q_{1} \in ℝ .$ Then $\begin{matrix} (χ_{ℝ} (\tilde{h}) \tilde{\circ} \tilde{l}) (q_{1}) & = ⋃_{q_{1} = qs} {χ_{ℝ} (\tilde{h}) (q) \cap \tilde{l} (s)} \\ = ⋃_{q_{1} = qs} {\tilde{l} (s)} \\ \subseteq ⋃_{q_{1} = qs} {\tilde{l} (qs)} = \tilde{l} (q_{1}), \\ (χ_{ℝ} (γ) \tilde{\circ} β) (q_{1}) & = ⋀_{q_{1} = qs} {χ_{ℝ} (γ) (q) \lor β (s)} \\ = ⋀_{q_{1} = qs} {β (s)} \\ \geq ⋀_{q_{1} = qs} {β (qs)} = β (q_{1}) . \end{matrix}$

Hence $χ_{ℝ} ({\tilde{h}}_{γ}) ⊙ {\tilde{l}}_{β} ⪡ {\tilde{l}}_{β}$ .

Remark 3.8. It is clear that if $0 \in ℝ$ , then for any hybrid ideals ${\tilde{l}}_{λ}$ and ${\tilde{h}}_{γ}$ in $ℝ$ , we have ${\tilde{l}}_{λ} ⪡ {\tilde{l}}_{λ} \oplus {\tilde{h}}_{γ}$ and ${\tilde{h}}_{γ} ⪡ {\tilde{l}}_{λ} \oplus {\tilde{h}}_{γ}$ . The following example demonstrates that the condition $0 \in ℝ$ in this remark is not superficial.

Example 3.9. Let $ℝ$ be the collection of positive integers. Then $ℝ$ is a semiring with no zero element in terms of standard addition and multiplication. For Q ≠ {φ} , define the hybrid structures ${\tilde{l}}_{λ}, {\tilde{h}}_{γ} \in H (ℝ)$ as follows: For $e \in ℝ,$ $\begin{matrix} \tilde{l} (e) = {\begin{matrix} Q & if e = 2 x for x \in ℝ \\ φ & otherwise, \end{matrix} \\ \tilde{h} (e) = {\begin{matrix} Q & if e = 3 x for x \in ℝ \\ φ & otherwise \end{matrix} \end{matrix}$ and λ, γ are constant mappings. Then ${\tilde{h}}_{γ}$ and ${\tilde{l}}_{λ}$ are hybrid ideals of $ℝ$ . Here $(\tilde{l} + \tilde{h}) (2) = φ = (\tilde{l} + \tilde{h}) (3)$ and $\tilde{l} (2) = Q = \tilde{h} (3) .$ So ${\tilde{l}}_{λ} ≪ {\tilde{l}}_{λ} \oplus {\tilde{h}}_{γ}$ and ${\tilde{h}}_{γ} ≪ {\tilde{l}}_{λ} \oplus {\tilde{h}}_{γ}$ .

Proposition 3.10. For ${\tilde{g}}_{β}, {\tilde{l}}_{γ} \in H (ℝ) .$

(i) If ${\tilde{g}}_{β}$ and ${\tilde{l}}_{γ}$ are, respectively, hybrid left and hybrid right ideals of $ℝ,$ then ${\tilde{l}}_{γ} ⊙ {\tilde{g}}_{β} ⪡ {\tilde{l}}_{γ} ⋒ {\tilde{g}}_{β} .$

(ii) If ${\tilde{g}}_{β}$ and ${\tilde{l}}_{γ}$ are hybrid right k-ideal and hybrid left k-ideal of $ℝ$ respectively, then ${\tilde{g}}_{β} ⊙_{k} {\tilde{l}}_{γ} ⪡ {\tilde{g}}_{β} ⋒ {\tilde{l}}_{γ} .$

Proof. (i) Since ${\tilde{l}}_{γ} ⊙ χ_{ℝ} ({\tilde{g}}_{β}) ⪡ {\tilde{l}}_{γ}$ and ${\tilde{g}}_{β} ⪡ χ_{ℝ} ({\tilde{g}}_{β}),$ by Theorem 2.9 of [8], ${\tilde{l}}_{γ} ⊙ {\tilde{g}}_{β} ⪡ {\tilde{l}}_{γ} ⊙ χ_{ℝ} ({\tilde{g}}_{β}) ⪡ {\tilde{l}}_{γ} .$ Similarly ${\tilde{l}}_{γ} ⊙ {\tilde{g}}_{β} ⪡ χ_{ℝ} ({\tilde{l}}_{γ}) ⊙ {\tilde{g}}_{β} ⪡ {\tilde{g}}_{β} .$ So ${\tilde{l}}_{γ} ⊙ {\tilde{g}}_{β} ⪡ {\tilde{l}}_{γ} ⋒ {\tilde{g}}_{β} .$ (ii) Consider the hybrid right k-ideal ${\tilde{g}}_{β}$ and hybrid left k-ideal ${\tilde{l}}_{γ}$ of $ℝ,$ and let $q \in ℝ .$

If $\exists a_{1}, a_{2}, b_{1}, b_{2} \in ℝ$ : q + a₁b₁ = a₂b₂ . Then $\tilde{g} (q) \supseteq \tilde{g} (a_{1} b_{1}) \cap \tilde{g} (a_{2} b_{2}) \supseteq \tilde{g} (a_{1}) \cap \tilde{g} (a_{2})$ and β (q) ≤ β (a₁b₁) ∨ β (a₂b₂) ≤ β (a₁) ∨ β (a₂) .

Similarly, we can get $\tilde{l} (q) \supseteq \tilde{l} (b_{1}) \cap \tilde{l} (b_{2})$ and γ (q) ≤ γ (b₁) ∨ γ (b₂) .

Now, $\begin{matrix} (\tilde{g} {\tilde{\circ}}_{k} & \tilde{l}) (q) \\ = ⋃_{q + a_{1} b_{1} = a_{2} b_{2}} {⋂ {\tilde{g} (a_{j}), \tilde{l} (b_{j}), j = 1, 2}} \\ \subseteq \tilde{g} (q) \cap \tilde{l} (q) = (\tilde{g} \cap \tilde{l}) (q), \end{matrix}$ $\begin{matrix} (β {\tilde{\circ}}_{k} & γ) (q) \\ = ⋀_{t + a_{1} b_{1} = a_{2} b_{2}} {⋁ {β (a_{j}), γ (b_{j}), j = 1, 2}} \\ \geq β (q) \lor γ (q) = (β \lor γ) (q) . \end{matrix}$

So, ${\tilde{g}}_{β} ⊙_{k} {\tilde{l}}_{γ} ⪡ {\tilde{g}}_{β} ⋒ {\tilde{l}}_{γ} .$

Proposition 3.11. Let ${\tilde{g}}_{β}, {\tilde{l}}_{γ} \in H (ℝ)$ . The below statements are hold for $ℝ$ .

(i) If ${\tilde{g}}_{β}$ and ${\tilde{l}}_{γ}$ are hybrid left (resp., right) ideals in $ℝ,$ then ${\tilde{g}}_{β} \oplus {\tilde{l}}_{γ}$ , ${\tilde{g}}_{β} ⊙ {\tilde{l}}_{γ}$ and ${\tilde{g}}_{β} ⋒ {\tilde{l}}_{γ}$ are also hybrid left (resp., right) ideals in $ℝ$ .

(ii) If ${\tilde{g}}_{β}$ and ${\tilde{l}}_{γ}$ are hybrid left (resp., right)k- ideals of $ℝ,$ then ${\tilde{g}}_{β} ⋒ {\tilde{l}}_{γ}$ is a hybrid left (resp., right) k-ideal in $ℝ$ .

Proof. (i) Consider the two hybrid left ideals ${\tilde{g}}_{β}$ and ${\tilde{l}}_{γ}$ of $ℝ,$ and let $e, s \in ℝ .$ If $(\tilde{g} \tilde{+} \tilde{l}) (e) = φ$ or $(\tilde{g} \tilde{+} \tilde{l}) (s) = φ,$ then $(\tilde{g} \tilde{+} \tilde{l}) (e) \cap (\tilde{g} \tilde{+} \tilde{l}) (s) = φ \subseteq (\tilde{g} \tilde{+} \tilde{l}) (e + s) .$

Also if $(β \tilde{+} γ) (e) = 1$ or $(β \tilde{+} γ) (s) = 1,$ then $(β \tilde{+} γ) (e) \lor (β \tilde{+} γ) (s) = 1 \geq (β \tilde{+} γ) (e + s) .$

Assume that $(\tilde{g} \tilde{+} \tilde{l}) (e) \neq φ;$ $(\tilde{g} \tilde{+} \tilde{l}) (s) \neq φ$ and $(β \tilde{+} γ) (e) \neq 1$ and $(β \tilde{+} γ) (s) \neq 1 .$ Then

$(\tilde{g} \tilde{+} \tilde{l}) (e) = ⋃_{e = q_{1} + s_{1}} {\tilde{g} (q_{1}) \cap \tilde{l} (s_{1})},$

$(β \tilde{+} γ) (e) = ⋀_{e = q_{1} + s_{1}} {β (q_{1}) \lor γ (s_{1})}$ and

$(\tilde{g} \tilde{+} \tilde{l}) (s) = ⋃_{s = q_{2} + s_{2}} {\tilde{g} (q_{2}) \cap \tilde{l} (s_{2})},$

$(β \tilde{+} γ) (s) = ⋀_{s = q_{2} + s_{2}} {β (q_{2}) \lor γ (s_{2})} .$

Now, $\begin{array}{l} (\tilde{g} {\tilde{°}}_{k} \tilde{l}) (q) = \underset{q + a_{1} b_{1} = a_{2} b_{2}}{\cup} {\cap {\tilde{g} (a_{j}), \tilde{l} (b_{j}), j = 1, 2}} \\ \subseteq \tilde{g} (q) \cap \tilde{l} (q) = (\tilde{g} \cap \tilde{l}) (q), \end{array}$ $\begin{matrix} (β {\tilde{°}}_{k} ν) (q) = \underset{t + a_{1} b_{1} = a_{2} b_{2}}{\land} {\lor {β (a_{j}), ν (b_{j}), j = 1, 2}} \\ \geq β (q) \lor ν (q) = (β \lor ν) (q) . \end{matrix}$

Also, if $(\tilde{g} \tilde{+} \tilde{l}) (e) = φ,$ then $(\tilde{g} \tilde{+} \tilde{l}) (e) = φ \subseteq (\tilde{g} \tilde{+} \tilde{l}) (se) .$ If $(β \tilde{+} γ) (e) = 1,$ then $(β \tilde{+} γ) (e) = 1 \geq (β \tilde{+} γ) (se) .$

So assume that $(\tilde{g} \tilde{+} \tilde{l}) (e) \neq φ$ and $(β \tilde{+} γ) (e) \neq 1 .$ Then

$\begin{matrix} (\tilde{g} \tilde{+} \tilde{l}) (e) & = ⋃_{e = q_{1} + s_{1}} {\tilde{g} (q_{1}) \cap \tilde{l} (s_{1})} \\ \subseteq ⋃_{e = q_{1} + s_{1}} {\tilde{g} ({sq}_{1}) \cap \tilde{l} ({ss}_{1})} \\ \subseteq ⋃_{se = q^{'} + s^{'}} {\tilde{g} (q^{'}) \cap \tilde{l} (s^{'})} = (\tilde{g} + \tilde{l}) (se), \\ (β \tilde{+} γ) (e) & = ⋀_{e = q_{1} + s_{1}} {β (q_{1}) \lor γ (s_{1})} \\ \geq ⋀_{e = q_{1} + s_{1}} {β ({sq}_{1}) \lor γ ({ss}_{1})} \\ \geq ⋀_{se = q^{'} + s^{'}} {β (q^{'}) \lor γ (s^{'})} = (β \tilde{+} γ) (se) . \end{matrix}$

Hence ${\tilde{g}}_{β} \oplus {\tilde{l}}_{γ}$ is hybrid left ideal.

Similarly, one can prove ${\tilde{g}}_{β} ⊙ {\tilde{l}}_{γ}$ is hybrid left ideal.

We now prove ${\tilde{g}}_{β} ⋒ {\tilde{l}}_{γ}$ is a hybrid left ideal in $ℝ .$ Let $q, w \in ℝ .$ Then $\begin{matrix} (\tilde{g} \tilde{\cap} \tilde{l}) (q + & w) = \tilde{g} (q + w) \cap \tilde{l} (q + w) \\ \supseteq (\tilde{g} (q) \cap \tilde{g} (w)) \cap (\tilde{l} (q) \cap \tilde{l} (w)) \\ = (\tilde{g} (q) \cap \tilde{l} (q)) \cap (\tilde{g} (w) \cap \tilde{l} (w)) \\ = (\tilde{g} \tilde{\cap} \tilde{l}) (q) \cap (\tilde{g} \tilde{\cap} \tilde{l}) (w), \end{matrix}$ $\begin{matrix} (β \lor γ) (q + & w) = β (q + w) \lor γ (q + w) \\ \leq {β (q) \lor β (w)} \lor {γ (q) \lor γ (w)} \\ = {β (q) \lor γ (q)} \lor {β (w) \lor γ (w)} \\ = (β \lor γ) (q) \lor (β \lor γ) (w) . \end{matrix}$ Also $\begin{matrix} (\tilde{g} \tilde{\cap} \tilde{l}) (qw) & = \tilde{g} (qw) \cap \tilde{l} (qw) \\ \supseteq \tilde{g} (w) \cap \tilde{l} (w) = (\tilde{g} \tilde{\cap} \tilde{l}) (w), \\ (β \lor γ) (qw) & = β (qw) \lor γ (qw) \\ \leq β (w) \lor γ (w) = (β \lor γ) (w) . \end{matrix}$

So ${\tilde{g}}_{β} ⋒ {\tilde{l}}_{γ}$ is hybrid left ideal.

(ii) Let ${\tilde{g}}_{β}$ and ${\tilde{l}}_{γ}$ be two hybrid left k-ideals of $ℝ$ , and let $q, u, e_{1} \in ℝ$ such that q + u = e₁ . Then $\begin{matrix} (\tilde{g} \tilde{\cap} \tilde{l}) (q) & = \tilde{g} (q) \cap \tilde{l} (q) \\ \supseteq (\tilde{g} (u) \cap \tilde{g} (e_{1})) \cap (\tilde{l} (u) \cap \tilde{l} (e_{1})) \\ = (\tilde{g} (u) \cap \tilde{l} (u)) \cap (\tilde{g} (e_{1}) \cap \tilde{l} (e_{1})) \\ = (\tilde{g} \tilde{\cap} \tilde{l}) (u) \cap (\tilde{g} \tilde{\cap} \tilde{l}) (e_{1}), \\ (β \lor γ) (q) & = β (q) \lor γ (q) \\ \leq {β (u) \lor β (e_{1})} \lor {γ (u) \lor γ (e_{1})} \\ = {β (u) \lor γ (u)} \lor {β (e_{1}) \lor γ (e_{1})} \\ = (β \lor γ) (u) \lor (β \lor γ) (e_{1}) . \end{matrix}$

So ${\tilde{g}}_{β} ⋒ {\tilde{l}}_{γ}$ is hybrid left k-ideal.

In general, for any hybrid k- ideals ${\tilde{l}}_{β}$ and ${\tilde{f}}_{γ}$ of $ℝ,$ ${\tilde{l}}_{β} \oplus {\tilde{f}}_{γ}$ ; ${\tilde{l}}_{β} ⊙ {\tilde{f}}_{γ}$ and ${\tilde{l}}_{β} ⊙_{k} {\tilde{f}}_{γ}$ are not necessarily hybrid k-ideals of $ℝ,$ as shows in the below example.

Example 3.12. Let N₀ represent a set of non-negative integers. Then N₀ forms a semiring in terms of usual addition and multiplication. For Q ≠ {φ} , define the hybrid structures ${\tilde{j}}_{β}, {\tilde{l}}_{γ} \in H (N_{0})$ as follows: For $e \in ℝ,$ $\begin{matrix} \tilde{j} (e) = {\begin{matrix} Q & if e = 2 x for x \in N_{0} \\ φ & otherwise, \end{matrix} \\ \tilde{l} (e) = {\begin{matrix} Q & if e = 3 x for x \in N_{0} \\ φ & otherwise \end{matrix} \end{matrix}$ and β, γ are constant mappings. Then ${\tilde{j}}_{β}$ and ${\tilde{l}}_{γ}$ are hybrid k-ideals of N₀.

(i) Since 3 = 2 +1, but $φ = (\tilde{j} + \tilde{l}) (1) ⊉ (\tilde{j} \tilde{+} \tilde{l}) (2) \cap (\tilde{j} \tilde{+} \tilde{l}) (3) = Q$ and $φ = (\tilde{j} {\tilde{\circ}}_{k} \tilde{l}) (1) ⊉ (\tilde{j} {\tilde{\circ}}_{k} \tilde{l}) (2) \cap (\tilde{j} {\tilde{\circ}}_{k} \tilde{l}) (3) = Q,$ so ${\tilde{j}}_{β} \oplus {\tilde{l}}_{γ}$ and ${\tilde{j}}_{β} ⊙_{k} {\tilde{l}}_{γ}$ are not hybrid k-ideals of N₀ .

(ii) Since 3 + 9 =12, but $φ = (\tilde{j} \tilde{\circ} \tilde{l}) (3) ⊉ (\tilde{j} \tilde{\circ} \tilde{l}) (9) \cap (\tilde{j} \tilde{\circ} \tilde{l}) (12) = Q,$ so ${\tilde{j}}_{β} ⊙ {\tilde{l}}_{γ}$ is not a hybrid k-ideal of N₀.

Proposition 3.13. For ${\tilde{l}}_{λ} \in H (ℝ),$ the assertions mentioned below are equivalent:

(i) ${\tilde{l}}_{λ}$ satisfies

(a) $(\begin{matrix} \tilde{l} (w + q) \supseteq \tilde{l} (w) \cap \tilde{l} (q) \\ λ (w + q) \leq λ (w) \lor λ (q) \end{matrix})$ and

(b) $w + s = t \Rightarrow (\begin{matrix} \tilde{l} (w) \supseteq \tilde{l} (s) \cap \tilde{l} (t) \\ λ (w) \leq λ (s) \lor λ (t) \end{matrix}),$

(ii) ${\tilde{l}}_{λ} \oplus_{k} {\tilde{l}}_{λ} ⪡ {\tilde{l}}_{λ} .$

Proof. (i) ⇒ (ii) Let $w \in ℝ .$ Then

$\begin{matrix} (\tilde{l} & {\tilde{+}}_{k} \tilde{l}) (w) \\ = ⋃_{w + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {\tilde{l} (e_{1}) \cap \tilde{l} (s_{2}) \cap \tilde{l} (e_{1}^{'}) \cap \tilde{l} (s_{2}^{'})} \\ \subseteq ⋃_{w + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {\tilde{l} (e_{1} + s_{2}) \cap \tilde{l} (e_{1}^{'} + s_{2}^{'})} \\ \subseteq \tilde{l} (w), \\ (λ & {\tilde{+}}_{k} λ) (w) \\ = ⋀_{w + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {λ (e_{1}) \lor λ (s_{2}) \\ \lor λ (e_{1}^{'}) \lor λ (s_{2}^{'})} \\ \geq ⋀_{w + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {λ (e_{1} + s_{2}) \lor λ (e_{1}^{'} + s_{2}^{'})} \\ \geq λ (w) . \end{matrix}$ Therefore ${\tilde{l}}_{λ} \oplus_{k} {\tilde{l}}_{λ} ⪡ {\tilde{l}}_{λ} .$

(ii) ⇒ (i) We first prove that $\tilde{l} (0) \supseteq \tilde{l} (s)$ and λ (0) ≤ λ (s) $\forall s \in ℝ .$

Let $s \in ℝ .$ Then $\begin{matrix} \tilde{l} (0) & \supseteq (\tilde{l} {\tilde{+}}_{k} \tilde{l}) (0) \\ = ⋃_{0 + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {\tilde{l} (e_{1}) \cap \tilde{l} (s_{2}) \\ \cap \tilde{l} (e_{1}^{'}) \cap \tilde{l} (s_{2}^{'})} \\ \supseteq \tilde{l} (s) \cap \tilde{l} (s) \cap \tilde{l} (s) \cap \tilde{l} (s) = \tilde{l} (s), \end{matrix}$ $\begin{matrix} λ (0) & \leq (λ {\tilde{+}}_{k} λ) (0) \\ = ⋀_{0 + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {λ (e_{1}) \lor λ (s_{2}) \\ \lor λ (e_{1}^{'}) \lor λ (s_{2}^{'})} \\ \leq λ (s) \lor λ (s) \lor λ (s) \lor λ (s) = λ (s) . \end{matrix}$

Now, for $s, q \in ℝ,$

$\begin{matrix} \tilde{l} (s + q) & \supseteq (\tilde{l} {\tilde{+}}_{k} \tilde{l}) (s + q) \\ = ⋃_{s + q + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {\tilde{l} (e_{1}) \cap \tilde{l} (s_{2}) \\ \cap \tilde{l} (e_{1}^{'}) \cap \tilde{l} (s_{2}^{'})} \\ \supseteq \tilde{l} (0) \cap \tilde{l} (0) \cap \tilde{l} (s) \cap \tilde{l} (q) \\ = \tilde{l} (s) \cap \tilde{l} (q), \\ λ (s + q) & \leq (λ \tilde{+_{k}} λ) (s + q) \\ = ⋀_{s + q + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {λ (e_{1}) \lor λ (s_{2}) \\ \lor λ (e_{1}^{'}) \lor λ (s_{2}^{'})} \\ \leq λ (0) \lor λ (0) \lor λ (s) \lor λ (q) \\ = λ (s) \lor λ (q) . \end{matrix}$

If s + s₁ = t, then s + s₁ + 0 = t + 0, so

$\begin{matrix} \tilde{l} (s) & \supseteq (\tilde{l} {\tilde{+}}_{k} \tilde{l}) (s) \\ = ⋃_{s + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {\tilde{l} (e_{1}) \cap \tilde{l} (s_{2}) \\ \cap \tilde{l} (e_{1}^{'}) \cap \tilde{l} (s_{2}^{'})} \\ \supseteq \tilde{l} (s_{1}) \cap \tilde{l} (0) \cap \tilde{l} (t) \cap \tilde{l} (0) = \tilde{l} (s_{1}) \cap \tilde{l} (t), \\ λ (s) & \leq (λ {\tilde{+}}_{k} λ) (s) \\ = ⋀_{s + (e_{1} + s_{2}) = e_{1}^{'} + s_{2}^{'}} {λ (e_{1}) \lor λ (s_{2}) \\ \lor λ (e_{1}^{'}) \lor λ (s_{2}^{'})} \\ \leq λ (s_{1}) \lor λ (0) \lor λ (t) \lor λ (0) = λ (s_{1}) \lor λ (t) . \end{matrix}$

Proposition 3.14. For ${\tilde{f}}_{λ} \in H (ℝ),$ the assertions mentioned below are equivalent:

(i) For any ${\tilde{h}}_{ν} \in H (ℝ),$

(a) ${\tilde{j}}_{λ} \oplus_{k} {\tilde{j}}_{λ} ⪡ {\tilde{j}}_{λ},$

(b) $χ_{ℝ} ({\tilde{h}}_{ν}) ⊙_{k} {\tilde{j}}_{λ} ⪡ {\tilde{j}}_{λ}$ (resp., ${\tilde{j}}_{λ} ⊙_{k} χ_{ℝ} ({\tilde{h}}_{ν}) ⪡ {\tilde{j}}_{λ}),$

(ii) ${\tilde{j}}_{λ}$ is hybrid left (resp., right) k-ideal.

Proof. Assume that the condition (i) holds. To prove ${\tilde{j}}_{λ}$ is a hybrid left k-ideal of $ℝ$ , by Proposition 3.13, it is enough to conclude that the conditions $\tilde{j} (pq) \supseteq \tilde{j} (q)$ and λ (pq) ≤ λ (q) are true.

Let $q, p \in ℝ .$ Then $\begin{matrix} \tilde{j} (pq) \supseteq (χ_{ℝ} (\tilde{h}) {\tilde{\circ}}_{k} \tilde{j}) (pq) \\ = ⋃_{pq + s_{1} e_{1} = s_{2} e_{2}} {χ_{ℝ} (\tilde{h}) (s_{1}) \cap \tilde{j} (e_{1}) \\ \cap χ_{ℝ} (\tilde{h}) (s_{2}) \cap \tilde{j} (e_{2})} \\ \supseteq ⋃_{pq + s_{1} e_{1} = s_{2} e_{2}} {\tilde{j} (s_{1} e_{1}) \cap \tilde{j} (s_{2} e_{2})} \\ \supseteq \tilde{j} (q) \cap \tilde{j} (q) since pq + 0 q = pq \\ = \tilde{j} (q), \end{matrix}$

$\begin{matrix} λ (pq) & \leq (χ_{ℝ} (ν) {\tilde{\circ}}_{k} λ) (pq) \\ = ⋀_{pq + s_{1} e_{1} = s_{2} e_{2}} {χ_{ℝ} (ν) (s_{1}) \lor λ (e_{1}) \\ \lor χ_{ℝ} (ν) (s_{2}) \lor λ (e_{2})} \\ \leq ⋀_{pq + s_{1} e_{1} = s_{2} e_{2}} {λ (s_{1} e_{1}) \lor λ (s_{2} e_{2})} \\ \leq λ (q) \lor λ (q) since pq + 0 q = pq \\ = λ (q) . \end{matrix}$

Hence ${\tilde{j}}_{λ}$ is hybrid left k-ideal.

(ii)⇒ (i) For a hybrid left k-ideal ${\tilde{j}}_{λ}$ of $ℝ$ , by Proposition 3.13, ${\tilde{j}}_{λ} \oplus_{k} {\tilde{j}}_{λ} ⪡ {\tilde{j}}_{λ} .$

We now prove (b). Let $c \in ℝ$ and ${\tilde{h}}_{ν} \in H (ℝ) .$

If c + s₁e₁ ≠ s₂e₂ for any $e_{1}, s_{1}, e_{2}, s_{2} \in ℝ,$ then $(χ_{ℝ} (\tilde{h}) \circ_{k} \tilde{j}) (c) = φ \subseteq \tilde{j} (c)$ and $(χ_{ℝ} (ν) \circ_{k} λ) (c) = 1 \leq λ (c) .$

Otherwise c + s₁e₁ = s₂e₂ for some $e_{1}, e_{2}, s_{1}, s_{2} \in ℝ .$ Then

$\begin{matrix} (χ_{ℝ} (\tilde{h}) {\tilde{\circ}}_{k} \tilde{j}) & (c) \\ = ⋃_{c + s_{1} e_{1} = s_{2} e_{2}} {χ_{ℝ} (\tilde{h}) (s_{1}) \cap \tilde{j} (e_{1}) \\ \cap χ_{ℝ} (\tilde{h}) (s_{2}) \cap \tilde{j} (e_{2})} \\ \subseteq ⋃_{c + s_{1} e_{1} = s_{2} e_{2}} {\tilde{j} (s_{1} e_{1}) \cap \tilde{j} (s_{2} e_{2})} \\ \subseteq ⋃_{c + s_{1} e_{1} = s_{2} e_{2}} \tilde{j} (c) = \tilde{j} (c), \\ (χ_{ℝ} (ν) {\tilde{\circ}}_{k} λ) (c) \\ = ⋀_{c + s_{1} e_{1} = s_{2} e_{2}} {χ_{ℝ} (ν) (s_{1}) \lor λ (e_{1}) \\ \lor χ_{ℝ} (ν) (s_{2}) \lor λ (e_{2})} \\ \geq ⋀_{c + s_{1} e_{1} = s_{2} e_{2}} {λ (s_{1} e_{1}) \lor λ (s_{2} e_{2})} \\ \geq ⋀_{c + s_{1} e_{1} = s_{2} e_{2}} λ (c) = λ (c) . \end{matrix}$

Therefore $χ_{ℝ} ({\tilde{h}}_{ν}) ⊙_{k} {\tilde{j}}_{λ} ⪡ {\tilde{j}}_{λ} .$

Proposition 3.15. [10] For a right k-ideal Q₁ and a left k-ideal Q₂ of $ℝ,$ the below assertions are hold:

(i) $\bar{Q_{1} Q_{2}} \supseteq Q_{1} \cap Q_{2} .$

(ii) $\bar{Q_{1} Q_{2}} = Q_{1} \cap Q_{2}$ if and only if $ℝ$ is k-regular.

Proposition 3.16. For ${\tilde{j}}_{ν}, {\tilde{h}}_{λ} \in H (ℝ),$ the below assertions are equivalent:

(i) $ℝ$ is k-regular,

(ii) If ${\tilde{j}}_{ν}$ and ${\tilde{h}}_{λ}$ are respectively hybrid right k-ideal and hybrid left k-ideal of $ℝ,$ then ${\tilde{j}}_{ν} ⋒ {\tilde{h}}_{λ} = {\tilde{j}}_{ν} ⊙_{k} {\tilde{h}}_{λ} .$

Proof. Consider that $ℝ$ is k-regular. By Proposition 3.10(ii) , ${\tilde{j}}_{ν} ⊙_{k} {\tilde{h}}_{λ} ⪡ {\tilde{j}}_{ν} ⋒ {\tilde{h}}_{λ} .$ Let $t \in ℝ .$ Then $\exists x_{1}, x_{2} \in ℝ : t + {tx}_{1} t = {tx}_{2} t$ .

Now $\begin{matrix} (\tilde{j} {\tilde{\circ}}_{k} \tilde{h}) (t) \\ = ⋃_{t + k_{1} e_{1} = k_{2} e_{2}} {\tilde{j} (k_{1}) \cap \tilde{h} (e_{1}) \cap \tilde{j} (k_{2}) \cap \tilde{h} (e_{2})} \\ \supseteq {\tilde{j} ({tx}_{1}) \cap \tilde{h} (t) \cap \tilde{j} ({tx}_{2}) \cap \tilde{h} (t)} \\ \supseteq \tilde{j} (t) \cap \tilde{h} (t) = (\tilde{j} \cap \tilde{h}) (t), \end{matrix}$ $\begin{matrix} (ν {\tilde{\circ}}_{k} λ) (t) \\ = ⋀_{e + s_{1} e_{1} = s_{2} e_{2}} {ν (s_{1}) \lor ν (s_{2}) \lor λ (e_{1}) \lor λ (e_{2})} \\ \leq {ν ({tx}_{1}) \lor λ (t) \lor ν ({tx}_{2}) \lor λ (t)} \\ \leq ν (t) \lor λ (t) = (ν \lor λ) (t) . \end{matrix}$ Thus ${\tilde{j}}_{ν} ⋒ {\tilde{h}}_{λ} ⪡ {\tilde{j}}_{ν} ⊙_{k} {\tilde{h}}_{λ}$ and hence ${\tilde{j}}_{ν} ⊙_{k} {\tilde{h}}_{λ} = {\tilde{j}}_{ν} ⋒ {\tilde{h}}_{λ} .$

Assume, on the other hand, the condition (ii) is true. For a right k-ideal A and a left k-ideal B of $ℝ,$ by Proposition 2.2 of [8], $χ_{A} ({\tilde{j}}_{ν})$ is a hybrid right k-ideal and $χ_{B} ({\tilde{h}}_{λ})$ is a hybrid left k-ideal of $ℝ .$ Clearly $\bar{AB} \subseteq A \cap B .$

Let e ∈ A ∩ B . Then $χ_{A} (\tilde{j}) (e) = 𝕩 = χ_{B} (\tilde{h}) (e)$ and χ_A (ν) (e) =0 = χ_B (λ) (e) . Thus $\begin{matrix} (χ_{A} (\tilde{j}) {\tilde{\circ}}_{k} χ_{B} (\tilde{h})) (e) = χ_{A} (\tilde{j}) (e) \cap χ_{B} (\tilde{h}) (e) = 𝕩, \\ (χ_{A} (ν) {\tilde{\circ}}_{k} χ_{B} (λ)) (e) = χ_{A} (ν) (e) \lor χ_{B} (λ) (e) = 0 . \end{matrix}$ This imply $⋂_{i = 1, 2} {χ_{A} (\tilde{j}) (x_{i}), χ_{B} (\tilde{h}) (y_{i})} = 𝕩$ and ⋁_i=1,2 {χ_A (ν) (x_i) , χ_B (λ) (y_i)} =0 for some $x_{i}, y_{i} \in ℝ$ satisfying e + x₁y₁ = x₂y₂ .

Since $χ_{A} (\tilde{j}) (x_{i}) = 𝕩 = χ_{B} (\tilde{h}) (y_{i})$ and χ_A (ν) (x_i) =0 = χ_B (λ) (y_i) , we have x_i ∈ A and y_i ∈ B . Thus $e \in \bar{AB}$ and hence $A \cap B = \bar{AB} .$ By Proposition 3.15(ii), $ℝ$ is k-regular.

Definition 3.17. A semiring $ℝ$ is described as left (resp., right) k-weakly regular if $q \in ℝ, \exists q_{1}, s_{1}, q_{1}^{'}, s_{1}^{'} \in ℝ : q + q_{1} {qs}_{1} q = q_{1}^{'} {qs}_{1}^{'} q$ (resp., $q + {qq}_{1} {qs}_{1} = {qq}_{1}^{'} {qs}_{1}^{'}) .$

Proposition 3.18. For a semiring $ℝ,$ the assertions mentioned below are equivalent:

(i) $ℝ$ is a left k-weakly regular,

(ii) For any hybrid left k-ideal ${\tilde{l}}_{β}$ of $ℝ,$ ${\tilde{l}}_{β} ⊙_{k} {\tilde{l}}_{β} = {\tilde{l}}_{β},$

(iii) For any hybrid left k-ideal ${\tilde{l}}_{β}$ and hybrid k-ideal ${\tilde{j}}_{μ}$ of $ℝ,$ ${\tilde{j}}_{μ} ⊙_{k} {\tilde{l}}_{β} = {\tilde{j}}_{μ} ⋒ {\tilde{l}}_{β} .$

Proof. (i) ⇒ (ii) For any hybrid left k-ideal ${\tilde{l}}_{β}$ of $ℝ,$ ${\tilde{l}}_{β} ⊙_{k} {\tilde{l}}_{β} ⪡ {\tilde{l}}_{β} .$

Let $w \in ℝ .$ Then $\exists q_{1}, e_{1}, q_{1}^{'}, e_{1}^{'} \in ℝ : w + q_{1} {we}_{1} w = q_{1}^{'} {we}_{1}^{'} w .$ Hence $\tilde{l} (w) = \tilde{l} (w) \cap \tilde{l} (w) \subseteq \tilde{l} (q_{1} w) \cap \tilde{l} (e_{1} w),$ β (w) = β (w) ∨ β (w) ≥ β (q₁w) ∨ β (e₁w). and

$\tilde{l} (w) = \tilde{l} (w) \cap \tilde{l} (w) \subseteq \tilde{l} (q_{1}^{'} w) \cap \tilde{l} (e_{1}^{'} w),$ $β (w) = β (w) \lor β (w) \geq β (q_{1}^{'} w) \lor β (e_{1}^{'} w)$ .

Now, $\begin{matrix} \tilde{l} (w) & \subseteq {\tilde{l} (q_{1} w) \cap \tilde{l} (e_{1} w)} \cap {\tilde{l} (q_{1}^{'} w) \cap \tilde{l} (e_{1}^{'} w)} \\ \subseteq ⋂_{w + q_{1} {we}_{1} w = q_{1}^{'} {we}_{1}^{'} w} {{\tilde{l} (q_{1} w) \cap \tilde{l} (e_{1} w)} \\ \cap {\tilde{l} (q_{1}^{'} w) \cap \tilde{l} (e_{1}^{'} w)}} \\ = (\tilde{l} {\tilde{\circ}}_{k} \tilde{l}) (w), \\ β (w) & \geq {β (q_{1} w) \lor β (e_{1} w)} \lor {β (q_{1}^{'} w) \lor β (e_{1}^{'} w)} \\ \geq ⋁_{w + q_{1} {we}_{1} w = q_{1}^{'} {we}_{1}^{'} w} {{β (q_{1} w) \lor β (e_{1} w)} \\ \lor {β (q_{1}^{'} w) \lor β (e_{1}^{'} w)}} \\ = (β {\tilde{\circ}}_{k} β) (w) . \end{matrix}$

Thus ${\tilde{l}}_{β} ⊙_{k} {\tilde{l}}_{β} ⪢ {\tilde{l}}_{β}$ and hence ${\tilde{l}}_{β} ⊙_{k} {\tilde{l}}_{β} = {\tilde{l}}_{β} .$

(ii) ⇒ (iii) For a hybrid left k-ideal ${\tilde{l}}_{β}$ and a hybrid k-ideal ${\tilde{f}}_{μ},$ ${\tilde{f}}_{μ} ⋒ {\tilde{l}}_{β}$ is a hybrid left k-ideal of $ℝ .$ By assumption, ${\tilde{f}}_{μ} ⋒ {\tilde{l}}_{β} = ({\tilde{f}}_{μ} ⋒ {\tilde{l}}_{β}) ⊙_{k} ({\tilde{f}}_{μ} ⋒ {\tilde{l}}_{β}) ⪡ {\tilde{f}}_{μ} ⊙_{k} {\tilde{l}}_{β} .$ Hence ${\tilde{f}}_{μ} ⋒ {\tilde{l}}_{β} = {\tilde{f}}_{μ} ⊙_{k} {\tilde{l}}_{β} .$

(iii) ⇒ (i) For a left k-ideal Q and k-ideal S of $ℝ,$ $χ_{Q} ({\tilde{l}}_{λ})$ and $χ_{S} ({\tilde{l}}_{λ})$ are hybrid left and hybrid ideals of $ℝ,$ respectively.

By assumption and Lemma 3.6, $\begin{matrix} χ_{Q} ({\tilde{l}}_{λ}) ⊙_{k} χ_{S} ({\tilde{l}}_{λ}) = χ_{Q} ({\tilde{l}}_{λ}) ⋒ χ_{S} ({\tilde{l}}_{λ}) \\ \Rightarrow χ_{\bar{QS}} ({\tilde{l}}_{λ}) = χ_{\bar{Q \cap S}} ({\tilde{l}}_{λ}) \\ \Rightarrow \bar{QS} = Q \cap S . \end{matrix}$ Then by Proposition 3.2 of [27], $ℝ$ is left k-weakly regular semiring.

Conclusion

We introduced and investigated the properties of hybrid k-sum and hybrid k-product of k-ideals in semirings in this paper. The properties of semirings were investigated by determining the relationships between hybrid k-sum and hybrid k-product of hybrid k-left (respectively, right) ideals. Future research will describe the concept of hybrid bi-ideals and hybrid k-bi-ideals in semirings and obtain their different properties and equivalent conditions for a hybrid bi-ideal to be a hybrid ideal in semirings using the concepts and findings of this paper.

Footnotes

Acknowledgments

The authors express their sincere thanks to the referees for their valuable comments and suggestions, which improved the paper a lot.

References

Ahsan

, Mordeson

J.N.

, Shabir

Fuzzy Semirings with Applications to Automata Theory. Studies in Fuzziness and Soft Computing 278. Springer, Berlin, Heidelberg, 2012.

Anis

, Khan

and Jun

Y.B.

, Hybrid ideals in semigroups, Cogent Mathematics 4 (2017), 1352117.

Baik

S.I.

and Kim

H.S.

, On fuzzy k-ideals in semirings, Kangweon-Kyungki Math Jour 8(2) (2000), 147–154.

Dixit

V.N.

, Kumar

and Ajmal

, On fuzzy rings, Fuzzy Sets and Systems 49(2) (1992), 205–213.

Elavarasan

, Jun

Y.B.

On hybrid ideals and hybrid biideals in semigroups, Iran J Math Sci Inform (to appear).

Elavarasan

and Jun

Y.B.

, Regularity of semigroups in terms of hybrid ideals and hybrid bi-ideals, Kragujev J Math 46(6) (2022), 857–864.

Elavarasan

, Muhiuddin

, Porselvi

and Jun

Y.B.

, Hybrid structures applied to ideals in near-rings, Complex Intell Syst 7(3) (2021), 1489–1498.

Elavarasan

and Jun

Y.B.

, Hybrid ideals in semirings, Advances in Mathematics: Scientific Journal 9(3) (2020), 1349–1357.

Elavarasan

, Porselvi

and Jun

Y.B.

, Hybrid generalized biideals in semigroups, International Journal of Mathematics and Computer Science 14(3) (2019), 601–612.

10.

Ghosh

Fuzzy k—ideals of semirings, Fuzzy Sets and Systems (95) (1998), 103—108.

11.

Jun

Y.B.

, Song

S.Z.

and Muhiuddin

, Hybrid structures and applications, Annals of Communications in Mathematics 1(1) (2018), 11–25.

12.

Jun

Y.B.

, Öztürk

M.A.

and Song

S.Z.

, On fuzzy h-ideals in hemirings, Inform Sci 162 (2004), 211–226.

13.

Jun

Y.B.

, Sapanci

and Ozturk

M.A

, Fuzzy ideals in gamma near-rings, Tr J of Mathematics 22 (1998), 449–459.

14.

Meenakshi

, Muhiuddin

, Elavarasan

and Al-Kadi

, Hybrid ideals in near-subtraction semigroups, AIMS Mathematics 7(7) (2022), 13493–13507.

15.

Maji

P.K.

, Roy

A.R.

and Biswas

, An application of soft sets in a decision making problem, Comput Math Appl 44 (2002), 1077–1083.

16.

Molodtsov

D.A.

, The description of a dependence with the help of soft sets, Journal of Computer and Systems Sciences International 40(6) (2001), 977–984.

17.

Molodtsov

, Soft set theory - First results, Comput Math Appl 37 (1999), 19–31.

18.

Muhiuddin

, Grace John

J.C.

, Elavarasan

, Jun

Y.B.

and Porselvi

, Hybrid Structures Applied to Modules over Semirings, J Intell Fuzzy Syst 42(3) (2022), 2521–2531.

19.

Muhiuddin

, Elavarasan

, Catherine

, John

, Porselvi

and Al-Kadi

, Properties of k-hybrid ideals in ternary semiring, J Intell Fuzzy Syst 42(6) (2021), 5799–5807.

20.

Muhiuddin

, Harizavi

, Jun

Y.B.

Bipolar-valued fuzzy soft hyper BCK ideals in hyper BCK algebras, Discrete Mathematics Algorithms and Applications, (2019) (Accepted). (ISSN: 1793-8309)

21.

Muhiuddin

, Rehman

and Jun

Y.B.

, A generalization of (∈; ∈ ∨ q)-fuzzy ideals in ternary semigroups, Annals of Communications in Mathematics 2(2) (2019), 73–83.

22.

Muhiuddin

and Aldhafeeri

, Characteristic fuzzy sets and conditional fuzzy subalgebras, J Comput Anal Appl 25(8) (2018), 1398–1409.

23.

Porselvi

, Muhiuddin

, Elavarasan

and Assiry

, Hybridnil radical of a ring, Symmetry 14(7) (2022), 1367.

24.

Porselvi

, Elavarasan

and Jun

Y.B.

, On hybrid interior ideals in ordered semigroups, New Math Nat Comput 18(01) (2022), 1–8.

25.

Porselvi

and Elavarasan

, On hybrid interior ideals in semigroups, Probl Anal Issues Anal 8(26)(3) (2019), 137–146.

26.

Rosenfeld

, Fuzzy groups, J Math Anal Appl 35 (1971), 512–517.

27.

Shabir

and Anjum

, Right k-weakly regular hemirings, Quasigroups and Related Systems 20 (2012), 97–112.

28.

Torra

and Narukawa

, On hesitant fuzzy sets and decision, The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea (2009), 1378–1382in.

29.

Zadeh

L.A.

On hesitant fuzzy sets and decision, in: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, (2009), 1378–1382.

30.

Zhan

and Jun

Y.B.

, Soft BL-algebras based on fuzzy sets, Comput Math Appl 59 (2010), 2037–2046.