On New Filters in Ordered Semigroups

Abstract

Ordered semigroups are understood through their subsets. The aim of this article is to study ordered semigroups through some new substructures. In this regard, quasi-filters and $(m,n)$-quasi-filters of ordered semigroups are introduced as new types of filters. Some properties of the new concepts are investigated, different examples are constructed, and the relations between quasi-filters and quasi-ideals as well as between $(m,n)$-quasi-filters and $(m,n)$-quasi-ideals are discussed.

1. Introduction and Preliminaries

Kehayopulu [1] was the first to investigate filters in $poe$-semigroups. Lee et al. [2] introduced and described the notion of left (resp. right) filters in $po$-semigroups in terms of prime right (resp. left) ideals. The notion of $\mathrm{\Gamma }$-filters in ordered $\mathrm{\Gamma }$-semigroups was developed by Hila [3], while Tang et al. [4] proposed the concept of filters in ordered semihypergroups. Khan et al. [5] introduced the notions of left-m-filters, right-n-filters, and $(m,n)$-filters in ordered semigroups as a generalization of the concept of left (right) filters of ordered semigroups. Fuzzy set theory was applied to filters of ordered semigroups by Kehayopulu and Tsingelis [6], and the notion of fuzzy filters in ordered semigroups was established. By generalizing the notion of fuzzy filters, Davvaz et al. [7] established the concept of $(\in ,\in \vee q)$-fuzzy filters in ordered semigroups. Ali [8] developed generalized rough approximations for fuzzy filters in ordered semigroups; in addition, in [9], Ali et al. proposed the notion of soft filters in soft ordered semigroups.

As novel forms of filters and in continuation of the work initiated in this regard, quasi-filters and $(m,n)$-quasi-filters of ordered semigroups are introduced herein. Some new concepts and characteristics are studied. Furthermore, relationships between quasi-filters (resp. $(m,n)$-quasi-filters) and quasi-ideals (resp. $(m,n)$-quasi-ideals) are discussed.

An ordered semigroup $(\mathrm{\Omega },·,\le )$ is a semigroup with a partial order relation ≤ that is compatible, i.e., $\vartheta \le \gamma$ implies $\vartheta \kappa \le \gamma \kappa \mathrm{and}\kappa \vartheta \le \kappa \gamma$ for all $\vartheta ,\gamma ,\kappa \in \mathrm{\Omega }$. For $\mathrm{{\rm Y}}\ne \varnothing \subseteq \mathrm{\Omega }$, we denote $\left(\mathrm{{\rm Y}}\right]=\{t\in \mathrm{\Omega }:t\le a\mathrm{for} \mathrm{some}a\in \mathrm{{\rm Y}}\}$ and $\left[\mathrm{{\rm Y}}\right)=\{t\in \mathrm{\Omega }:a\le t\mathrm{for} \mathrm{some}a\in \mathrm{{\rm Y}}\}$.

Ordered semigroups have been studied through their subsets (see [5,10,11,12,13].) A subset $\mathrm{{\rm Y}}\ne \varnothing$ of $\mathrm{\Omega }$ is called a subsemigroup of $\mathrm{\Omega }$ if $\mathrm{{\rm Y}}\mathrm{{\rm Y}}\subseteq \mathrm{{\rm Y}}$, and $\mathrm{{\rm Y}}$ is called the left (resp. right) ideal of $\mathrm{\Omega }$ if $\mathrm{\Omega }\mathrm{{\rm Y}}\subseteq \mathrm{{\rm Y}}(\mathrm{{\rm Y}}\mathrm{\Omega }\subseteq \mathrm{{\rm Y}})$ and $\left(\mathrm{{\rm Y}}\right]\subseteq \mathrm{{\rm Y}}$. If a subset $\mathrm{{\rm Y}}$ is both a left ideal and a right ideal of $\mathrm{\Omega }$, it is called an ideal of $\mathrm{\Omega }$. A subsemigroup $\mathrm{{\rm Y}}$ of $\mathrm{\Omega }$ is called a bi-ideal of $\mathrm{\Omega }$ if $\mathrm{{\rm Y}}\mathrm{\Omega }\mathrm{{\rm Y}}\subseteq \mathrm{{\rm Y}}$ and $\left(\mathrm{{\rm Y}}\right]\subseteq \mathrm{{\rm Y}}$. A non-empty subset $\mathrm{{\rm Y}}$ of $\mathrm{\Omega }$ is called a quasi-ideal of $\mathrm{\Omega }$ if $\left(\mathrm{{\rm Y}}\mathrm{\Omega }\right]\cap \left(\mathrm{\Omega }\mathrm{{\rm Y}}\right]\subseteq \mathrm{{\rm Y}}$ and $\left(\mathrm{{\rm Y}}\right]\subseteq \mathrm{{\rm Y}}$. Furthermore, a subsemigroup $\mathrm{{\rm Y}}$ of $\mathrm{\Omega }$ is called a left filter (resp. right filter) of $\mathrm{\Omega }$ if for all $a,b\in \mathrm{\Omega }$, $ab\in \mathrm{{\rm Y}}$ implies $a\in \mathrm{{\rm Y}}$ (resp. $b\in \mathrm{{\rm Y}}$) and $\left[\mathrm{{\rm Y}}\right)\subseteq \mathrm{{\rm Y}}$. It is a filter if it is both a left filter and a right filter of $\mathrm{\Omega }$. For positive integers m and n, a subsemigroup Q of $\mathrm{\Omega }$ is an $(m,n)$-quasi-ideal of $\mathrm{\Omega }$ of $\left(Q\right]\subseteq Q$ and $\left(Q^m\mathrm{\Omega }\right]\cap \left(\mathrm{\Omega }{Q}^n\right]\subseteq Q$.

An ordered semigroup $\mathrm{\Omega }$ is called $(m,n)$-regular if for all $\vartheta \in \mathrm{\Omega }$, there exists $\gamma \in \mathrm{\Omega }$ such that $\vartheta \le {\vartheta }^m\gamma {\vartheta }^n$. For more related details, we refer to [14].

2. Quasi-Filters of Ordered Semigroups: Redefined

In [15], Jirojkul and Chinram introduced quasi-filters of ordered semigroups. In addition, in [16], Yaqoob and Tang used a similar definition to introduce quasi-hyperfilters. Their definition was based on a non-general definition of a quasi-ideal. In this section, we redefine quasi-filters of ordered semigroups in a more general way. Furthermore, we explore some of their properties and relate them to quasi-ideals.

Definition 1.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q\ne \varnothing \subseteq \mathrm{\Omega }$. Then, Q is a quasi-filter of Ω if the following conditions hold for all $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\Omega }$.

 (1) 

$Q·Q\subseteq Q$;

 (2) 

$\left[Q\right)\subseteq Q$;

 (3) 

If $x\in Q$ and for some $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\Omega }$, $x\le \alpha ·\beta$ and $x\le \gamma ·\delta$, then, $\{\alpha ,\delta \}\cap Q\ne \varnothing$ and $\{\beta ,\gamma \}\cap Q\ne \varnothing$.

If we drop the subsemigroup condition in Definition 1, we obtain Q as a generalized quasi-filter of $\mathrm{\Omega }$.

Remark 1.

A quasi-filter Q in a semigroup Ω is a subsemigroup of Ω satisfying the following condition for all $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\Omega }$.

$\alpha ·\beta =\gamma ·\delta \in Q$ implies $\{\alpha ,\delta \}\cap Q\ne \varnothing$ and $\{\beta ,\gamma \}\cap Q\ne \varnothing$.

Example 1.

Let $({\mathbb{Z}}^{+}\cup \left\{0\right\},·,\le )$ be the semigroup of non-negative integers under standard multiplication and the usual order of numbers. Then, ${\mathbb{Z}}^{+}$ is a proper quasi-filter of ${\mathbb{Z}}^{+}\cup \left\{0\right\}$.

Example 2.

Consider the ordered semigroup ${\mathrm{\Omega }}_1=\{\vartheta ,\kappa ,\varpi \}$, with operation “${·}_1$” and order “${\le }_1$” described as follows:

${·}_1$	$\vartheta$	$\kappa$	$\varpi$
$\vartheta$	$\vartheta$	$\vartheta$	$\vartheta$
$\kappa$	$\kappa$	$\kappa$	$\kappa$
$\varpi$	$\varpi$	$\varpi$	$\varpi$

$${\le }_1:=\{(\vartheta ,\vartheta ),(\kappa ,\kappa ),(\gamma ,\gamma ),(\vartheta ,\kappa ),(\vartheta ,\varpi )\}.$$

One can easily see that $\{\kappa ,\varpi \}$ is a quasi-filter of ${\mathrm{\Omega }}_1$.

Example 3.

Let $(M_2\left(\mathbb{Z}\right),·,{\le }_t)$ be the ordered semigroup of two by two matrices with integer coefficients under the standard multiplication of matrices and trivial order. Then, $\mho =\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):(b,c,d)\ne (0,0,0)\}$ is a generalized quasi-filter of $M_2\left(\mathbb{Z}\right)$, and it is not a quasi-filter of $M_2\left(\mathbb{Z}\right)$ as ${\mho }^2$ is not a subset of $\mho .$ This is clear as $\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right)\left(\begin{array}{cc}1& 0\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}2& 0\\ 0& 0\end{array}\right)\notin \mho .$

Now, to show that ℧ is a generalized quasi-filter of $M_2\left(\mathbb{Z}\right)$, it suffices to show that for all $\alpha ,\beta ,\gamma ,\delta \in M_2\left(\mathbb{Z}\right)$, if $\alpha \beta =\gamma \delta \in \mho$, thus, $\{\alpha ,\delta \}\cap \mho \ne \varnothing$ and $\{\beta ,\gamma \}\cap \mho \ne \varnothing$. Without loss of generality, suppose that $\{\alpha ,\delta \}\cap \mho =\varnothing .$ Then, there exist $a,b\in \mathbb{Z}$ with $\alpha =\left(\begin{array}{cc}a& 0\\ 0& 0\end{array}\right)$ and $\delta =\left(\begin{array}{cc}b& 0\\ 0& 0\end{array}\right).$

Let $\beta =\left(\begin{array}{cc}c& d\\ e& f\end{array}\right)$ and $\gamma =\left(\begin{array}{cc}g& h\\ i& j\end{array}\right).$ Having $\alpha \beta =\left(\begin{array}{cc}ac& ad\\ 0& 0\end{array}\right)=\left(\begin{array}{cc}gb& 0\\ ib& 0\end{array}\right)=\gamma \delta$ implies that $ad=ib=0.$ The latter implies that $\alpha \beta =\gamma \delta \notin \mho .$

Lemma 1.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q\ne \varnothing \subseteq \mathrm{\Omega }$. If Q is a (generalized) quasi-filter of Ω, then, Q is prime.

Proof. 

Let $\alpha ,\beta \in \mathrm{\Omega }$ with $\alpha ·\beta \in Q$. By setting $x=\alpha ·\beta$, we see that $x\in Q$ satisfies $x\le \alpha ·\beta$. Having Q a (generalized) quasi-filter of $\mathrm{\Omega }$ implies that $\{\alpha ,\beta \}\cap Q\ne \varnothing$ and, hence, $\alpha \in Q$ or $\beta \in Q$. □

Lemma 2.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup, $p_1,p_2,\dots ,p_k\in \mathrm{\Omega }$ and $Q\ne \varnothing \subseteq \mathrm{\Omega }$ be a (generalized) quasi-filter of Ω. If $p_1{p}_2\dots p_k\in Q$, then, $\{p_1,p_2,\dots ,p_k\}\cap Q\ne \varnothing$.

Proof. 

Let $p_1(p_2\dots p_k)\in Q$. Having Q be a quasi-filter of $\mathrm{\Omega }$ implies that Q is prime (by Lemma 1) and, hence, $p_1\in Q$ or $p_2\dots p_k\in Q$. If $p_1\in Q$, we are finished. Otherwise, $p_2(p_3\dots p_k)\in Q$ implies that $p_2\in Q$ or $p_3\dots p_k\in Q$. If $p_2\in Q$, we are finished. Otherwise, $p_3\dots p_k\in Q$. Repeating the same procedure, we see that $\{p_1,p_2,\dots ,p_k\}\cap Q\ne \varnothing$. □

Proposition 1.

Let $(\mathrm{\Omega },·,\le )$ be a commutative ordered semigroup and $Q\ne \varnothing \subseteq \mathrm{\Omega }$. Then, Q is a quasi-filter of Ω if and only if Q is a filter of Ω.

Proof. 

Let Q be a filter of $\mathrm{\Omega }$ and $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\Omega }$. Let $x\le \alpha ·\beta$ and $x\le \gamma ·\delta$, with $x\in Q$. Then, $\alpha ·\beta \in Q$ and $\gamma ·\delta \in Q$ as $\left[Q\right)\subseteq Q$. Having Q be a filter of $\mathrm{\Omega }$ implies that $\alpha ,\beta ,\gamma ,\delta \in Q$ and, hence, Q is a quasi-filter of $\mathrm{\Omega }$.

Conversely, let Q be a quasi-filter of $\mathrm{\Omega }$ and $\alpha ·\beta \in Q$. By setting $x=\alpha ·\beta$ and $x\in Q$, we see that $x\le \alpha ·\beta$ and $x\le \beta ·\alpha$. aving Q be a quasi-filter of $\mathrm{\Omega }$ implies that $\left\{\alpha \right\}\cap Q\ne \varnothing$ and $\left\{\beta \right\}\cap Q\ne \varnothing$. The latter implies that $\{\alpha ,\beta \}\subseteq Q$ and, hence, Q is a filter of $\mathrm{\Omega }$. □

Proposition 2.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q\ne \varnothing \subseteq \mathrm{\Omega }$. If Q is a left filter (right filter) of Ω, then, Q is a quasi-filter of Ω.

Proof. 

Let Q be a left filter of $\mathrm{\Omega }$ and $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\Omega }$. Let $x\in Q$ with $x\le \alpha ·\beta$ and $x\le \gamma ·\delta$. Having Q be a left filter of $\mathrm{\Omega }$ implies that $\alpha ,\gamma \in Q$ and, hence, $\{\alpha ,\delta \}\cap \mathrm{\Omega }\ne \varnothing$, and $\{\beta ,\gamma \}\cap \mathrm{\Omega }\ne \varnothing$. Therefore, Q is a quasi-filter of $\mathrm{\Omega }$. The case Q is a right filter of $\mathrm{\Omega }$ can be handled similarly. □

Proposition 3.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q,F\ne \varnothing \subseteq \mathrm{\Omega }$. If Q is a (generalized) quasi-filter of Ω, and F is a filter of Ω, then, $Q\cap F$ is either empty or a (generalized) quasi-filter of Ω.

Proof. 

Let $x\in Q\cap F\ne \varnothing$. One can easily see that $Q\cap F$ is a subsemigroup of $\mathrm{\Omega }$ and that $[Q\cap F)\subseteq Q\cap F$. Suppose that there exist $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\Omega }$ with $x\le \alpha ·\beta$ and $x\le \gamma ·\delta$. Having $x\in F$ and $\left[F\right)\subseteq F$ implies that $\alpha ·\beta \in F,\gamma ·\delta \in F$ and, hence, $\{\alpha ,\beta ,\gamma ,\delta \}\subseteq F$. In addition, having Q be a quasi-filter of $\mathrm{\Omega }$ implies that $\{\alpha ,\delta \}\cap Q\ne \varnothing$ and $\{\beta ,\gamma \}\cap Q\ne \varnothing$. We see now that $\{\alpha ,\delta \}\cap (Q\cap F)\ne \varnothing$ and $\{\beta ,\gamma \}\cap (Q\cap F)\ne \varnothing$. Therefore, $Q\cap F$ is a quasi-filter of $\mathrm{\Omega }$. □

Lemma 3.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q_1,Q_2\ne \varnothing \subseteq \mathrm{\Omega }$ be (generalized) quasi-filters of Ω. Then, $Q_1\cup Q_2$ is a generalized quasi-filter of Ω.

Proof. 

One can easily see that $[Q_1\cup Q_2)\subseteq Q_1\cup Q_2$. Let $x\in Q_1\cup Q_2$, with $x\le \alpha ·\beta$ and $x\le \gamma ·\delta$ for some $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\Omega }$. We have two cases: $x\in Q_1$ and $x\in Q_2$. We deal with the case $x\in Q_1$, and the case $x\in Q_2$ is handled similarly. Since $Q_1$ is a (generalized) quasi-filter of $\mathrm{\Omega }$, it follows that $\{\alpha ,\delta \}\cap Q_1\ne \varnothing$ and $\{\beta ,\gamma \}\cap Q_1\ne \varnothing$. The latter implies that $\{\alpha ,\delta \}\cap (Q_1\cup Q_2)\ne \varnothing$ and $\{\beta ,\gamma \}\cap (Q_1\cup Q_2)\ne \varnothing$ and, hence, $Q_1\cup Q_2$ is a generalized quasi-filter of $\mathrm{\Omega }$. □

Lemma 4.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q_1,Q_2\ne \varnothing \subseteq \mathrm{\Omega }$ be quasi-filters of Ω. Then, $Q_1\cup Q_2$ is a quasi-filter of Ω if and only if $Q_1\cup Q_2$ is a subsemigroup of Ω.

Proof. 

The proof can be easily executed using Lemma 3. □

Lemma 5.

Let $(\mathrm{\Omega },·,\le )$ be an ordered monoid with identity 1 and $Q\ne \varnothing \subseteq \mathrm{\Omega }$ be a (generalized) quasi-filter of Ω. Then, $1\in Q$.

Proof. 

Let $x\in Q\ne \varnothing$; then, $x\le x·1$ and $x\le 1·x$. Since Q is a (generalized) quasi-filter of $\mathrm{\Omega }$, it follows that $1\in Q$. □

Corollary 1.

Let $(\mathrm{\Omega },·,\le )$ be an ordered group. Then, $Q\ne \varnothing \subseteq \mathrm{\Omega }$ is a (generalized) quasi-filter of Ω if and only if $Q=\mathrm{\Omega }$.

Proof. 

Let $Q\ne \varnothing$ be a (generalized) quasi-filter of $\mathrm{\Omega }$. Lemma 5 asserts that $1\in Q$. Having $1=x·x^{-1}$ for all $x\in \mathrm{\Omega }$ implies that $1\le x·x^{-1}$ and $1\le x^{-1}·x$. Since Q is a (generalized) quasi-filter of $\mathrm{\Omega }$, it follows that $x\in Q$ and, hence, $Q=\mathrm{\Omega }$. □

Proposition 4.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup with $0\in \mathrm{\Omega }$ satisfying $0·\theta =\theta ·0=0$ for all $\theta \in \mathrm{\Omega }$ and Q be a (generalized) quasi-filter of Ω. If $0\in Q$, then, $Q=\mathrm{\Omega }$.

Proof. 

For all $x\in \mathrm{\Omega }$, we have $0\le 0·x$ and $0\le x·0$. Having Q be a (generalized) quasi-filter of $\mathrm{\Omega }$ implies that $x\in Q$ and, hence, $Q=\mathrm{\Omega }$. □

Lemma 6.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q\ne \varnothing$ be a proper subset of Ω. Then, $\left[Q\right)\subseteq Q$ if and only if $(\mathrm{\Omega }-Q]\subseteq \mathrm{\Omega }-Q$.

Lemma 7.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q\ne \varnothing$ be a proper subset of Ω. Then, Q is a subsemigroup of Ω if and only if $\mathrm{\Omega }-Q$ is a prime subset of $\mathrm{\Omega }$.

In [2], Lee SK and Lee SS proved that a non-empty proper subset F of $\mathrm{\Omega }$ was a left (right) filter of $\mathrm{\Omega }$ if and only if $\mathrm{\Omega }-F$ was a right (left) ideal of $\mathrm{\Omega }$. The following theorem presents a similar result regarding quasi-filters.

Theorem 1.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q\ne \varnothing$ be a proper subset of Ω. Then, Q is a quasi-filter of Ω if and only if $\mathrm{\Omega }-Q$ is a prime quasi-ideal of Ω.

Proof. 

Let Q be a quasi-filter of $\mathrm{\Omega }$ and $x\in \left(\right(\mathrm{\Omega }-Q\left)\mathrm{\Omega }\right]\cap \left(\mathrm{\Omega }\right(\mathrm{\Omega }-Q\left]\right)$. If $x\notin \mathrm{\Omega }-Q$, then, $x\in Q$ and, hence, there exist $\alpha ,\delta \in \mathrm{\Omega }-Q,\beta ,\gamma \in \mathrm{\Omega }$ such that $x\le \alpha ·\beta$ and $x\le \gamma ·\delta$. Having Q be a quasi-filter of $\mathrm{\Omega }$ implies that $\{\alpha ,\delta \}\cap Q\ne \varnothing$ and $\{\beta ,\gamma \}\cap Q\ne \varnothing$. The latter contradicts the fact that $\{\alpha ,\delta \}\subseteq \mathrm{\Omega }-Q$. Lemmas 6 and 7 complete the proof.

Conversely, suppose that $\mathrm{\Omega }-Q$ is a quasi-ideal of $\mathrm{\Omega }$, and let $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\Omega }$. Let $x\le \alpha \beta$ and $x\le \gamma ·\delta$, with $x\in Q$. Then, $\{\alpha ,\delta \}\cap Q=\varnothing$ or $\{\beta ,\gamma \}\cap Q\ne \varnothing$. Otherwise, $x\in \left(\right(\mathrm{\Omega }-Q\left)\mathrm{\Omega }\right]\cap \left(\mathrm{\Omega }\right(\mathrm{\Omega }-Q\left]\right)\subseteq \mathrm{\Omega }-Q$ contradicts the fact that $x\in Q$. Thus, Q is a quasi-filter of $\mathrm{\Omega }$. Lemmas 6 and 7 complete the proof. □

Corollary 2.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup. Then, Ω has no proper quasi-filters if and only if Ω has no proper prime quasi-ideals.

Proof. 

The proof is an immediate consequence of Theorem 1. □

Remark 2.

An ordered semigroup may have proper quasi-ideals but still has no proper quasi-filters. See Example 4.

Example 4.

Let ${\mathrm{\Omega }}_2=\{0,\mathring{a}\}$, with operation “${·}_2$” and order “ ${\le }_2$ ” described as follows:

${·}_2$	0	$\mathring{a}$
0	0	0
$\mathring{a}$	0	0

$${\le }_2:=\{(0,0),(0,\mathring{a}),(\mathring{a},\mathring{a})\}.$$

It is clear that ${\mathrm{\Omega }}_2$ is an ordered semigroup, and that $\left\{0\right\}$ is a proper quasi-ideal of ${\mathrm{\Omega }}_2$. Moreover, ${\mathrm{\Omega }}_2$ has no proper quasi-filters.

Theorem 2.

Let $(\mathrm{\Omega },·,\le )$ be an ordered semigroup and $Q\ne \varnothing$ be a proper subset of Ω. Then, Q is a generalized quasi-filter of Ω if and only if $\mathrm{\Omega }-Q$ is a quasi-ideal of Ω.

Proof. 

The proof is similar to that of Theorem 1. □

3. (m, n)-Quasi-Filters of Ordered Semigroups

In this section, we generalize the concept of (generalized) quasi-filters of ordered semigroups to (generalized) $(m,n)$-quasi-filters of ordered semigroups. Moreover, we present some non-trivial examples of the new concept and relate them to (generalized) $(m,n)$-quasi-ideals of ordered semigroups.

Throughout this section, m and n are positive integers.

Definition 2.

Let $m,n>0$ be integers, $(\mathrm{\Omega },·,\le )$ be an ordered semigroup, and $Q\ne \varnothing \subseteq \mathrm{\Omega }$. Then, Q is an $(m,n)$-quasi-filter of Ω if the following conditions hold for all $p_1,\dots ,p_{m+1},q_1,\dots ,q_{n+1}\in \mathrm{\Omega }$.

 (1) 

$Q·Q\subseteq Q$;

 (2) 

$\left[Q\right)\subseteq Q$;

 (3) 

If $x\in Q$, $x\le p_1\dots p_m{p}_{m+1}$ and $x\le q_1\dots q_n{q}_{n+1}$, then, $\{p_1,\dots ,p_m,q_2,\dots ,q_{n+1}\}\cap Q\ne \varnothing$, and $\{p_2,\dots ,p_{m+1},q_1,\dots ,q_n\}\cap Q\ne \varnothing$.

If we drop the subsemigroup condition in Definition 2, we obtain Q as a generalized $(m,n)$-quasi-filter.

Proposition 5.

Let $m>0$ be an integer, $(\mathrm{\Omega },·,\le )$ be an ordered semigroup, and $Q\ne \varnothing \subseteq \mathrm{\Omega }$ be an $(m,m)$-quasi filter (generalized $(m,m)$-quasi filter) of Ω. Then, for all $p_1,\dots ,p_m,p_{m+1}\in \mathrm{\Omega }$, $p_1\dots p_m{p}_{m+1}\in Q$ implies that $p_i\in Q$ for some $i\in \{1,\dots ,m+1\}$.

Proof. 

Let $x=p_1\dots p_m{p}_{m+1}\in Q$. Then, $x\le p_1\dots p_m{p}_{m+1}\in Q$ implies that $\{p_1,\dots ,p_m,$ $p_{m+1}\} \cap Q\ne \varnothing$. □

Proposition 6.

Let $m,n>0$ be integers, $(\mathrm{\Omega },·,\le )$ be an ordered semigroup, and $Q\ne \varnothing \subseteq \mathrm{\Omega }$ be a (generalized) quasi-filter of Ω. Then, Q is a (generalized) $(m,n)$-quasi-filter of Ω.

Proof. 

Let $x\in Q$, $x\le p_1\dots p_m{p}_{m+1}$, and $x\le q_1\dots q_n{q}_{n+1}$. Having Q be a (generalized) quasi-filter of $\mathrm{\Omega }$, $x\le p_1(p_2\dots p_m{p}_{m+1})$, and $x\le q_1(q_2\dots q_n{q}_{n+1})$ implies that $\{p_1,q_2\dots q_n{q}_{n+1}\}\cap Q\ne \varnothing$ and $\{q_1,p_2\dots p_m{p}_{m+1}\}\cap Q\ne \varnothing$. We have the following four cases.

Case $p_1,q_1\in Q$. Having

$$q_1\in \{q_1,\dots ,q_n,p_2,\dots p_m,p_{m+1}\}\cap Q\mathrm{and}{p}_1\in \{p_1,\dots ,p_m,q_2,\dots ,q_n,q_{n+1}\}\cap Q$$

implies that

$$\{p_1,\dots ,p_m,q_2,\dots ,q_n,q_{n+1}\}\cap Q\ne \varnothing \mathrm{and}\{q_1,\dots ,q_n,p_2,\dots p_m,p_{m+1}\}\cap Q\ne \varnothing .$$

Case $p_1,p_2\dots p_m{p}_{m+1}\in Q$. Having $p_2\dots p_m{p}_{m+1}\in Q$ and Q be a (generalized) quasi-filter of $\mathrm{\Omega }$, we see that $\{p_2,\dots ,p_m,p_{m+1}\}\cap Q\ne \varnothing$ (by Lemma 2). We see now that

$$p_1\in \{p_1,\dots ,p_m,q_2,\dots ,q_{n+1}\}\cap Q\mathrm{and}\{p_2,\dots ,p_m,p_{m+1}\}\cap Q.$$

The latter implies that

$$\{p_1,\dots ,p_m,q_2,\dots ,q_n,q_{n+1}\}\cap Q\ne \varnothing \mathrm{and}\{q_1,\dots ,q_n,p_2,\dots p_m,p_{m+1}\}\cap Q\ne \varnothing .$$

Case $q_1,q_2\dots q_n{q}_{n+1}\in Q$. Having $q_2\dots q_n{q}_{n+1}\in Q$ and Q a (generalized) quasi-filter of $\mathrm{\Omega }$, we see that $\{q_2,\dots ,q_n,q_{n+1}\}\cap Q\ne \varnothing$. We see now that

$$\{q_2,\dots ,q_n,q_{n+1}\}\cap Q\subseteq \{p_1,\dots ,p_m,q_2,\dots ,q_{n+1}\}\cap Q\mathrm{and}{q}_1\in \{q_1,\dots ,q_n,p_2,\dots ,p_{m+1}\}.$$

The latter implies that

$$\{p_1,\dots ,p_m,q_2,\dots ,q_n,q_{n+1}\}\cap Q\ne \varnothing \mathrm{and}\{q_1,\dots ,q_n,p_2,\dots p_m,p_{m+1}\}\cap Q\ne \varnothing .$$

Case $p_2\dots p_m{p}_{m+1},q_2\dots q_n{q}_{n+1}\in Q$. Having $p_2\dots p_m{p}_{m+1},q_2\dots q_n{q}_{n+1}\in Q$ and Q a (generalized) quasi-filter of $\mathrm{\Omega }$, we see that $\{p_2,\dots ,p_{m+1}\}\cap Q\ne \varnothing$ and $\{q_2,\dots ,q_{n+1}\}\cap Q\ne \varnothing$. We see now that

$$\{q_2,\dots ,q_n,q_{n+1}\}\cap Q\subseteq \{p_1,\dots ,p_m,q_2,\dots ,q_{n+1}\}\cap Q$$

and

$$\{p_2,\dots ,p_{m+1}\}\cap Q\subseteq \{q_1,\dots ,q_n,p_2,\dots ,p_{m+1}\}\cap Q.$$

The latter implies that

$$\{p_1,\dots ,p_m,q_2,\dots ,q_n,q_{n+1}\}\cap Q\ne \varnothing \mathrm{and}\{q_1,\dots ,q_n,p_2,\dots p_m,p_{m+1}\}\cap Q\ne \varnothing .$$

□

Remark 3.

An $(m,n)$-quasi-filter of an ordered semigroup may fail to be a quasi-filter. See Example 5.

Example 5.

Let ${\mathrm{\Omega }}_3=\{\dot{a},\dot{b},\dot{c},\dot{d}\}$, with operation “${·}_3$” and order relation “${\le }_3$” described as follows:

${·}_3$	$\dot{a}$	$\dot{b}$	$\dot{c}$	$\dot{d}$
$\dot{a}$	$\dot{a}$	$\dot{a}$	$\dot{a}$	$\dot{a}$
$\dot{b}$	$\dot{b}$	$\dot{b}$	$\dot{b}$	$\dot{b}$
$\dot{c}$	$\dot{c}$	$\dot{c}$	$\dot{c}$	$\dot{c}$
$\dot{d}$	$\dot{a}$	$\dot{b}$	$\dot{b}$	$\dot{a}$

${\le }_3:=\{(\dot{a},\dot{a}),(\dot{a},\dot{b}),(\dot{b},\dot{b}),(\dot{b},\dot{c}),(\dot{c},\dot{c}),(\dot{d},\dot{d})\}.$ One can easily see that $({\mathrm{\Omega }}_3,{·}_3,{\le }_3)$ is an ordered semigroup, and that $\{\dot{b},\dot{c}\}$ is a $(2,2)$-quasi-filter of ${\mathrm{\Omega }}_3$ that is not a quasi-filter of ${\mathrm{\Omega }}_3$. This is clear because $\dot{b}\le \dot{b}{·}_3\dot{d}=\dot{d}{·}_{3}b\in \{b,c\}$ and $d\notin \{b,c\}.$

Theorem 3.

Let $m,n>0$ be integers, $(\mathrm{\Omega },·,\le )$ be an ordered semigroup, and $Q\ne \varnothing$ be a proper subset of Ω. Then, Q is an $(m,n)$-quasi-filter of Ω if and only if $\mathrm{\Omega }-Q$ is a prime generalized $(m,n)$-quasi-ideal of Ω.

Proof. 

Let Q be an $(m,n)$-quasi-filter of $\mathrm{\Omega }$ and $x\in \left({(\mathrm{\Omega }-Q)}^m\mathrm{\Omega }\right]\cap \left(\mathrm{\Omega }{(\mathrm{\Omega }-Q)}^n\right]$. Then, there exist $p_1,\dots ,p_m,q_1,\dots ,q_n\in \mathrm{\Omega }-Q$, $\alpha ,\beta \in \mathrm{\Omega }$ such that $x\le p_1\dots p_m\alpha$ and $x\le \beta q_1\dots q_n$. If $x\notin \mathrm{\Omega }-Q$, then, $x\in Q$. Having Q an $(m,n)$-quasi-filter of $\mathrm{\Omega }$ implies that $\{p_1,\dots ,p_m,q_1,\dots ,q_n\}\cap Q\ne \varnothing$ and $\{p_2,\dots ,p_m,\alpha ,\beta ,q_1,\dots ,q_{n-1}\}\cap Q\ne \varnothing$. Having $\{p_1,\dots ,p_m,q_1,\dots ,q_n\}\cap Q\ne \varnothing$ contradicts the fact that $\{p_1,\dots ,p_m,q_1,\dots ,q_n\}\subseteq \mathrm{\Omega }-Q$. Lemmas 6 and 7 complete the proof.

Conversely, let $\mathrm{\Omega }-Q$ be a prime generalized $(m,n)$-quasi-filter of $\mathrm{\Omega }$ and $x\in Q$, with $x\le p_1\dots p_m{p}_{m+1}$ and $x\le q_1\dots q_n{q}_{n+1}$ for some $p_1,\dots ,p_m,p_{m+1},q_1,\dots ,q_n,q_{n+1}\in \mathrm{\Omega }$. Suppose that $\{p_1,\dots ,p_m,q_2,\dots ,q_{n+1}\}\cap Q=\varnothing$ or $\{p_2,\dots ,p_{m+1},q_2,\dots ,q_n\}\cap Q=\varnothing$. Then, $p_1\dots p_m{p}_{m+1}\in \left({(\mathrm{\Omega }-Q)}^m\mathrm{\Omega }\right]$, and $q_1\dots q_n{q}_{n+1}\in \left(\mathrm{\Omega }{(\mathrm{\Omega }-Q)}^n\right]$. Having $\mathrm{\Omega }-Q$ be a prime generalized $(m,n)$-quasi-ideal of $\mathrm{\Omega }$ implies that $x\in \left({(\mathrm{\Omega }-Q)}^m\mathrm{\Omega }\right]\cap \left(\mathrm{\Omega }{(\mathrm{\Omega }-Q)}^n\right]\subseteq \mathrm{\Omega }-Q$, which contradicts the fact that $x\in Q$. Lemmas 6 and 7 complete the proof. □

Theorem 4.

Let $m,n>0$ be integers, $(\mathrm{\Omega },·,\le )$ be an ordered semigroup, and $Q\ne \varnothing$ be a proper subset of Ω. Then, Q is a generalized $(m,n)$-quasi-filter of Ω if and only if $\mathrm{\Omega }-Q$ is a generalized $(m,n)$-quasi-ideal of Ω.

Proof. 

The proof is similar to that of Theorem 3. □

Theorem 5.

Let $m,n>0$ be integers, $(\mathrm{\Omega },·,\le )$ be an $(m,n)$-regular semigroup, and $Q\ne \varnothing \subseteq \mathrm{\Omega }$. Then, Q is a (generalized) $(m,n)$-quasi-filter of Ω if and only if Q is a (generalized) quasi-filter of Ω.

Proof. 

Let Q be a (generalized) $(m,n)$-quasi-filter of $\mathrm{\Omega }$ and $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\Omega }$. Since $\mathrm{\Omega }$ is an $(m,n)$-regular semigroup, it follows that there exist $z_1,z_2,z_3,z_4\in \mathrm{\Omega }$ such that $\alpha \le {\alpha }^m{z}_1{\alpha }^n$, $\beta \le {\beta }^m{z}_2{\beta }^n$, $\gamma \le {\gamma }^m{z}_3{\gamma }^n$, and $\delta \le {\delta }^m{z}_4{\delta }^n$. Let $x\in Q$, with $x\le \alpha ·\beta$ and $x\le \gamma ·\delta$. Then, $x\le {\alpha }^m\left(z_1{\alpha }^n{\beta }^m{z}_2\right){\beta }^n$, and $x\le {\gamma }^m\left(z_3{\gamma }^n{\delta }^m{z}_4\right){\delta }^n$. Having Q be a (generalized) $(m,n)$-quasi-filter of $\mathrm{\Omega }$ implies that $\{\alpha ,\delta \}\cap Q\ne \varnothing$ and $\{\beta ,\gamma \}\cap Q\ne \varnothing$ and, hence, Q is a (generalized) quasi-filter of $\mathrm{\Omega }$. □

4. Conclusions and Future Directions

This paper introduced new types of filters in ordered semigroups. More precisely, it discussed quasi-filters and $(m,n)$-quasi-filters of ordered semigroups by exploring their properties and finding their relationships with quasi-ideals and $(m,n)$-quasi-ideals. The main results were formulated in five theorems, i.e., Theorems 1–5.

For future work, we raise the following ideas:

• Study quasi-filters in certain special ordered semigroups, such as regular semigroups and intra-regular semigroups.
• Define quasi-filters and $(m,n)$-quasi-filters in other ordered algebraic structures, such as ordered semirings.