A Note on the w-Pseudo-Orders in Ordered (Semi)Hyperrings

Abstract

In this work, we study the notion of $\mathbf{w}$-pseudo-order on an ordered (semi)hyperring and give some explicit examples. In addition, we give some examples to compare weak pseudo-order relations with pseudo-order relations. Finally, we construct ordered (semi)hyperrings using regular relations.

1. Introduction

In 1934, Marty [1] introduced the concept of hypergroups. More details about hyperrings can be found in [2]. In 2011, Heidari and Davvaz [3] introduced the concept of ordered semihypergroups. A semihypergroup $(S,\circ )$ with a (partial) order relation ≤ is called an ordered semihypergroup if for all $m,n,a\in S$ for which $m\le n$ it follows that $a\circ m\le a\circ n$ and $m\circ a\le n\circ a$. Here, a relation $a\circ m\le a\circ n$ is only possible if for any $u\in a\circ m$ there exists $v\in a\circ n$ such that $u\le v$.

The concept of pseudo-order on an ordered semigroup $(S,·,\le )$ first appeared in the work of Kehayopula and Tsingelis [4,5] and was extended for ordered semihypergroups by Davvaz et al. [6]. In [6,7], the relation between ordered semihypergroups and ordered semi(hyper)groups was established. In further studies, Gu and Tang [8] and Tang et al. [9] completely solved an open problem on ordered semihypergroups [6]. Ordered structures have applications in various fields, such as computer science, physics, coding theory and topological spaces, etc. Omidi and Davvaz in [10,11,12] and Rao et al. in [13,14,15] obtained some results on ordered hyperstructures connected with pseudo-orders and k-hyperideals.

In this work, we study ordered (semi)hyperrings to find a regular relation such that the constructed quotient structure is an ordered (semi)hyperring. Indeed, we try to answer the open problem that has been asked in [10], by defining a new notion called “weak pseudo-order relation”. We construct a regular relation on an ordered hyperstructure in such a way that the quotient structure is an ordered hyperstructure. We provide a way to produce ordered hyperstructures using weak pseudo-orders ($\mathbf{w}$-pseudo-orders). In this note, we extend the notion of $\mathbf{w}$-pseudo-orders to ordered (semi)hyperrings and investigate some properties of them. Some examples to compare $\mathbf{w}$-pseudo-order relations with pseudo-order relations are given. A construction of an ordered (semi)hyperring via a $\mathbf{w}$-pseudo-order is given. In fact, we present a method to produce ordered (semi)hyperrings using $\mathbf{w}$-pseudo-orders. In this work, we aim to find good examples of ordered (semi)hyperring deal with $\mathbf{w}$-pseudo-orders.

2. Preliminaries

In this section we review definitions for ordered (semi)hyperrings, pseudo-orders and regular relations, which will be used in this paper.

A triple $(R,+,·)$ is a Krasner hyperring [16] if the following hold: (1) $(R,+)$ is a canonical hypergroup; (2) $(R,·)$ is a semigroup such that $x·0=0·x=0$; (3) · is distributive with respect to +. Also, R is a semihyperring [17] if it satisfies (3) and $(R,+)$ and $(R,·)$ are semihypergroups.

By an ordered (semi)hyperring $(R,+,·,\le )$, we mean a (semi)hyperring R with a (partial) order relation ≤ such that if $m\le n$, then $m+x\le n+x$, $x+m\le x+n$, $m·x\le n·x$ and $x·m\le x·n$ for all $m,n,x\in R$. Here, $P\le Q$ meaning that for any $p\in P$, there exists $q\in Q$ such that $p\le q$, where $\varnothing \ne P,Q\subseteq S$. For further details and examples of ordered (semi)hyperrings, see, for instance, ref. [11].

Example 1.

Let $R=\{0,a,b,c,d\}$. Define the symmetrical hyperoperations +, · and (partial) order relation ${\le }_R$ on R as follows:

$$\begin{array}{|cccccc|}\hline +& 0& a& b& c& d\\ 0& 0& a& b& c& d\\ a& a& \{0,a\}& b& c& d\\ b& b& b& \{0,a\}& d& c\\ c& c& c& d& \{0,a\}& b\\ d& d& d& c& b& \{0,a\}\\ \hline\end{array}$$

$$\begin{array}{|cccccc|}\hline ·& 0& a& b& c& d\\ 0& 0& 0& 0& 0& 0\\ a& 0& \{0,a\}& \{0,a\}& \{0,a\}& \{0,a\}\\ b& 0& \{0,a\}& \{0,a\}& \{0,a\}& \{0,a\}\\ c& 0& \{0,a\}& \{0,a\}& \{0,a\}& \{0,a\}\\ d& 0& \{0,a\}& \{0,a\}& \{0,a\}& \{0,a\}\\ \hline\end{array}$$

$$\begin{array}{c}{\le }_R:=\{(0,0),(0,a),(a,a),(b,b),(c,c),(d,d)\}.\hfill \end{array}$$

Then $(R,+,·,\le )$ is an ordered hyperring [11].

For a relation $\sigma$ on R and $\varnothing \ne P,Q\subseteq R$, we have

(1)

$P\overrightarrow{\sigma }Q\iff \forall p\in P, \exists q\in Q; p\sigma q$.

(2)

$P\overleftarrow{\sigma }Q\iff \forall q^{\prime }\in Q, \exists p^{\prime }\in P; p^{\prime }\sigma q^{\prime }$.

(3)

$P\tilde{\sigma }Q\iff \forall p\in P, \exists q\in Q; p\sigma q$ and $q\sigma p$, $\forall q^{\prime }\in Q, \exists p^{\prime }\in P; q^{\prime }\sigma p^{\prime }$ and $p^{\prime }\sigma q^{\prime }$.

(4)

$P\overline{\sigma }Q\iff P\overrightarrow{\sigma }Q$ and $P\overleftarrow{\sigma }Q$.

(5)

$P\overline{\overline{\sigma }}Q\iff \forall p\in P, \forall q\in Q; p\sigma q$.

A relation $\sigma$ on R is called regular [2,11] if for all $m,n,x\in R$, (1) $m\sigma n\Rightarrow (m+x)\overline{\sigma }(n+x)\mathrm{and}(x+m)\overline{\sigma }(x+n)$; (2) $m\sigma n\Rightarrow (m·x)\overline{\sigma }(n·x)\mathrm{and}(x·m)\overline{\sigma }(x·n)$. $\sigma$ is said to be strongly regular [2,11] if (1) $m\sigma n\Rightarrow (m+x)\overline{\overline{\sigma }}(n+x)\mathrm{and}(x+m)\overline{\overline{\sigma }}(x+n)$; (2) $m\sigma n\Rightarrow (m·x)\overline{\overline{\sigma }}(n·x)\mathrm{and}(x·m)\overline{\overline{\sigma }}(x·n)$.

A relation $\sigma$ on $(R,+,·,\le )$ is called a pseudo-order [11] if (1) $\le \subseteq \sigma$; (2) $u\sigma v$ and $v\sigma w$ imply $u\sigma w$; (3) $u\sigma v$ implies $(u+m)\overline{\overline{\sigma }}(v+m)$ and $(m+u)\overline{\overline{\sigma }}(m+v)$ and (4) $u\sigma v$ implies $(u·m)\overline{\overline{\sigma }}(v·m)$ and $(m·u)\overline{\overline{\sigma }}(m·v)$ for all $u,v,m\in R$.

Here, we give an example of a pseudo-order.

Example 2.

In Example 1,

$$\begin{gathered} \begin{array}{cc}\sigma \hfill & =\left\{\right(0,0),(0,a),(0,b),(a,0),(a,a),(a,b),(b,0),\hfill \end{array} \\ (b,a),(b,b),(c,c),(c,d),(d,c),(d,d)\} \end{gathered}$$

is a pseudo-order.

3. Construction of Ordered (Semi)Hyperrings via $\mathbf{w}$-Pseudo-Orders

$\varnothing \ne U\subseteq R$ is a 2-hyperideal if (1) $U+R,R+U\subseteq U$; (2) $U·R,R·U\subseteq U$; (3) $\left(U\right]:=\{x\in R\mid x\le m\mathrm{for}\mathrm{some}m\in U\}\subseteq U$. A 2-hyperideal I, which is not R, is called a proper 2-hyperideal.

Example 3.

We consider the ordered semihyperring $(R,+,·,{\le }_R)$ in Example 1 of [18] with two symmetrical hyperoperations + and ·. Since $x·y=\{b,d\}$ for all $x,y\in R$ and $\left(\right\{b,d\left\}\right]=R⊈\{b,d\}$, we observe that there do not exist proper 2-hyperideals in R.

Definition 1.

A relation σ on an ordered (semi)hyperring $(R,+,·,{\le }_R)$ is called a $\mathbf{w}$-pseudo-order if for all $u,v,m\in R$,

(1) 

${\le }_R\subseteq \sigma$;

(2) 

$u\sigma v$ and $v\sigma m$ imply $u\sigma m$;

(3) 

$u\sigma v$ implies $(u+m)\overrightarrow{\sigma }(v+m)$ and $(m+u)\overrightarrow{\sigma }(m+v)$;

(4) 

$u\sigma v$ implies $(u·m)\overrightarrow{\sigma }(v·m)$ and $(m·u)\overrightarrow{\sigma }(m·v)$;

(5) 

$u\sigma v$ and $v\sigma u$ imply $(u+m)\tilde{\sigma }(v+m)$ and $(m+u)\tilde{\sigma }(m+v)$;

(6) 

$u\sigma v$ and $v\sigma u$ imply $(u·m)\tilde{\sigma }(v·m)$ and $(m·u)\tilde{\sigma }(m·v)$.

Note that ${\le }_R$ is a $\mathbf{w}$-pseudo-order. Every pseudo-order is a $\mathbf{w}$-pseudo-order. The converse is not true, in general. Here are some examples.

Example 4.

In Example 3,

$$\begin{gathered} \begin{array}{cc}\sigma \hfill & =\left\{\right(a,a),(a,b),(b,a),(b,b),(c,b),(c,c),(c,d),\hfill \end{array} \\ (d,c),(d,d),(e,d),(e,e)\} \end{gathered}$$

is a w-pseudo-order on R, but it is not a pseudo-order, since $(a+a)\overline{\overline{\sigma }}(b+a)$ does not hold. Indeed:

$$\begin{array}{c}\hfill (a,b)\in \sigma ,a+a=\{b,c\}\mathrm{and}b+a=\{b,d\}\mathrm{but}(b,d)\notin \sigma .\end{array}$$

Example 5.

Consider the ordered Krasner hyperring $(R,+,·,{\le }_R)$ as follows:

$$\begin{array}{|ccccc|}\hline +& 0& a& b& c\\ 0& 0& a& b& c\\ a& a& \{0,a\}& c& \{b,c\}\\ b& b& c& \{0,b\}& \{a,c\}\\ c& c& \{b,c\}& \{a,c\}& R\\ \hline\end{array}\begin{array}{|ccccc|}\hline ·& 0& a& b& c\\ 0& 0& 0& 0& 0\\ a& 0& 0& 0& 0\\ b& 0& a& b& c\\ c& 0& a& b& c\\ \hline\end{array}$$

$$\begin{array}{c}{\le }_R:=\{(0,0),(a,a),(b,b),(c,c),(0,a),(b,c)\}.\hfill \end{array}$$

Set

$$\begin{array}{c}\sigma :=\left\{\right(0,0),(0,a),(a,0),(a,a),(b,b),(b,c),(c,b),(c,c\left)\right\}.\hfill \end{array}$$

Then, σ is a $\mathbf{w}$-pseudo-order on R, but it is clearly not a pseudo-order, since $(b+b)\overline{\overline{\sigma }}(c+b)$ does not hold. Indeed:

$$(b,c)\in \sigma ,b+b=\{0,b\}\mathrm{and}c+b=\{a,c\}\mathrm{but}(0,c),(b,a)\notin \sigma .$$

Lemma 1.

Let $\left\{{\sigma }_i\right|i\in \mathsf{\Lambda }\}$ be a family of $\mathbf{w}$-pseudo-orders on $(R,+,·,{\le }_R)$. Then, $\sigma =\bigcap _{\begin{array}{c}i\in \mathsf{\Lambda }\end{array}}{\sigma }_i$ is a $\mathbf{w}$-pseudo-order.

Proof. 

Clearly, ${\le }_R\subseteq \sigma$. Let $(u,v)\in \sigma$ and $(v,m)\in \sigma$. Then $(u,v)\in {\sigma }_i$ and $(v,m)\in {\sigma }_i$ for all $i\in \mathsf{\Lambda }$. Since ${\sigma }_i$ is a $\mathbf{w}$-pseudo-order, we obtain $(u,m)\in {\sigma }_i$ for all $i\in \mathsf{\Lambda }$. Thus $(u,m)\in \sigma$.

Now, let $(u,v)\in \sigma$ and $m\in R$. Then $(u,v)\in {\sigma }_i$ for all $i\in \mathsf{\Lambda }$. Since each ${\sigma }_i$ is a $\mathbf{w}$-pseudo-order on R, we have $(u+m)\overrightarrow{{\sigma }_i}(v+m)$ and $(m+u)\overrightarrow{{\sigma }_i}(m+v)$. So, for every $p\in u+m$, there exists $q\in v+m$ such that $p{\sigma }_{i}q$ for all $i\in \mathsf{\Lambda }$. So, $p\sigma q$ and hence $(u+m)\overrightarrow{\sigma }(v+m)$. Similarly, $(m+u)\overrightarrow{\sigma }(m+v)$, $(u·m)\overrightarrow{\sigma }(v·m)$ and $(m·u)\overrightarrow{\sigma }(m·v)$.

Let $(u,v)\in \sigma$ and $(v,u)\in \sigma$. Then $(u,v)\in {\sigma }_i$ and $(v,u)\in {\sigma }_i$ for all $i\in \mathsf{\Lambda }$. Since ${\sigma }_i$ is a $\mathbf{w}$-pseudo-order on R, we obtain $(u+m)\widetilde{{\sigma }_i}(v+m)$ and $(m+u)\widetilde{{\sigma }_i}(m+v)$. So, for every $p\in u+m$, there exists $q\in v+m$ such that $p{\sigma }_{i}q$ and $q{\sigma }_{i}p$ and for every $q^{\prime }\in v+m$, there exists $p^{\prime }\in u+m$ such that $q^{\prime }{\sigma }_i{p}^{\prime }$ and $p^{\prime }{\sigma }_i{q}^{\prime }$ for all $i\in \mathsf{\Lambda }$. It follows that $(u+m)\tilde{\sigma }(v+m)$. Similarly, $(m+u)\tilde{\sigma }(m+v)$, $(u·m)\tilde{\sigma }(v·m)$ and $(m·u)\tilde{\sigma }(m·v)$. Therefore, σ is a $\mathbf{w}$-pseudo-order. □

Theorem 1.

If σ is a $\mathbf{w}$-pseudo-order on an ordered (semi)hyperring $(R,+,·,{\le }_R)$, then

$${\sigma }^{*}=\{(p,q)\in R\times R|p\sigma q\mathrm{and}q\sigma p\}$$

is a regular relation on R such that $(R/{\sigma }^{*},⪯)$ is an ordered (semi)hyperring, where

$$⪯:=\{({\sigma }^{*}\left(m\right),{\sigma }^{*}\left(n\right))\in R/{\sigma }^{*}\times R/{\sigma }^{*}|\exists p\in {\sigma }^{*}\left(m\right),\exists q\in {\sigma }^{*}\left(n\right)\mathrm{such}\mathrm{that}(p,q)\in \sigma \}.$$

Proof. 

The proof is similar to the proof of Theorem 3.8 in [9]. □

Corollary 1.

Let us follow the notations used in Theorem 1. Then, ${\sigma }^{*}$ is a regular relation on R.

Proof. 

By Theorem 1, $(R/{\sigma }^{*},\oplus ,\odot ,⪯)$ is an ordered (semi)hyperring. Let $(u,v)\in \le$. Since σ is a $\mathbf{w}$-pseudo-order, we obtain $(u,v)\in \sigma$. So, ${\sigma }^{*}\left(u\right)⪯{\sigma }^{*}\left(v\right)$. □

In the following, some explicit examples are given in order to clarify Theorem 1.

Example 6.

In Example 4,

$${\sigma }^{*}=\{(a,a),(a,b),(b,a),(b,b),(c,c),(c,d),(d,c),(d,d),(e,e)\}$$

and $R/{\sigma }^{*}=\{m_1,m_2,m_3\}$, where $m_1=\{a,b\}$, $m_2=\{c,d\}$ and $m_3=\left\{e\right\}$. By Theorem 1, $(R/{\sigma }^{*},\oplus ,\odot ,{⪯}_R)$ is an ordered semihyperring, where

$$\begin{array}{cccc}\oplus & m_1& m_2& m_3\\ m_1& \{m_1,m_2\}& \{m_1,m_2\}& m_3\\ m_2& \{m_1,m_2\}& \{m_1,m_2\}& m_3\\ m_3& \{m_1,m_2\}& \{m_1,m_2\}& m_3\end{array}\begin{array}{cccc}\odot & m_1& m_2& m_3\\ m_1& \{m_1,m_2\}& \{m_1,m_2\}& \{m_1,m_2\}\\ m_2& \{m_1,m_2\}& \{m_1,m_2\}& \{m_1,m_2\}\\ m_3& \{m_1,m_2\}& \{m_1,m_2\}& \{m_1,m_2\}\end{array}$$

$${⪯}_R=\{(m_1,m_1),(m_2,m_1),(m_2,m_2),(m_3,m_2),(m_3,m_3)\}.$$

Example 7.

In Example 5, ${\sigma }^{*}=\sigma$ and $R/{\sigma }^{*}=\{m_1,m_2\}$, where $m_1=\{0,a\}$ and $m_2=\{b,c\}$. By Theorem 1, $(R/{\sigma }^{*},\oplus ,\odot ,{⪯}_R)$ is an ordered Krasner hyperring, where

$$\begin{array}{ccc}\oplus & m_1& m_2\\ m_1& m_1& m_2\\ m_2& m_2& \{m_1,m_2\}\end{array}\begin{array}{ccc}\odot & m_1& m_2\\ m_1& m_1& m_1\\ m_2& m_1& m_2\end{array}$$

and ${⪯}_R=\{(m_1,m_1),(m_2,m_2)\}$.

4. Conclusions

In this paper, we studied and extended the notion of weak pseudo-orders ($\mathbf{w}$-pseudo-orders) to ordered (semi)hyperrings and investigated some properties of them. A construction of an ordered (semi)hyperring via a $\mathbf{w}$-pseudo-order was given. We tried to solve an open problem in ordered hyperring theory. In the future, we will study $\mathbf{w}$-pseudo-orders in ordered superrings.