Hybrid structures applied to modules over semirings

Abstract

The concept of a hybrid structure in $X$ -semimodules, where $X$ is a semiring, is introduced in this paper. The notions of hybrid subsemimodule and hybrid right (resp., left) ideals are defined and discussed in semirings. We investigate the representations of hybrid subsemimodules and hybrid ideals using hybrid products. We also get some interesting results on t-pure hybrid ideals in $X$ . Furthermore, we show how hybrid products and hybrid intersections are linked. Finally, the characterization theorem is proved in terms of hybrid structures for fully idempotent semirings.

Keywords

Hybrid semirings hybrid X-semimodules fully idempotent hybrid ideals weakly regular

1 Introduction

Semirings are well-known structures that can be found in both mathematics and computer science. Modules over a ring are a useful method for characterising the properties of the ring. Then, as a generalisation of modules over rings, semimodules over semi-rings are normal (See [7 , 16]). Yusuf [34] introduced the inverse semimodule over a semiring in 1966 and discovered numerous inverse semimodule analogues to module theory. Semimodules over semirings are useful in both theoretical and cryptographic computer science [21] and are found in many fields of mathematics. A semimodule over a semiring studied by many researchers (See [1 , 33]). In a similar line to set theory, Zadeh [35] presented the idea of fuzzy subsets and their characteristics in 1965, and it has a wide range of applications in computer engineering, machine learning, control theory, operational science, business administration, robotics, and many other fields. The term “fuzzy subgroups” was introduced by Rosenfeld [31] to describe the concept of fuzzy algebraic structures. Following that, a number of researchers developed algebraic structure and applied it to a variety of pure and applied mathematics fields (See [2–4 , 24]).

Molodtsov [23] established soft set theory as a universal mathematical tool for modelling uncertainty. In the absence of partial information, researchers can select the type of parameters they require, greatly simplifying and increasing efficiency. Probability theory, interval mathematics, fuzzy set theory, and other mathematical tools can also be used to model complex problems. However, each of these methods has its own set of issues. In fuzzy set theory, determining the value of the membership function has always been difficult. Furthermore, none of these techniques have parameterizable tools, so they can not be used to solve problems in areas such as finance, the environment, or social issues.

Jun et al. [17] defined a hybrid structure by combining the concepts of fuzzy sets and soft sets, and it was applied to BCK/BCI-algebras and linear spaces. Anis et al. [5] defined hybrid ideals concepts in semigroups. Many authors developed hybrid concepts in algebraic structures, (See [10–13 , 30]). In this paper, we introduce and investigate the properties of hybrid subsemimodules over semirings, and we obtain some semiring characterizations of the right t-pure hybrid ideals. Furthermore, we present equivalent conditions for a semiring to be completely idempotent.

2 Preliminaries

This section gathers a few ideas and observations that would be helpful in our key findings. We recall Golan’s definition of a semiring from his monograph [14].

Definition 2.1. A semiring is an algebraic structure made up of a nonempty set $X$ and two binary operations (addition and multiplication) that meet the following criteria:

(i) $(X, +)$ is a commutative semigroup with identity element 0 .

(ii) $(X, \cdot)$ is a commutative semigroup with identity element 1 (≠0) .

(iii) n · (u + w) = n · u + n · w and (n + u) . w = n · w + u · w, for all $n, u, w \in X$ .

(iv) 0 . u = u . 0 =0 $\forall u \in X$ .

A ring is obviously a semiring with an additive inverse for each element. A module over a ring is a generalisation of vector space over a field in which the corresponding scalars are elements of an arbitrarily chosen ring (with identity) and the elements of the ring and the elements of the module are multiplied (on the left and/or on the right).

Definition 2.2. For an additive commutative semigroup $M$ with an identity element 0 and a semiring $X$ , a right $X$ -semimodule, $M_{X},$ is a function $φ : M \times X \to M$ such that if φ (m, u) is referred to as mu, then the following conditions must be met:

(i) (m₁ + m₂) u = m₁u + m₂u,

(ii) m₁ (u₁ + u₂) = m₁u₁ + m₁u₂,

(iii) m₁ (u₁u₂) = (m₁u₁) u₂,

(iv) m₁ · 1 = m₁,

(v) 0 · u₁ = m₁ · 0 = 0 $\forall m_{1}, m_{2} \in M$ ; $u_{1}, u_{2} \in X .$

Similarly, a left $X$ -semimodule $X_{M}$ can be established. Clearly every semiring $X$ is a right (resp., left) $X$ semimodule over itself.

In this paper, $X$ denotes a semiring, $M$ a right $X$ -semimodule, and $P (V)$ the power set of a set V.

Definition 2.3. For a right $X$ -module $M$ and $N \in P (M),$ if $k_{1}^{'} + k_{2}^{'} \in N$ and ${uk}_{1}^{'} \in N$ $\forall k_{1}^{'}, k_{2}^{'} \in N$ and $u \in X$ , then N is known as a submodule of $M$ .

Naturally, N has become a right $X$ -module of its own way, with the same addition and scalar multiplication as $M .$

Any ring is clearly a semiring, and thus a left module over a ring $X$ is a left semimodule over $X$ .

Definition 2.4. For $N \in P (X),$ if N is a subsemimodule of $X_{X} ({resp .,}_{X} X)$ , then N is called as a right (resp., left) ideal of $X .$

If N is both a left and a right ideal of $X,$ then N is called an ideal of $X .$

Definition 2.5. An additive idempotent element s of a semiring $X$ is one that satisfies the condition “s+s=s”.

If every element s of $X$ satisfies the condition s + s = s, then $X$ is referred to as an idempotent semiring.

Definition 2.6. A semiring $X$ is referred to as a weakly regular if for $s \in X$ ,

$\exists a_{1}, a_{2} \in X$ : s = sa₁sa₂ .

3 Hybrid subsemimodule

Definition 3.1. [5] Let I = [0, 1] and $V$ be the universal set. For a nonempty set Q, a mapping that defines a hybrid structure in Q over $V$ is as follows: ${\tilde{g}}_{δ} : = (\tilde{g}, δ) : Q \to P (V) \times I, q \mapsto (\tilde{g} (q), δ (q))$

where $\tilde{g} : Q \to P (V)$ and δ : Q → I are mappings. The set of all hybrid structures in Q over $V$ is denoted by $ℍ (Q)$ . Define an order relation ⪡ on $ℍ (Q)$ as follows: $(\forall {\tilde{g}}_{σ}, {\tilde{ψ}}_{ν} \in ℍ (Q)) ({\tilde{g}}_{σ} ⪡ {\tilde{ψ}}_{ν} \Leftrightarrow \tilde{g} \tilde{\subseteq} \tilde{ψ}, σ \geq ν)$ where $\tilde{g} \tilde{\subseteq} \tilde{ψ}$ means $\tilde{g} (q) \subseteq \tilde{ψ} (q)$ and σ ≥ ν means that σ (q) ≥ ν (q) ∀q ∈ Q . $(ℍ (Q), ⪡)$ is clearly a poset.

Definition 3.2. Let ${\tilde{h}}_{δ} \in ℍ (Q)$ . Then for $S \in P (V)$ and t ∈ I, the set $\begin{matrix} {\tilde{h}}_{δ} [S, t] & : = {q \in Q ∣ \tilde{h} (q) \supseteq S and δ (q) \leq t} \end{matrix}$ is referred to as [S, t] -hybrid cut of ${\tilde{h}}_{δ}$ .

Definition 3.3. Let ${\tilde{h}}_{δ}, {\tilde{g}}_{λ} \in ℍ (Q)$ . Then denotes the hybrid intersection of ${\tilde{h}}_{δ}$ and ${\tilde{g}}_{λ}$ and its hybrid structure defined as ${\tilde{h}}_{δ} ⋒ {\tilde{g}}_{λ} : Q \to P (V) \times I, q \mapsto ((\tilde{h} \tilde{\cap} \tilde{g}) (q), (δ \lor λ) (q))$ where $\begin{matrix} \tilde{h} \tilde{\cap} \tilde{g} : Q \to P (V), q \mapsto \tilde{h} (q) \cap \tilde{g} (q), \\ δ \lor λ : Q \to I, q \mapsto δ (q) \lor λ (q) . \end{matrix}$

Definition 3.4. Let ${\tilde{h}}_{δ} \in ℍ (M)$ . ${\tilde{h}}_{δ}$ is called a hybrid subsemimodule of a right semimodule $M_{X}$ if the below conditions hold:

(i) $(\forall q, z \in M) (\begin{matrix} \tilde{h} (q + z) \supseteq \tilde{h} (q) \cap \tilde{h} (z) \\ δ (q + z) ⩽ δ (q) \lor δ (z) \end{matrix}) .$

(ii) $(\forall r \in X; q \in M$ ) $(\begin{matrix} \tilde{h} (qr) \supseteq \tilde{h} (q) \\ δ (qr) ⩽ δ (q) \end{matrix}) .$

It is clear that for any hybrid subsemimodule ${\tilde{h}}_{δ}$ , we have $\tilde{h} (0_{M}) \supseteq \tilde{h} (m)$ and $δ (0_{M}) \leq δ (m)$ for any $m \in M .$

Example 3.5. Consider the semiring

X = {0, 1, 2, 3}

with two binary operations + and . on

X

, which are described in the tables below.

+	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

.	1	2	3
0	0	0	0
1	1	2	3
2	2	0	2
3	3	2	1

Define a hybrid structure ${\tilde{ψ}}_{σ}$ in the semiring $X$ over $V = {u_{1}, u_{2}, u_{3}, u_{4}}$ as follows:

$X$	$\tilde{ψ}$	σ
0	$V$	0
1	{u₁}	0.9
2	{u₁, u₃}	0.7
3	{u₁, u₂, u₃}	0.5

Then $M = {0, 2}$ is a right $X$ -semimodule and ${\tilde{ψ}}_{σ}$ is a hybrid subsemimodule in $M$ over $V$ .

Definition 3.6. Let ${\tilde{h}}_{δ} \in ℍ (X) .$ If ${\tilde{h}}_{δ}$ is a hybrid subsemimodule of right semimodule of $X_{X},$ then ${\tilde{h}}_{δ}$ is referred to as a hybrid right ideal in $X$ over $V .$ If ${\tilde{h}}_{δ}$ is a hybrid subsemimodule of left semimodule of $X_{} X,$ then ${\tilde{h}}_{δ}$ is referred to as a hybrid left ideal in $X$ over $V .$ If ${\tilde{h}}_{δ}$ is both a hybrid left and a hybrid right ideal in $X$ , then ${\tilde{h}}_{δ}$ is referred to as a hybrid ideal in $X$ .

Let ${\tilde{h}}_{δ} \in ℍ (M)$ and ${\tilde{g}}_{λ} \in ℍ (X) .$ For $m \in M,$ if m = bc for $b \in M$ and $c \in X,$ then $\tilde{h} (b) \cap \tilde{g} (c) \in P (V)$ and δ (b) ∨ λ (c) ∈ I .

Theorem 3.7. Let ${\tilde{h}}_{δ} \in ℍ (M)$ . The below statements are equivalent:

(a) ${\tilde{h}}_{δ}$ is a hybrid subsemimodule of $M,$

(b) For $S \in P (V)$ and t ∈ I, ${\tilde{h}}_{δ} [S, t] (\neq φ)$ is a subsemimodule of $M .$

Proof. Let $q, z \in {\tilde{h}}_{δ} [S, t] .$ Then $\tilde{h} (q + z) \supseteq \tilde{h} (q) \cap \tilde{h} (z) \supseteq S$ and δ (q + z) ≤ δ (q) ∨ δ (z) ≤ t . So $q + z \in {\tilde{h}}_{δ} [S, t] .$

Also for $r \in X, \tilde{h} (qr) \supseteq \tilde{h} (q) \supseteq S$ and δ (qr) ≤ δ (q) ≤ t .

Therefore ${\tilde{h}}_{δ} [S, t]$ is a subsemimodule of $M .$

In the other hand, suppose that for any $S \in P (V)$ and t ∈ I, ${\tilde{h}}_{δ} [S, t] (\neq φ)$ is a subsemimodule of $M_{X}$ and let $x, z \in M .$ Then $\tilde{h} (x) = Q_{1}; δ (x) = t_{1}$ and $\tilde{h} (z) = Q_{2}; δ (z) = t_{2}$ for some $Q_{1}, Q_{2} \in P (V)$ and t₁, t₂ ∈ I .

If Q : = Q₁ ∩ Q₂ and t : = t₁ ∨ t₂, then $\tilde{h} (x) \supseteq Q; δ (x) \leq t$ and $\tilde{h} (z) \supseteq Q; δ (z) \leq t,$ so $x, z \in {\tilde{h}}_{δ} [S, t] .$ Since ${\tilde{h}}_{δ} [Q, t]$ is subsemimodule, we have $x + z \in {\tilde{h}}_{δ} [S, t]$ which gives $\tilde{h} (x + z) \supseteq Q = Q_{1} \cap Q_{2} = \tilde{h} (x) \cap \tilde{h} (z)$ and δ (x + z) ≤ t = t₁ ∨ t₂ ⩽ δ (x) ∨ δ (z) .

Also for $r \in X,$ we have $xr \in {\tilde{h}}_{δ} [Q_{1}, t_{1}]$ which implies $\tilde{h} (xr) \supseteq Q_{1} = \tilde{h} (x)$ and δ (xr) ≤ t₁ = δ (x) . So ${\tilde{h}}_{δ}$ is a hybrid subsemimodule of $M .$

As a result of the preceding result, we have the below corollary.

Corollary 3.8. Let $B (\neq φ) \in P (M)$ and ${\tilde{h}}_{δ} \in ℍ (M)$ be a hybrid structure in $M$ characterized as follows:

For $S, V \in P (V)$ and ω, t ∈ I, $\begin{matrix} \tilde{h} (z) : = {\begin{matrix} S & if z \in B \\ V & otherwise, \end{matrix} δ (z) : = {\begin{matrix} ω & if z \in B \\ t & otherwise, \end{matrix} \end{matrix}$ where S ⊃ V in $P (V)$ and ω < t in I . Then the statements mentioned below are equivalent:

(a) ${\tilde{h}}_{δ}$ is a hybrid subsemimodule in $M,$

(b) B is a subsemimodule of $M .$

Definition 3.9. Let ${\tilde{h}}_{β} \in H (M)$ and for $φ \neq L \subseteq M$ . $χ_{L} ({\tilde{h}}_{β})$ is a hybrid characteristic structure in $M$ over $V$ which is described as follows: $\begin{matrix} χ_{L} ({\tilde{h}}_{β}) = (χ_{L} (\tilde{h}), χ_{L} (β)) : M ⟶ P (V) \times I, \\ e \mapsto (χ_{L} (\tilde{h}) (e), χ_{L} (β) (e)), \end{matrix}$ where $\begin{matrix} χ_{L} (\tilde{h}) & : M \to P (V), e \mapsto {\begin{matrix} V & if e \in L \\ φ & otherwise, \end{matrix} \\ χ_{L} (β) & : M \to I, e \mapsto {\begin{matrix} 0 & if e \in L \\ 1 & otherwise \end{matrix} \end{matrix}$ for any $e \in M .$

Corollary 3.10. Let $B (\neq φ) \in P (M)$ and ${\tilde{h}}_{δ} \in ℍ (M) .$ Then the below assertions are equivalent:

(a) $χ_{B} ({\tilde{h}}_{δ})$ is a hybrid subsemimodule in $M,$

(b) B is a subsemimodule of $M .$

Proof. It is implied by Corollary 3.8.

Definition 3.11. Let ${\tilde{h}}_{β}, {\tilde{g}}_{λ} \in ℍ (M) .$ Then

(i) ${\tilde{h}}_{β} \oplus {\tilde{g}}_{λ}$ , a hybrid sum of ${\tilde{h}}_{β}$ and ${\tilde{g}}_{λ}$ , which can be defined as ${\tilde{h}}_{β} \oplus {\tilde{g}}_{λ} = (\tilde{h} \tilde{+} \tilde{g}, β \tilde{+} λ),$ where, $\forall z \in M,$ $\begin{matrix} (\tilde{h} \tilde{+} \tilde{g}) (z) = {\begin{matrix} ⋃_{z = w + q} {\tilde{h} (w) \cap \tilde{g} (q)} & if z = w + q \\ φ & otherwise, \end{matrix} \\ (β \tilde{+} λ) (z) = {\begin{matrix} ⋀_{z = w + q} {β (w) \lor λ (q)} & if z = w + q \\ 1 & otherwise \end{matrix} \end{matrix}$ for $w, q \in M .$

(ii) ${\tilde{h}}_{β} ⊙ {\tilde{g}}_{λ}$ , a hybrid product of ${\tilde{h}}_{β}$ and ${\tilde{g}}_{λ},$ which can be defined as $({\tilde{h}}_{β} ⊙ {\tilde{g}}_{λ}) : = (\tilde{h} \tilde{\circ} \tilde{g}, β \tilde{\circ} λ),$ where $\forall z \in M$ $\begin{matrix} (\tilde{h} \tilde{\circ} \tilde{g}) (z) = {\begin{matrix} ⋃_{z = ac} {\tilde{h} (a) \cap \tilde{g} (c)} & if z = ac \\ φ & otherwise, \end{matrix} \\ (β \tilde{\circ} λ) (z) = {\begin{matrix} ⋀_{z = ac} {β (a) \lor λ (c)} & if z = ac \\ 1 & otherwise \end{matrix} \end{matrix}$ for $a \in M$ and $c \in X .$

Theorem 3.12. Let ${\tilde{h}}_{δ}, {\tilde{g}}_{γ} \in ℍ (M) .$ If ${\tilde{h}}_{δ}$ and ${\tilde{g}}_{γ}$ are hybrid subsemimodules of $M,$ then ${\tilde{h}}_{δ} \oplus {\tilde{g}}_{γ}$ is a hybrid subsemimodule in $M .$

Proof. Consider $x, z \in M .$ Then for q, q^′, w, w^′ ∈ M, we get

\begin{matrix} (\tilde{h} \tilde{+} \tilde{g}) (x) \cap (\tilde{h} \tilde{+} \tilde{g}) (z) = [\underset{x = w + q}{\cup} {\tilde{h} (w) \cap \tilde{g} (q)}] \\ \cap [\underset{z = w^{'} + q^{'}}{\cup} {\tilde{h} (w^{'}) \cap \tilde{g} (q^{'})}] \\ = \underset{z = w^{'} + q^{'}}{\underset{x = w + q}{\cup}} {{\tilde{h} (w) \cap \tilde{g} (q)} \cap {\tilde{h} (w^{'}) \cap \tilde{g} (q^{'})}} \\ = \underset{z = w^{'} + q^{'}}{\underset{x = w + q}{\cup}} {{\tilde{h} (w) \cap \tilde{h} (w)} \cap {\tilde{g} (q) \cap \tilde{g} (q)}} \\ \subseteq \underset{x + z = (w + w^{'}) + (q + q^{'})}{\cup} {\tilde{h} (w + w^{'}) \cap \tilde{g} (q + q^{'})} \\ = (\tilde{h} \tilde{+} \tilde{g}) (x + z), \end{matrix}

\begin{matrix} (δ \tilde{+} γ) (x) \lor (δ \tilde{+} γ) (z) = [\underset{x = w + q}{\land} {δ (w) \lor γ (q)}] \\ \lor [\underset{z = w^{'} + q^{'}}{\land} {δ (w^{'}) \lor γ (q^{'})}] \\ = \underset{z = w^{'} + q^{'}}{\underset{x = w + q}{\land}} {{δ (w) \lor γ (q)} \lor {δ (w^{'}) \lor γ (q^{'})}} \\ = \underset{z = w^{'} + q^{'}}{\underset{x = w + q}{\land}} {δ (w) \lor δ (w^{'})} \lor {γ (q) \lor γ (q^{'})}} \\ = \underset{x + z = (w + w^{'}) + (q + q^{'})}{\land} {δ (w + w^{'}) \lor γ (q + q^{'})} \\ = (δ \tilde{+} γ) (x + z) \end{matrix}

Also, for $z \in M$ and $r \in X,$ we have $\begin{matrix} (\tilde{h} \tilde{+} \tilde{g}) (z) = \underset{z = q + w}{\cup} {\tilde{h} (q) \cap \tilde{g} (w)} \\ \subseteq \underset{z r = q r + c r}{\cup} {\tilde{h} (q r) \cap \tilde{g} (c r)} \\ \subseteq \underset{z r = q^{'} + w^{'}}{\cup} {\tilde{h} (q) \cap \tilde{g} (w)} \\ = (\tilde{h} \tilde{+} \tilde{g}) (z r), \\ (δ \tilde{+} γ) (z) = \underset{z = q + w}{\land} {δ (q) \lor γ (w)} \\ \geq \underset{z r = q r + w r}{\land} {δ (q r) \lor γ (w r)} \\ \geq \underset{z r = q^{'} + w^{'}}{\land} {δ (q^{'}) \lor γ (w^{'})} \\ = (δ \tilde{+} γ) (z r) \end{matrix}$ Hence ${\tilde{h}}_{δ} \oplus {\tilde{g}}_{γ}$ is a hybrid subsemimodule in $M .$

Theorem 3.13. For ${\tilde{g}}_{γ} \in ℍ (X)$ and ${\tilde{h}}_{δ} \in ℍ (M) .$ If ${\tilde{g}}_{γ}$ is a hybrid ideal in $X$ , then ${\tilde{h}}_{δ} ⊙ {\tilde{g}}_{γ}$ is a hybrid subsemimodule in $M .$

Proof. Consider $x, z \in M$ . If ∃ $w, d \in M$ and $c, e \in X : x = wc$ and z = de, then

$(\tilde{h} \tilde{\circ} \tilde{g}) (x) \tilde{\cap} (\tilde{h} \tilde{\circ} \tilde{g}) (z) = [⋃_{x = wc} {\tilde{h} (w) \cap \tilde{g} (c)}] \cap [⋃_{z = de} {\tilde{h} (d) \cap \tilde{g} (e)}] = ⋃_{x = wc} {\tilde{h} (w) \cap \tilde{g} (c)} \cap ⋃_{z = de} {\tilde{h} (d) \cap \tilde{g} (e)} = ⋃_{\binom{x = wc}{z = de}} {\tilde{h} (w) \cap \tilde{g} (c) \cap \tilde{h} (d) \cap \tilde{g} (e)} \subseteq ⋃_{x + z = y + s} {\tilde{h} (y) \cap \tilde{g} (s)} = (\tilde{h} \tilde{\circ} \tilde{g}) (x + z),$

$(δ \tilde{\circ} γ) (x) \lor (δ \tilde{\circ} γ) (z) = [⋀_{x = wc} {δ (w) \lor γ (c)}] \lor [⋀_{z = de} {δ (d) \lor γ (e)}] = ⋀_{x = wc} {δ (w) \lor γ (c)} \lor ⋀_{z = de} {δ (d) \lor γ (e)} = ⋀_{\binom{x = wc}{z = de}} {δ (w) \lor γ (c) \lor δ (d) \lor γ (e)} \geq ⋀_{x + z = y + s} {δ (y) \lor γ (s)} = (δ \tilde{\circ} γ) (x + z) .$

Also, for $r \in X$ , $\begin{matrix} (\tilde{h} \tilde{\circ} \tilde{g}) (z) & = ⋃_{z = wc} {\tilde{h} (w) \cap \tilde{g} (c)} \\ \subseteq ⋃_{z = wc} {\tilde{h} (w) \cap \tilde{g} (cr)} \\ \subseteq ⋃_{zr = w^{″} c^{″}} {\tilde{h} (w^{″}) \cap \tilde{g} (c^{″})} \\ = (\tilde{h} \tilde{\circ} \tilde{g}) (zr), \end{matrix}$ $\begin{matrix} (δ \tilde{\circ} γ) (z) & = ⋀_{z = wc} {δ (w) \lor γ (c)} \\ \geq ⋀_{z = wc} {δ (w) \lor γ (cr)} \end{matrix}$ $\begin{matrix} \geq \underset{z r = w^{″} c^{″}}{\land} {δ (w^{″}) \cap γ (c^{″})} \\ = (δ \tilde{°} γ) (z r) \end{matrix}$

So $(\tilde{h} \tilde{\circ} \tilde{g}) (zr) \supseteq (\tilde{h} \tilde{\circ} \tilde{g}) (z)$ and $(δ \tilde{\circ} γ) (zr) \leq (δ \tilde{\circ} γ) (z)$ for $z \in M$ and $r \in X .$

Therefore $({\tilde{h}}_{δ} ⊙ {\tilde{g}}_{γ})$ is hybrid subsemimodule.

Corollary 3.14. If ${\tilde{h}}_{δ}$ and ${\tilde{g}}_{γ}$ are hybrid ideals in $X$ , then ${\tilde{h}}_{δ} ⊙ {\tilde{g}}_{γ}$ and ${\tilde{h}}_{δ} \oplus {\tilde{g}}_{γ}$ are hybrid ideals in $X$ .

Remark 3.15. It is simple to verify that ${\tilde{g}}_{γ} ⋒ {\tilde{h}}_{δ}$ is a hybrid ideal in $X$ for any hybrid ideals ${\tilde{g}}_{γ}$ and ${\tilde{h}}_{δ}$ in $X$ .

Definition 3.16. [3] In $X$ , an ideal J is said to be right t-pure if for t ∈ J, ∃ b ∈ J : t = tb .

Theorem 3.17. [3] For a two-sided ideal J of $X$ , the conditions mentioned below are equivalent:

(i) J is right t-pure,

(ii) for any right ideal K of $X$ , K ∩ J = KJ.

Definition 3.18. A subsemimodule N of $M$ is said to be pure in $M$ if $N \cap M I = NI$ for any ideal I of $X$ . If each subsemimodule of $M$ is pure in $M$ , then $M$ is referred to as normal.

Definition 3.19. A hybrid right ideal ${\tilde{h}}_{δ}$ in $X$ is referred to as a hybrid right t-pure ideal in $X$ if ${\tilde{g}}_{γ} ⋒ {\tilde{h}}_{δ} = {\tilde{g}}_{γ} ⊙ {\tilde{h}}_{δ}$ for every hybrid right ideal ${\tilde{g}}_{γ}$ in $X$ .

Theorem 3.20. [11] For $C, D \in P (X)$ and ${\tilde{g}}_{σ} \in ℍ (X),$ the below conditions are hold:

(i) $χ_{C} ({\tilde{g}}_{σ}) ⋒ χ_{D} ({\tilde{g}}_{σ}) = χ_{C \cap D} ({\tilde{g}}_{σ}) .$

(ii) $χ_{C} ({\tilde{g}}_{σ}) ⊙ χ_{D} ({\tilde{g}}_{σ}) = χ_{CD} ({\tilde{g}}_{σ}) .$

Theorem 3.21. [11] For any ${\tilde{ψ}}_{σ} \in ℍ (X)$ and $Q \in P (X),$ the conditions mentioned below are equivalent:

(a) $χ_{Q} ({\tilde{ψ}}_{σ})$ is hybrid left (resp., right) ideal in $X,$

(b) Q is a left (resp., right) ideal of $X$ .

Theorem 3.22. For an ideal C of $X$ and ${\tilde{h}}_{δ} \in ℍ (X)$ , the conditions mentioned below are equivalent:

(i) $χ_{C} ({\tilde{h}}_{δ})$ is a hybrid right t-pure ideal in $X$ ,

(ii) C is a right t-pure ideal of $X$ .

Proof. (i) ⇒ (ii) Suppose that $χ_{C} ({\tilde{h}}_{δ})$ is a hybrid right t-pure ideal of $X$ . Then by Theorem 3.21, C is a right ideal of $X$ .

For any right ideal D of $X$ , according to the hypothesis, $χ_{D} ({\tilde{h}}_{δ}) ⋒ χ_{C} ({\tilde{h}}_{δ}) = χ_{D} ({\tilde{h}}_{δ}) ⊙ χ_{C} ({\tilde{h}}_{δ})$ and by Theorem 3.20, we have $χ_{D \cap C} ({\tilde{h}}_{δ}) = χ_{DC} ({\tilde{h}}_{δ})$ which implies D ∩ C = DC. Therefore C is a right t -pure ideal of $X$ .

(ii) ⇒ (i) For a right t-pure ideal C in $X,$ $χ_{C} ({\tilde{h}}_{δ})$ is a hybrid ideal of $X$ by Theorem 3.21. Let ${\tilde{g}}_{γ}$ be a hybrid right ideal of $X$ .

We now prove that ${\tilde{g}}_{γ} ⋒ χ_{C} ({\tilde{h}}_{δ}) = {\tilde{g}}_{γ} ⊙ χ_{C} ({\tilde{h}}_{δ}) .$

Let $w \in X$ . Then $\begin{matrix} (\tilde{g} \tilde{\circ} χ_{C} (\tilde{h})) (w) & = ⋃_{w = ad} {\tilde{g} (a) \cap χ_{C} (\tilde{h}) (d)} \\ \subseteq ⋃_{w = ad} {\tilde{g} (ad) \cap χ_{C} (\tilde{h}) (ad)} \\ \subseteq ⋃_{w = ad} {\tilde{g} (w) \cap χ_{C} (\tilde{h}) (w)} \\ = \tilde{g} (w) \cap χ_{C} (\tilde{h}) (w) \\ = (\tilde{g} \cap χ_{C} (\tilde{h})) (w), \\ (γ \tilde{\circ} χ_{C} (δ)) (w) & = ⋀_{w = ad} {γ (a) \lor χ_{C} (δ) (d)} \\ ⩾ ⋀_{w = ad} {γ (ad) \lor χ_{C} (δ) (ad)} \\ ⩾ ⋀_{w = ad} {γ (w) \lor χ_{C} (δ) (w)} \\ = γ (w) \lor χ_{C} (δ) (w) \\ = (γ \lor χ_{C} (δ)) (w) . \end{matrix}$ Thus ${\tilde{g}}_{γ} ⊙ χ_{C} ({\tilde{h}}_{δ}) ⪡ {\tilde{g}}_{γ} ⋒ χ_{C} ({\tilde{h}}_{δ}) .$

Conversely, if w ∉ C, then

$(\tilde{g} \cap χ_{C} (\tilde{h})) (w) = \tilde{g} (w) \cap χ_{C} (\tilde{h}) (w) = φ \subseteq (\tilde{g} \tilde{\circ} χ_{C} (\tilde{h})) (w)$ and

$(γ \lor χ_{C} (δ)) (w) = γ (w) \lor χ_{C} (δ) (w) = 1 ⩾ (γ \tilde{\circ} χ_{C} (δ)) (w) .$

So ${\tilde{g}}_{γ} ⋒ χ_{C} ({\tilde{h}}_{δ}) ⪡ {\tilde{g}}_{γ} ⊙ χ_{C} ({\tilde{h}}_{δ}) .$

If w ∈ C, then ∃t ∈ C : w = wt. Now, $\begin{matrix} (\tilde{g} \cap χ_{C} (\tilde{h})) (w) & = \tilde{g} (w) \cap χ_{C} (\tilde{h}) (w) \\ = \tilde{g} (w) \cap χ_{C} (\tilde{h}) (wt) \\ \subseteq ⋃_{w = ad} {\tilde{g} (a) \cap χ_{C} (\tilde{h}) (d)} \\ = \tilde{g} \tilde{\circ} χ_{C} (\tilde{h}) (w), \end{matrix}$ $\begin{matrix} (γ \lor χ_{C} (δ)) (w) & = γ (w) \lor χ_{C} (δ) (w) \\ = γ (w) \lor χ_{C} (δ) (wt) \\ ⩾ ⋀_{w = ad} {γ (a) \lor χ_{C} (δ) (d)} \\ = γ \tilde{\circ} χ_{C} (δ) (w) . \end{matrix}$

Thus ${\tilde{g}}_{γ} ⋒ χ_{C} ({\tilde{h}}_{δ}) ⪡ {\tilde{g}}_{γ} ⊙ χ_{C} ({\tilde{h}}_{δ})$ and hence $χ_{C} ({\tilde{h}}_{δ})$ is a hybrid right t-pure ideal of $X$ .

We now explain that every arbitrary hybrid subset of a right $X$ -semimodule must satisfy a necessary condition in order to be a hybrid subsemimodule.

Theorem 3.23. For any semiring $X$ , the below statements hold.

(a) For hybrid t -pure ideals ${\tilde{h}}_{δ}$ and ${\tilde{g}}_{γ}$ of $X$ , ${\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}$ is a hybrid t-pure ideal of $X$ .

(b) If ${{\tilde{h}}_{i δ_{i}} : i \in Δ}$ is a family of hybrid right t-pure ideals of $X,$ then ${\tilde{g}}_{γ} ⋒ (\cup_{i \in Δ} {\tilde{h}}_{i δ_{i}}) = {\tilde{g}}_{γ} ⊙ (\cup_{i \in Δ} {\tilde{h}}_{i δ_{i}})$ for each hybrid right ideal ${\tilde{g}}_{γ}$ of $X$ , where ${\tilde{h}}_{i δ_{i}} = ({\tilde{h}}_{i})_{δ_{i}}$

Proof. (a) Let ${\tilde{h}}_{δ}$ and ${\tilde{g}}_{γ}$ be hybrid t-pure ideals of $X,$ and ${\tilde{ψ}}_{σ}$ be a hybrid right ideal of $X$ . In order to prove ${\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}$ is a t-pure hybrid ideal of $X,$ we have to show that ${\tilde{ψ}}_{σ} ⋒ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) = {\tilde{ψ}}_{σ} ⊙ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) .$

Since ${\tilde{h}}_{δ}$ and ${\tilde{g}}_{γ}$ are hybrid t-pure ideals, we have ${\tilde{ψ}}_{σ} ⋒ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) = ({\tilde{ψ}}_{σ} ⋒ {\tilde{h}}_{δ}) ⋒ ({\tilde{ψ}}_{σ} ⋒ {\tilde{g}}_{γ}) = ({\tilde{ψ}}_{σ} ⊙ {\tilde{h}}_{δ}) ⋒ ({\tilde{ψ}}_{σ} ⊙ {\tilde{g}}_{γ})$

Also, ${\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ} ⪡ {\tilde{h}}_{δ}$ and ${\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ} ⪡ {\tilde{g}}_{γ},$ we get ${\tilde{ψ}}_{σ} ⊙ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) ⪡ {\tilde{ψ}}_{σ} ⊙ {\tilde{h}}_{δ}$ and ${\tilde{ψ}}_{σ} ⊙ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) ⪡ {\tilde{ψ}}_{σ} ⊙ {\tilde{g}}_{γ}$ . Thus ${\tilde{ψ}}_{σ} ⊙ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) ⪡ ({\tilde{ψ}}_{σ} ⊙ {\tilde{h}}_{δ}) ⋒ ({\tilde{ψ}}_{σ} ⊙ {\tilde{g}}_{γ}) .$

On the other hand, since ${\tilde{h}}_{δ}$ and ${\tilde{g}}_{γ}$ are t-pure hybrid ideals, for a right hybrid ideal ${\tilde{ψ}}_{σ}$ , we have ${\tilde{ψ}}_{σ} ⊙ {\tilde{h}}_{δ}$ is a hybrid right ideal. Also,

${\tilde{ψ}}_{σ} ⋒ {\tilde{h}}_{δ} = {\tilde{ψ}}_{σ} ⊙ {\tilde{h}}_{δ}$ and ${\tilde{ψ}}_{σ} ⋒ {\tilde{g}}_{γ} = {\tilde{ψ}}_{σ} ⊙ {\tilde{g}}_{γ}$ .

${\tilde{ψ}}_{σ} ⋒ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) = ({\tilde{ψ}}_{σ} ⋒ {\tilde{h}}_{δ}) ⋒ {\tilde{g}}_{γ} = {\tilde{ψ}}_{σ} ⊙ {\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ} = ({\tilde{ψ}}_{σ} ⊙ {\tilde{h}}_{δ}) ⊙ {\tilde{g}}_{γ}$ by the associativity property of the operation, we have ${\tilde{ψ}}_{σ} ⋒ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) = {\tilde{ψ}}_{σ} ⊙ ({\tilde{h}}_{δ} ⊙ {\tilde{g}}_{γ}) .$

Since ${\tilde{h}}_{δ} ⊙ {\tilde{g}}_{γ} ⪡ {\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ},$ we have ${\tilde{ψ}}_{σ} ⊙ ({\tilde{h}}_{δ} ⊙ {\tilde{g}}_{γ}) ⪡ {\tilde{ψ}}_{σ} ⊙ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ})$ . Thus ${\tilde{ψ}}_{σ} ⋒ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) ⪡ {\tilde{ψ}}_{σ} ⊙ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ})$ and hence ${\tilde{ψ}}_{σ} ⋒ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ}) = {\tilde{ψ}}_{σ} ⊙ ({\tilde{h}}_{δ} ⋒ {\tilde{g}}_{γ})$ .

(b) Let ${{\tilde{h}}_{i δ_{i}} : i \in Δ}$ be a family of hybrid right t-pure ideals in $X$ and ${\tilde{g}}_{γ}$ be any hybrid right ideal in $X$ . Since ${\tilde{h}}_{i δ_{i}}$ ’s are hybrid right t-pure, we have

${\tilde{g}}_{γ} ⋒ (⋃_{i \in Δ} {\tilde{h}}_{i δ_{i}}) = ⋃_{i \in Δ} ({\tilde{g}}_{γ} ⋒ {\tilde{h}}_{i δ_{i}}) = ⋃_{i \in Δ} ({\tilde{g}}_{γ} ⊙ {\tilde{h}}_{i δ_{i}}) .$

As, ${\tilde{g}}_{γ} ⊙ {\tilde{h}}_{i δ_{i}} ⪡ {\tilde{g}}_{γ} ⊙ (⋃_{i \in Δ} {\tilde{h}}_{i δ_{i}})$ , we have $⋃_{i \in Δ} ({\tilde{g}}_{γ} ⊙ {\tilde{h}}_{i δ_{i}}) ⪡ {\tilde{g}}_{γ} ⊙ (⋃_{i \in Δ} {\tilde{h}}_{i δ_{i}}) .$

Thus ${\tilde{g}}_{γ} ⋒ (⋃_{i \in Δ} {\tilde{h}}_{i δ_{i}}) ⪡ {\tilde{g}}_{γ} ⊙ (⋃_{i \in Δ} {\tilde{h}}_{i δ_{i}}) .$

On the other hand for each $r \in X,$ we have $\begin{matrix} (\tilde{g} \tilde{\circ} ⋃_{i \in Δ} {\tilde{h}}_{i}) (r) & = ⋃_{r = wc} {\tilde{g} (w) \cap (⋃_{i \in Δ} {\tilde{h}}_{i} (c))} \\ \subseteq ⋃_{r = wc} {\tilde{g} (wc) \cap (⋃_{i \in Δ} {\tilde{h}}_{i}) (wc)} \\ = ⋃_{r = wc} {\tilde{g} (r) \cap (⋃_{i \in Δ} {\tilde{h}}_{i}) (r)} \\ = (\tilde{g} \cap (⋃_{i \in Δ} {\tilde{h}}_{i})) (r), \\ (γ \tilde{\circ} (⋁_{i \in Δ} δ_{i})) (r) & = ⋀_{r = wc} {γ (w) \lor (⋁_{i \in Δ} δ_{i} (c))} \\ ⩾ ⋀_{r = wc} {γ (wc) \lor (⋁_{i \in Δ} δ_{i} (wc))} \\ = ⋀_{r = wc} {γ (r) \lor (⋁_{i \in Δ} δ_{i} (r))} \\ = {γ \lor (⋁_{i \in Δ} δ_{i})} (r) . \end{matrix}$ Thus ${\tilde{g}}_{γ} ⊙ (⋃_{i \in Δ} {\tilde{h}}_{i δ_{i}}) ⪡ {\tilde{g}}_{γ} ⋒ (⋃_{i \in Δ} {\tilde{h}}_{i δ_{i}})$ and hence ${\tilde{g}}_{γ} ⊙ (⋃_{i \in Δ} {\tilde{h}}_{i δ_{i}}) = {\tilde{g}}_{γ} ⋒ (⋃_{i \in Δ} {\tilde{h}}_{i δ_{i}}) .$

4 Fully idempotent semirings

Definition 4.1. An ideal I of $X$ is idempotent if I² = I. If all of $X$ ’s ideals are idempotent, then $X$ is said to be fully idempotent. It is easy to verify that every weakly regular semiring is fully idempotent. A hybrid ideal ${\tilde{h}}_{μ}$ in $X$ is called idempotent if ${\tilde{h}}_{μ} ⊙ {\tilde{h}}_{μ} = ({\tilde{h}}_{μ})^{2} = {\tilde{h}}_{μ}$ .

The following characterization theorem is proven for these semiring.

Theorem 4.2. For a semiring $X$ , the below statements are equivalent.

(i) $X$ is fully idempotent,

(ii) every hybrid ideal in $X$ is idempotent,

(iii) For every pair of hybrid ideals ${\tilde{h}}_{μ}$ and ${\tilde{g}}_{γ}$ of $X$ , ${\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ} = {\tilde{h}}_{μ} ⊙ {\tilde{g}}_{γ}$ . If $X$ is commutative, then the preceding statements are equivalent to:

(iv) $X$ is weakly regular.

Proof. (i) ⇒ (ii) Let ${\tilde{h}}_{μ}$ be a hybrid ideal of $X$ . Then for any $s \in X$ , $\begin{matrix} {\tilde{h}}^{2} (s) & = (\tilde{h} \tilde{\circ} \tilde{h}) (s) \\ = ⋃_{s = ud} {\tilde{h} (u) \cap \tilde{h} (d)} \\ \subseteq ⋃_{s = ud} {\tilde{h} (ud) \cap \tilde{h} (ud)} \\ = ⋃_{s = ud} {\tilde{h} (s) \cap \tilde{h} (s)} \\ = \tilde{h} (s), \\ μ^{2} (s) & = (μ \circ μ} (s) \\ = ⋀_{s = ud} {μ (u) \lor μ (d)) \\ ⩾ ⋀_{s = ud} {μ (ud) \lor μ (ud)} \\ = ⋀_{s = ud} {μ (s) \lor μ (s)} \\ = μ (s) \end{matrix} .$

So ${\tilde{h}}_{μ} ⊙ {\tilde{h}}_{μ} ⪡ {\tilde{h}}_{μ} .$ As $X$ is fully idempotent, we have $z \in < z > = < z >^{2} = X z X X z X$ , so $\exists r_{1}, w_{1}, r_{2}, w_{2} \in X : z = r_{1} {zw}_{1} r_{2} {zw}_{2}$ . Now

$\tilde{h} (z) = \tilde{h} (z) \cap \tilde{h} (z) \subseteq \tilde{h} (r_{1} {zw}_{1}) \cap \tilde{h} (r_{2} {zw}_{2}) \subseteq ⋃_{z = (r_{1} {zw}_{1}) (r_{2} {zw}_{2})} {\tilde{h} (r_{1} {zw}_{1}) \cap \tilde{h} (r_{2} {zw}_{2})} \subseteq ⋃_{z = ud} {\tilde{h} (u) \cap \tilde{h} (d)} = (\tilde{h} \tilde{\circ} \tilde{h}) (z) = {\tilde{h}}^{2} (z),$

$μ (z) = μ (z) \lor μ (z) ⩾ μ (r_{1} {zw}_{1}) \lor μ (r_{2} {zw}_{2}) ⩾ ⋀_{z = (r_{1} {zw}_{1}) (r_{2} {zw}_{2})} {μ (r_{1} {zw}_{1}) \lor μ (r_{2} {zw}_{2})} ⩾ ⋀_{z = ud} {μ (u) \lor μ (d)} = (μ \tilde{\circ} μ) (z) = μ^{2} (z) .$

Thus ${\tilde{h}}_{μ} ⪡ {\tilde{h}}_{μ} ⊙ {\tilde{h}}_{μ}$ and hence ${\tilde{h}}_{μ}^{2} = {\tilde{h}}_{μ}$ .

(ii) ⇒ (i) If ${\tilde{h}}_{μ}$ is a hybrid ideal in $X$ and C is an ideal of $X$ , then $χ_{C} ({\tilde{h}}_{μ})$ is a hybrid ideal of $X,$ so $χ_{C} ({\tilde{h}}_{μ}) ⊙ χ_{C} ({\tilde{h}}_{μ}) = χ_{C} ({\tilde{h}}_{μ})$ which gives ${\tilde{h}}_{μ}^{2} = {\tilde{h}}_{μ}$ .

(i) ⇒ (iii) Let ${\tilde{h}}_{μ}$ and ${\tilde{g}}_{γ}$ be any two hybrid ideals in $X$ . For any $s \in X$ , $\begin{matrix} (\tilde{h} \tilde{\circ} \tilde{g}) (s) & = ⋃_{s = ud} {\tilde{h} (u) \cap \tilde{g} (d)} \\ \subseteq ⋃_{s = ud} {\tilde{h} (ud) \cap \tilde{g} (ud)} \\ = ⋃_{s = ud} {\tilde{h} (s) \cap \tilde{g} (s)} \\ = \tilde{h} (s) \cap \tilde{g} (s) \\ = (\tilde{h} \tilde{\cap} \tilde{g}) (s), \\ (μ \tilde{\circ} γ) (s) & = ⋀_{s = ud} {μ (u) \lor γ (d)} \\ ⩾ ⋀_{s = ud} {μ (ud) \lor γ (ud)} \\ = ⋀_{s = ud} {μ (s) \lor γ (s)} \\ = μ (s) \lor γ (s) \\ = (μ \lor γ) (s) . \end{matrix}$ So ${\tilde{h}}_{μ} ⊙ {\tilde{g}}_{γ} ⪡ {\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ}$ .

Since $X$ is fully idempotent, (s) = (s) ² for any $s \in X$ . We have, as stated in the first part of this theorem’s proof, $\begin{matrix} (\tilde{h} \tilde{\cap} \tilde{g}) (s) & = \tilde{h} (s) \cap \tilde{g} (s) \\ \subseteq ⋃_{s = ud} {\tilde{h} (u) \cap \tilde{g} (d)} \\ = (\tilde{h} \tilde{\circ} \tilde{g}) (s), \\ (μ \lor γ) (s) & = μ (s) \lor γ (s) \\ ⩾ ⋀_{s = ud} {μ (u) \lor γ (d)} \\ = (μ \tilde{\circ} γ) (s) . \end{matrix}$

Thus ${\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ} ⪡ {\tilde{h}}_{μ} ⊙ {\tilde{g}}_{γ}$ and hence ${\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ} = {\tilde{h}}_{μ} ⊙ {\tilde{g}}_{γ}$ .

(iii) ⇒ (i) Let ${\tilde{h}}_{μ}$ be hybrid ideal of $X$ . Then ${\tilde{h}}_{μ} ⊙ {\tilde{h}}_{μ} = {\tilde{h}}_{μ} ⋒ {\tilde{h}}_{μ} = {\tilde{h}}_{μ} .$

Since (i) ⇔ (ii) and (i) ⇔ (iii) , we have (i) ⇔ (ii) ⇔ (iii).

It is simple to prove (i) ⇔ (iv) if $X$ is commutative.

For a semiring $X,$ the gathering of all hybrid ideals in $X$ over $V$ is represented by $ℍ_{I} (L)$ .

Theorem 4.3. The below assertions are equivalent for a semiring $X$ :

(i) $X$ is fully idempotent,

(ii) Under the hybrid sum and hybrid intersection of hybrid ideals with ${\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ} = {\tilde{h}}_{μ} ⊙ {\tilde{g}}_{γ}$ for any two hybrid ideals ${\tilde{h}}_{μ}$ and ${\tilde{g}}_{γ}$ in $X,$ $(ℍ_{I} (L), ⪡)$ is a distributive lattice.

Proof. (i) ⇒ (ii) It is clearly that $(ℍ_{I} (L), ⪡)$ is a lattice under the hybrid sum and hybrid intersection of hybrid ideals with ${\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ} = {\tilde{h}}_{μ} ⊙ {\tilde{g}}_{γ}$ for any two hybrid ideals ${\tilde{h}}_{μ}$ and ${\tilde{g}}_{γ}$ of $X .$ Furthermore, since the semiring $X$ is fully idempotent, using Theorem 4.2(iii), we get ${\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ} = {\tilde{h}}_{μ} ⊙ {\tilde{g}}_{γ}$ .

Now we prove that $ℍ_{I} (L)$ is a distributive lattice,

i . e . , for hybrid ideals ${\tilde{h}}_{μ}, {\tilde{g}}_{γ}$ and ${\tilde{ψ}}_{σ}$ of $X$ , we prove that $({\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ}) \oplus {\tilde{ψ}}_{σ} = ({\tilde{h}}_{μ} \oplus {\tilde{ψ}}_{σ}) ⋒ ({\tilde{g}}_{γ} \oplus {\tilde{ψ}}_{σ}) .$

For any $u \in X$ , we have

$((\tilde{h} \tilde{\cap} \tilde{g}) \tilde{+} \tilde{ψ}) (u) = ⋃_{u = s + d} {(\tilde{h} \tilde{\cap} \tilde{g}) (s) \cap \tilde{ψ} (d)} = ⋃_{u = s + d} {(\tilde{h} \tilde{\cap} \tilde{g}) (s) \cap \tilde{ψ} (d) \cap \tilde{ψ} (d)} = ⋃_{u = s + d} {\tilde{h} (s) \cap \tilde{ψ} (d)} \cap {\tilde{g} (s) \cap \tilde{ψ} (d)} \subseteq ⋃_{u = s + d} {(\tilde{h} + \tilde{ψ}) (u) \cap (\tilde{g} + \tilde{ψ}) (u)} (as, for s + d = u, \tilde{h} (s) \cap \tilde{ψ} (d) \subseteq (\tilde{h} + \tilde{ψ}) (u) similarly, \tilde{g} (s) \cap \tilde{ψ} (d) \subseteq (\tilde{g} + \tilde{ψ}) (u)) = (\tilde{h} + \tilde{ψ}) (u) \cap (\tilde{g} + \tilde{ψ}) (u) = ((\tilde{h} + \tilde{ψ}) \cap (\tilde{g} + \tilde{ψ})) (u), ((μ \cap γ) \tilde{+} σ) (u) = ⋀_{u = s + d} {(μ \lor γ) (s) \lor σ (d)} = ⋀_{u = s + d} {μ (s) \lor γ (s) \lor σ (d) \lor σ (d)} = ⋀_{u = s + d} {μ (s) \lor σ (d)} \lor {γ (s) \lor σ (d)} ⩾ ⋀_{u = s + d} {(μ + σ) (u) \lor (γ + σ) (u)} (because, for u = s + d, μ (s) \lor σ (d) ⩾ (μ + σ) (u) similarly, γ (s) \lor σ (d) ⩾ (γ + σ) (u) = (μ + σ) (u) \lor (γ + σ) (u) = ((μ + σ) \lor (γ + σ)) (u) .$

So $({\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ}) \oplus {\tilde{ψ}}_{σ} ⪡ ({\tilde{h}}_{μ} \oplus {\tilde{ψ}}_{σ}) ⋒ ({\tilde{g}}_{γ} \oplus {\tilde{ψ}}_{σ}) .$

Also,

$((\tilde{h} \tilde{+} \tilde{ψ}) \tilde{\cap} (\tilde{g} \tilde{+} \tilde{ψ})) (u) = ((\tilde{h} \tilde{+} \tilde{ψ}) \tilde{\circ} (\tilde{g} \tilde{+} \tilde{ψ})) (u) = ⋃_{u = ab} {(\tilde{h} \tilde{+} \tilde{ψ}) (a) \cap (\tilde{g} \tilde{+} \tilde{ψ}) (b)} = ⋃_{u = ab} [{⋃_{a = d + e} (\tilde{h} (d) \cap \tilde{ψ} (e))} \cap {⋃_{b = c + v} (\tilde{g} (c) \cap \tilde{ψ} (v))}] = ⋃_{u = ab} [⋃_{\binom{a = d + e}{b = c + v}} {\tilde{h} (d) \cap \tilde{ψ} (e) \cap \tilde{g} (c) \cap \tilde{ψ} (v)}] = ⋃_{u = ab} [⋃_{\binom{a = d + e}{b = c + v}} {\tilde{h} (d) \cap \tilde{ψ} (e) \cap \tilde{ψ} (e) \cap \tilde{g} (c) \cap \tilde{ψ} (v)}] \subseteq ⋃_{u = ab} [⋃_{\binom{a = d + e}{b = c + v}} {\tilde{h} (dc) \cap \tilde{g} (dc) \cap \tilde{ψ} (ec) \cap \tilde{ψ} (ev) \cap \tilde{ψ} (dv)}] \subseteq ⋃_{u = ab} [⋃_{\binom{a = d + e}{b = c + v}} {(\tilde{h} \tilde{\cap} \tilde{g}) (dc) \cap \tilde{ψ} (ec + ev + dv)}] \subseteq ⋃_{u = ab} {((\tilde{h} \tilde{\cap} \tilde{g}) \tilde{+} \tilde{ψ}) (ab)} \subseteq ((\tilde{h} \tilde{\cap} \tilde{g}) \tilde{+} \tilde{ψ}) (u), ((μ \tilde{+} σ) \lor (γ \tilde{+} σ)) (u) = ((μ \tilde{+} σ) \circ (γ \tilde{+} σ)) (u) = ⋀_{u = ab} {(μ \tilde{+} σ) (a) \lor (γ \tilde{+} σ) (b)} = ⋀_{u = ab} [{⋀_{a = d + e} (μ (d) \lor σ (e))} \lor {⋀_{b = c + v} (γ (c) \lor σ (v))}] = ⋀_{u = ab} [⋀_{\binom{a = d + e}{b = c + v}} {μ (d) \lor σ (e) \lor γ (c) \lor σ (v)}] = ⋀_{u = ab} [⋀_{\binom{a = d + e}{b = c + v}} {μ (d) \lor σ (e) \lor σ (e) \lor γ (c) \lor σ (v)}] ⩾ ⋀_{u = ab} [⋀_{\binom{a = d + e}{b = c + v}} {μ (dc) \lor γ (dc) \lor σ (ec) \lor σ (ev) \lor σ (dv)}] ⩾ ⋀_{u = ab} [⋀_{\binom{a = d + e}{b = c + v}} {(μ \lor γ) (dc) \lor σ (ec + ev + dv)}] ⩾ ⋀_{u = ab} {((μ \lor γ) + σ) (ab)} = ((μ \lor γ) \tilde{+} σ) (u) .$

Thus $({\tilde{h}}_{μ} \oplus {\tilde{ψ}}_{σ}) ⋒ ({\tilde{g}}_{γ} \oplus {\tilde{ψ}}_{σ}) ⪡ ({\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ}) \oplus {\tilde{ψ}}_{σ}$ and hence $({\tilde{h}}_{μ} ⋒ {\tilde{g}}_{γ}) \oplus {\tilde{ψ}}_{σ} = ({\tilde{h}}_{μ} \oplus {\tilde{ψ}}_{σ}) ⋒ ({\tilde{g}}_{γ} \oplus {\tilde{ψ}}_{σ})$ .

Therefore $(ℍ_{I} (L), ⪡)$ is a distributive lattice.

(ii) ⇒ (i) For any hybrid ideal ${\tilde{h}}_{μ}$ of $X$ , we have ${\tilde{h}}_{μ} ⊙ {\tilde{h}}_{μ} = {\tilde{h}}_{μ} ⋒ {\tilde{h}}_{μ} = {\tilde{h}}_{μ}$ . By Theorem 4.2, $X$ is fully idempotent.

Theorem 4.4. If $X$ is weakly regular, then for any hybrid subsemimodule ${\tilde{h}}_{δ}$ in $M$ , hybrid ideal ${\tilde{g}}_{σ}$ in $X$ and $x \in M,$

$(\tilde{h} \tilde{\circ} \tilde{g}) (x) = {\begin{matrix} ⋃_{x = zu} {\tilde{h} (zu) \cap \tilde{g} (u)} & if x = zu \\ φ & otherwise, \end{matrix}$ $(δ \tilde{\circ} σ) (x) = {\begin{matrix} ⋀_{x = zu} {δ (zu) \lor σ (u)} & if x = zu \\ 1 & otherwise . \end{matrix}$

for some $z \in M$ and $u \in X .$ Proof. Let $x \in M$ . By definition, we have

$(\tilde{h} \tilde{\circ} \tilde{g}) (x) = {\begin{matrix} ⋃_{x = zu} {\tilde{h} (z) \cap \tilde{g} (u)} & if x = zu \\ φ & otherwise \end{matrix}$ $(δ \tilde{\circ} σ) (x) = {\begin{matrix} ⋀_{x = zu} {δ (z) \lor σ (u)} & if x = zu \\ 1 & otherwise \end{matrix}$ for some $z \in M$ and $u \in X .$

Since $X$ is weakly regular, for $u \in X, \exists w_{1}, w_{2} \in X : u = {uw}_{1} {uw}_{2} .$

It is clear that $\tilde{h} (zu) \subseteq$ $\tilde{h} (zu w_{1}) \subseteq \tilde{h} (zu w_{1} {uw}_{2}) = \tilde{h} (zu)$ . So, $\tilde{h} (zu) = \tilde{h} (zu w_{1}) .$

Further $\tilde{g} (u) \subseteq \tilde{g} (u w_{1}) \subseteq \tilde{g} (u w_{1} {uw}_{2}) = \tilde{g} (u) .$ Thus $\tilde{g} (u) = \tilde{g} (u w_{1}) .$

Also, δ (zu)⩾ δ (zuw₁) ⩾ δ (zuw₁uw₂) = δ (zu) which gives δ (zu) = δ (zuw₁) and σ (u) ⩾ σ (uw₁) ⩾ σ (uw₁uw₂) = σ (u) which imply σ (u) = σ (uw₁) .

Now, $\begin{matrix} (\tilde{h} \tilde{\circ} \tilde{g}) (x) & \subseteq ⋃_{x = zu} {\tilde{h} (zu) \cap \tilde{g} (u)} \\ = ⋃_{x = zu} {\tilde{h} (zu w_{1}) \cap \tilde{g} (u w_{1})} \\ \subseteq ⋃_{x = ys} {\tilde{h} (y) \cap \tilde{g} (s)} \\ = (\tilde{h} \tilde{\circ} \tilde{g}) (x), \end{matrix}$ $\begin{matrix} (δ \tilde{\circ} σ) (x) & ⩾ ⋀_{x = zu} {δ (zu) \lor σ (u)} \\ = ⋀_{x = zu} {δ (zu w_{1}) \lor σ (u w_{1})} \\ ⩾ ⋀_{x = ys} {δ (y) \lor σ (s)} \\ = (δ \tilde{\circ} σ) (x) . \end{matrix}$

So, $\forall x \in M,$ $\begin{matrix} (\tilde{h} \tilde{\circ} \tilde{g}) (x) = {\begin{matrix} ⋃_{x = zu} {\tilde{h} (zu) \cap \tilde{g} (u)} & if x = zu \\ φ & otherwise, \end{matrix} \end{matrix}$ $\begin{matrix} (δ \tilde{\circ} σ) (x) & = {\begin{matrix} ⋀_{x = zu} {δ (zu) \lor σ (u)} & if x = zu \\ 1 & otherwise . \end{matrix} \end{matrix}$ for some $z \in M, u \in X$ .

5 Conclusion

In this study, we applied hybrid structures to semiring modules, the notion of hybrid subsemimodules was established, and some of their fundamental aspects were examined. The hybrid cut set of hybrid subsemimodules in the right module $M$ had been shown to be a subsemimodule of $M$ . We also outlined the idea of hybrid t-pure ideals in a semiring and discussed the various relationships between hybrid subsemimodules and hybrid t-pure ideals in a semiring. Finally, for fully idempotent semirings, we proved the characterization theorem.

Footnotes

Acknowledgment

The authors would like to express their gratitude to the referees for their insightful comments and suggestions for improving the paper.

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