On almost (m, n)-ideals and fuzzy almost (m, n)-ideals in semigroups

Abstract

In this paper, we define almost $(m,n)$-ideals of semigroups by using the concepts of $(m,n)$-ideals and almost ideals of semigroups. An almost $(m,n)$-ideal is a generalization of $(m,n)$-ideals and a generalization of almost one-sided ideals. We investigate properties of almost $(m,n)$-ideals of semigroups. Moreover, we define fuzzy almost $(m,n)$-ideals of semigroups and give relationships between almost $(m,n)$-ideals and fuzzy almost $(m,n)$-ideals.

Keywords:

• (m, n)-ideals
• almost (m, n)-ideals
• fuzzy almost $(m.n)$-ideals

2010 Mathematics Subject Classifications:

• 20M12
• 03E72

1. Introduction and preliminaries

This notion of $(m,n)$-ideals of semigroups was first introduced and studied by Lajos in [1]. He investigated remarkable properties of $(m,n)$-ideals of semigroups in [2–6]. Let m and n be non-negative integers. A subsemigroup A of a semigroup S is called an $(m,n)$-ideal of S if $A^{m}S{A}^n\subseteq A.$ Note that a left ideal of a semigroup S is a (0,1)-ideal of S and a right ideal of S is a (1,0)-ideal of S. An $(m,n)$-ideal is a one of generalizations of one-sided ideals. Furthermore, the theory of $(m,n)$-ideals in other structures have also been studied by many authors, for example, $(m,n)$-ideals in ordered semigroups were studied by Bussaban and Changphas in [7] and in LA-semigroups were studied by Akram et al. in [8], etc. In [9], Omidi and Davaz defined $(m,n)$-hyperideals and $(m,n)$-bi-hyperideals in ordered semihyperrings and investigate some of their related properties. Recently, Khan and Mahboob characterized $(m,n)$-filters of $(m,n)$-regular ordered semigroups in terms of its prime generalized $(m,n)$-ideals in [10].

In 1965, Zadeh introduced the fundamental fuzzy set concept in [11]. Since then, fuzzy sets are now applied in various fields. A fuzzy subset of S is a function from S into the closed interval $[0,1]$. For any two fuzzy subsets f and g of S,

1. $f\cap g$ is a fuzzy subset of S defined by $(f\cap g)\left(x\right)=min\left\{f\right(x),g(x\left)\right\}=f\left(x\right)\wedge g\left(x\right)$ for all $x\in S$,

2. $f\cup g$ is a fuzzy subset of S defined by $(f\cup g)\left(x\right)=max\left\{f\right(x),g(x\left)\right\}=f\left(x\right)\vee g\left(x\right)$ for all $x\in S$ and

3. $f\subseteq g$ if $f\left(x\right)\le g\left(x\right)$ for all $x\in S$.

For a fuzzy subset f of S, the support of f is defined by $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)=\{x\in S\mid f(x)\ne 0\}.$ The characteristic mapping of a subset A of S is a fuzzy subset of S defined by $C_A\left(x\right)=\left\{\begin{array}{ll}1,& x\in A,\\ 0,& x\notin A.\end{array}\right.$ The definition of fuzzy points was given by Pu and Liu [12]. For $x\in S$ and $\alpha \in (0,1]$, a fuzzy point $x_{\alpha }$ of a set S is a fuzzy subset of a set S defined by $x_{\alpha }\left(y\right)=\left\{\begin{array}{ll}\alpha ,& y=x,\\ 0,& y\ne x.\end{array}\right.$ Some interesting topics of fuzzy points were studied in [13–15]. Let $F\left(S\right)$ be the set of all fuzzy subsets in a semigroup S. The semigroup S itself is a fuzzy subset of S such that $S\left(x\right)=1$ for all $x\in S,$ denoted also by S. For each $f,g\in F\left(S\right)$, the product of f and g is a fuzzy subset $f\circ g$ defined as follows: $\begin{array}{rl}& (f\circ g)\left(x\right)\\ & =\left\{\begin{array}{ll}\underset{x=ab}{\bigvee }f\left(a\right)\wedge g\left(b\right)& \mathrm{i}\mathrm{f}x=ab\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}a,b\in S,\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}\right.\end{array}$ for all $x\in S$. Then $F\left(S\right)$ is a semigroup with the product °. An introductory definition of left, right, two-sided almost ideals of semigroups were launched in 1980 by Grosek and Satko [16]. They characterized these ideals when a semigroup S contains no proper left, right, two-sided almost ideals in [16], and afterwards they discovered minimal almost ideals and smallest almost ideals of semigroups in [17,18], respectively. A nonempty subset A of a semigroup S is called a left almost ideal of S if $sA\cap A\ne \mathrm{\varnothing }$ for any $s\in S.$ A right almost ideal of a semigroup S is defined analogously. A nonempty subset A of a semigroup S is called an almost ideal of S if $sA\cap A\ne \mathrm{\varnothing }$ and $At\cap A\ne \mathrm{\varnothing }$ for all $s,t\in S.$ In 1981, Bogdanovic [19] introduced the notion of almost bi-ideals in semigroups by using the concepts of almost ideals and bi-ideals of semigroups. Likewise, Wattanatripop, Chinram and Changphas examined quasi-almost-ideals of semigroups and gave properties of quasi-almost-ideals in [20]. Furthermore, they defined fuzzy almost ideals of semigroups in [20] and fuzzy almost bi-ideals of semigroups in [21] and provided relationship between almost ideals and fuzzy almost ideals of semigroups. Recently, Gaketem generalized results in [21] to study interval-valued fuzzy almost bi-ideals of semigroups in [22]. In [23], Solano, Suebsung and Chinram extended this idea to study almost ideals of n-ary semigroups.

Our purpose of this paper is to define the notion of almost $(m,n)$- ideals of semigroups by using the concepts of $(m,n)$-ideals and almost ideals of semigroups and study them. Moreover, we define the notion of fuzzy almost $(m,n)$-ideals of semigroups and give relationships between almost $(m,n)$-ideals and fuzzy almost $(m,n)$-ideals of semigroups.

2. Almost $(m,n)$-ideals

Let m and n be non-negative integers. Let A be a non-empty subset of a semigroup S and $s\in S$. Note that $A^{0}s{A}^0:=\left\{s\right\}$. For $k\in \mathbb{N}$, let $A^{k}s{A}^0:=A^{k}s$ and $A^{0}s{A}^k:=s{A}^k$. Firstly, we define an almost $(m,n)$-ideal of semigroup by using the concepts of $(m,n)$-ideals defined in [1] and almost ideals of semigroups defined in [16].

Definition 2.1

Let S be a semigroup. A non-empty subset A of S is called an almost $(m,n)$-ideal of S if $A^{m}s{A}^n\cap A\ne \mathrm{\varnothing }\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}s\in S.$

Remark 2.1

The following statements hold.

1. An almost $(1,0)$-ideal of a semigroup S is a right almost ideal of S defined in [15].

2. An almost $(0,1)$-ideal of a semigroup S is a left almost ideal of S defined in [15].

3. Every $(m,n)$-ideal of a semigroup S is an almost $(m,n)$-ideal of S.

4. Consider the semigroup ${\mathbb{Z}}_6$ under the usual addition. We have $A=\{\overline{1},\overline{4},\overline{5}\}$ is a $(1,0)$-ideal of ${\mathbb{Z}}_6$ but A is not a subsemigroup of ${\mathbb{Z}}_6$. Therefore, an almost $(m,n)$-ideal of a semigroup S need not be a subsemigroup of S and need not be an $(m,n)$-ideal of S.

Proposition 2.2

If A is an almost $(m,n)$-ideal of a semigroup S, then every subset H of S such that $A\subseteq H$ is an almost $(m,n)$-ideal of S.

Proof.

Assume that A is an almost $(m,n)$-ideal of S and H is a subset of S with $A\subseteq H.$ Then $\mathrm{\varnothing }\ne A^{m}s{A}^n\cap A\subseteq H^{m}s{H}^n\cap H$ for all $s\in S$. Therefore H is an almost $(m,n)$-ideal of S.

Corollary 2.3

The union of two almost $(m,n)$-ideals of a semigroup S is an almost $(m,n)$-ideal of S.

Proof.

Let $A_1$ and $A_2$ be any two almost $(m,n)$-ideals of S. Then $A_1\subseteq A_1\cup A_2$. By Proposition 2.2, $A_1\cup A_2$ is an almost $(m,n)$-ideal of S.

Note that in the proof of Corollary 2.3 is true if $A_1$ or $A_2$ is an almost $(m,n)$-ideal of S.

Example 2.4

Consider the semigroup ${\mathbb{Z}}_6$ under the usual addition. We have $A_1=\{\overline{1},\overline{4},\overline{5}\}$ and $A_2=\{\overline{1},\overline{2},\overline{5}\}$ are almost (1,0)-ideals of ${\mathbb{Z}}_6$ but $A_1\cap A_2=\{\overline{1},\overline{5}\}$ is not an almost (1,0)-ideal of S.

Example 2.4 implies that, in general, the intersection of two almost $(m,n)$-ideals of a semigroup S need not be an almost $(m,n)$-ideal of S.

Theorem 2.5

Let S be a semigroup such that $|S|>1$. A semigroup S has no proper almost $(m,n)$-ideal if and only if for any $a\in S$ there exists $s_a\in S$ such that $(S\setminus \{a\left\}{)}^m{s}_a\right(S\setminus \left\{a\right\}{)}^n=\left\{a\right\}$.

Proof.

Assume that S has no proper almost $(m,n)$-ideal and let $a\in S$. Then $S\setminus \left\{a\right\}$ is not an almost $(m,n)$-ideal of S. Then there exists $s_a\in S$ such that $\left[\right(S\setminus \left\{a\right\}{)}^m{s}_a(S\setminus \{a\left\}{)}^n\right]\cap (S\setminus \{a\left\}\right)=\mathrm{\varnothing }.$ This implies that $(S\setminus \{a\left\}{)}^m{s}_a\right(S\setminus \left\{a\right\}{)}^n]=\{a\}.$ Conversely, let $a\in S$. Then there exists $s_a\in S$ such that $(S\setminus \{a\left\}{)}^m{s}_a\right(S\setminus \left\{a\right\}{)}^n=\left\{a\right\}.$ This implies that $\left[\right(S\setminus \left\{a\right\}{)}^m{s}_a(S\setminus \{a\left\}{)}^n\right]\cap (S\setminus \{a\left\}\right)=\mathrm{\varnothing }.$ Hence, $S\setminus \left\{a\right\}$ is not an almost $(m,n)$-ideal of S for all $a\in S.$ Suppose that S has a proper almost $(m,n)$-ideal B. Then $B\subseteq S\setminus \left\{a^{\prime }\right\}$ for some $a^{\prime }\in S$. By Theorem 2.2, $S\setminus \left\{a^{\prime }\right\}$ is also an almost $(m,n)$-ideal of S, this is contradiction. Therefore S has no proper almost $(m,n)$-ideal.

Theorem 2.6

Let S be a semigroup such that $|S|>1$ and $a\in S$. If $S\setminus \left\{a\right\}$ is not an almost $(m,n)$-ideal of S, then at least one of them is true.

1. $a=a^{m+n+1}$.

2. $a=a^{(m+n{)}^3+1}$.

3. $a=a^{(m+n+1{)}^{m+n}+1}$.

Proof.

Assume that $S\setminus \left\{a\right\}$ is not an almost $(m,n)$-ideal of S. Then there exists $s_a\in S$ such that $\left[\right(S\setminus \left\{a\right\}{)}^m{s}_a(S\setminus \{a\left\}{)}^n\right]\cap (S\setminus \{a\left\}\right)=\mathrm{\varnothing }$.

Case 1: $s_a\ne a.$ Then $s_a\in S\setminus \left\{a\right\}$. This implies that $\left(s_a{)}^m{s}_a\right(s_a{)}^n=a$. So $a=(s_a{)}^{m+n+1}$. Suppose that $a\ne a^{m+n+1}$. Then $a^{m+n+1}\in S\setminus \left\{a\right\}$, so $\left(a^{m+n+1}{)}^m{s}_a\right(a^{m+n+1}{)}^n=a$. Hence $a=(a^{m+n+1}{)}^{m+n}{s}_a$.

Case 1.1: If $a^{(m+n{)}^2}{s}_a=a$, then $a=(a^{m+n+1}{)}^{m+n}{s}_a=a^{m+n}{a}^{(m+n{)}^2}{s}_a=a^{m+n+1}$ which is a contradiction.

Case 1.2: If $a^{(m+n{)}^2}{s}_a\ne a$, then $a^{(m+n{)}^2}{s}_a\in S\setminus \left\{a\right\}$. Thus $\left(a^{(m+n{)}^2}{s}_a{)}^m{s}_a\right(a^{(m+n{)}^2}{s}_a{)}^n=a.$ This implies that $a=(s_a{)}^{m+n+1}{a}^{(m+n{)}^3}=a^{(m+n{)}^3+1}.$

Case 2: $s_a=a.$ Suppose that $a\ne a^{m+n+1}$. Then $a^{m+n+1}\in S\setminus \left\{a\right\}$. So $\left(a^{m+n+1}{)}^m{s}_a\right(a^{m+n+1}{)}^n=a$. Therefore, $a=\left(a^{m+n+1}{)}^{m}a\right(a^{m+n+1}{)}^n=a^{(m+n+1{)}^{m+n}+1}.$

Corollary 2.7

Let S be a semigroup such that $|S|>1$ and $a\in S$. If $S\setminus \left\{a\right\}$ is not a left $[$right$]$ almost ideal of S, then $a=a^2$ or $a=a^3$.

Proof.

By Theorem 2.6, choose m=0 and n=1 [choose m=1 and n=0].

3. Fuzzy almost $(m,n)$-ideals

In this section, we define and study fuzzy almost $(m,n)$-ideal and give relationships between fuzzy almost $(m,n)$-ideals and almost $(m,n)$-ideals. Let m and n be non-negative integers. Let f be a fuzzy subset and $(x{)}_{\alpha }$ be a fuzzy point of a semigroup S. Note that $f^0:=S$ and $f^0\circ (x{)}_{\alpha }\circ f^0:=(x{)}_{\alpha }$. For $k\in \mathbb{N}$, let

1. $f^k:=\underset{k\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}{\underbrace{f\circ f\circ \dots \circ f}}$,

2. $f^k\circ (x{)}_{\alpha }\circ f^0:=f^k\circ (x{)}_{\alpha }$ and

3. $f^0\circ (x{)}_{\alpha }\circ f^k:=(x{)}_{\alpha }\circ f^k.$

Proposition 3.1

Let f,g and h be fuzzy subsets of S.

1. If $f\subseteq g,$ then $f^n\subseteq g^n$ for all $n\in \mathbb{N}\cup \left\{0\right\}$.

2. If $f\subseteq g,$ then $f\circ h\subseteq g\circ h$.

3. If $f\subseteq g,$ then $f\cap h\subseteq g\cap h$.

Proof.

The proof is straightforward.

Definition 3.2

A fuzzy subset f of a semigroup S is called a fuzzy almost $(m,n)$-ideal of S if $(f^m\circ (x{)}_{\alpha }\circ f^n)\cap f\ne 0$ for all fuzzy point $(x{)}_{\alpha }$ of S.

This implies that f is a fuzzy almost $(m,n)$-ideal of S if for all fuzzy point $(x{)}_{\alpha }$ of S, there exists $y\in S$ such that $\left[\right(f^m\circ (x{)}_{\alpha }\circ f^n)\cap f\left]\right(y)\ne 0$.

Proposition 3.3

Let f be a fuzzy almost $(m,n)$-ideal of S and g be a fuzzy subset of S such that $f\subseteq g$. Then g is a fuzzy almost $(m,n)$-ideal of S.

Proof.

Assume that f is a fuzzy almost $(m,n)$-ideal of S and g is a fuzzy subset of S such that $f\subseteq g.$ Let $(x{)}_{\alpha }$ be a fuzzy point in S. We have $0\ne (f^m\circ (x{)}_{\alpha }\circ f^n)\cap f\subseteq (g^m\circ (x{)}_{\alpha }\circ g^n)\cap g.$ Therefore, g is a fuzzy almost $(m,n)$-ideal of S.

Corollary 3.4

Let f and g be fuzzy almost $(m,n)$-ideals of S. Then $f\cup g$ is a fuzzy almost $(m,n)$-ideal of S.

Proof.

Since $f\subseteq f\cup g$, by Proposition 3.3, $f\cup g$ is a fuzzy almost $(m,n)$-ideal of S.

Note that in the proof of Corollary 3.4 is true if f or g is a fuzzy almost $(m,n)$-ideal of S.

Example 3.5

Consider n=1,m=0 and the semigroup ${\mathbb{Z}}_6$ under the usual addition, $f:{\mathbb{Z}}_6\to [0,1]$ is defined by $\begin{array}{rl}& f\left(\overline{0}\right)=0,f\left(\overline{1}\right)=0.2,f\left(\overline{2}\right)=0,f\left(\overline{3}\right)=0,\\ & f\left(\overline{4}\right)=0.5,f\left(\overline{5}\right)=0.3\end{array}$ and $g:{\mathbb{Z}}_6\to [0,1]$ defined by $\begin{array}{rl}& g\left(\overline{0}\right)=0,g\left(\overline{1}\right)=0.8,g\left(\overline{2}\right)=0.4,g\left(\overline{3}\right)=0,\\ & g\left(\overline{4}\right)=0,g\left(\overline{5}\right)=\mathrm{0.3.}\end{array}$ We have f and g are fuzzy almost (1,0)-ideals of ${\mathbb{Z}}_6$ but $f\cap g$ is not a fuzzy almost (1,0)-ideal of ${\mathbb{Z}}_6$.

Example 3.5 implies that, in general, the intersection of two fuzzy almost $(m,n)$-ideals of S need not be a fuzzy almost $(m,n)$-ideal of S.

Note that for a subset A of S, define $A^0:=S$.

Lemma 3.6

Let A be a subset of S and $n\in \mathbb{N}\cup \left\{0\right\}$. Then $(C_A{)}^n=C_{A^n}$.

Proof.

The proof is straightforward.

Theorem 3.7

Let A be a nonempty subset of a semigroup S. Then A is an almost $(m,n)$-ideal of S if and only if $C_A$ is a fuzzy almost $(m,n)$-ideal of S.

Proof.

Assume that A is an almost $(m,n)$-ideal of S. Then $A^{m}s{A}^n\cap A\ne \mathrm{\varnothing }$ for all $s\in S$. Let $s\in S$ and $\alpha \in (0,1]$. Thus there exists $x\in A^{m}s{A}^n\cap A.$ So $\left[\right[C_{A^m}\circ (s{)}_{\alpha }\circ C_{A^n}]\cap C_A\left]\right(x)\ne 0.$ By Lemma 3.6, we have $\left[\right[(C_A{)}^m\circ (s{)}_{\alpha }\circ \left(C_A{)}^n\right]\cap C_A\left]\right(x)\ne 0.$ Hence, $C_A$ is a fuzzy almost $(m,n)$-ideal of S.

Conversely, assume that $C_A$ is a fuzzy almost $(m,n)$-ideal of S. Let $s\in S$ and $\alpha \in (0,1]$. Thus $\left[\right(C_A{)}^m\circ (s{)}_{\alpha }\circ (C_A{)}^n]\cap C_A\ne 0.$ Then there exists $x\in S$ such that $\left[\right[(C_A{)}^m\circ (s{)}_{\alpha }\circ \left(C_A{)}^n\right]\cap C_A\left]\right(x)\ne 0.$ By Lemma 3.6, we have $\left[\right[C_{A^m}\circ (s{)}_{\alpha }\circ C_{A^n}]\cap C_A\left]\right(x)\ne 0.$ Hence, $x\in A^{m}s{A}^n\cap A.$ Eventually, $A^{m}s{A}^n\cap A\ne \mathrm{\varnothing }$.

Theorem 3.8

Let f be a fuzzy subset of S. Then f is a fuzzy almost $(m,n)$-ideal of S if and only if $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)$ is an almost $(m,n)$-ideal of S.

Proof.

Assume that f is a fuzzy almost $(m,n)$-ideal of S. Let $x\in S$. Then for any $\alpha \in (0,1],$ we have $(f^m\circ (x{)}_{\alpha }\circ f^n)\cap f\ne 0.$ Thus, there exists $y\in S$ such that $\left[\right(f^m\circ (x{)}_{\alpha }\circ f^n)\cap f\left]\right(y)\ne 0.$ So, $f\left(y\right)\ne 0$ and $y=a_1{a}_2\cdots a_{m}x{b}_1{b}_2\cdots b_n$ for some $a_1,a_2,\dots ,a_m,b_1,b_2,\dots ,b_n\in S$ such that $\begin{array}{rl}& f\left(a_1\right)\ne 0,f\left(a_2\right)\ne 0,\dots ,f\left(a_m\right)\ne 0,f\left(b_1\right)\ne 0,\\ & f\left(b_2\right)\ne 0,\dots ,f\left(b_n\right)\ne 0.\end{array}$ This implies that $a_1,a_2,\dots ,a_m,b_1,b_2,\dots ,b_n,y\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)$. Thus, $\left[\right(C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^m\circ (x{)}_{\alpha }\circ (C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^n\left]\right(y)\ne 0$ and $C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}\left(y\right)\ne 0$. Hence, $\left[\right(C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^m\circ (x{)}_{\alpha }\circ (C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^n\cap C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}\left]\right(y)\ne 0.$ So, $C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}$ is a fuzzy almost $(m,n)$-ideal of S. By Theorem 3.7, $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)$ is an almost $(m,n)$-ideal of S.

Conversely, assume that $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)$ is an almost $(m,n)$-ideal of S. By Theorem 3.7, $C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}$ is a fuzzy almost $(m,n)$-ideal of S. Let $(x{)}_{\alpha }$ be a fuzzy point in S. Then $\left[\right((C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^m\circ (x{)}_{\alpha }\circ \left(C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^n\right)\cap C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}]\ne 0.$ Then there exists $y\in S$ such that $\left[\right((C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^m\circ (x{)}_{\alpha }\circ \left(C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^n\right)\cap C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}\left]\right(y)\ne 0.$ Hence, $(C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^m\circ (x{)}_{\alpha }\circ \left(C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}{)}^n\right)\left)\right(y)\ne 0$ and $C_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)}\left(y\right)\ne 0.$ Then there exist $a_1,a_2,\dots ,a_m,b_1,b_2,\dots ,b_n\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)$ and $y=a_1{a}_2\cdots a_{m}x{b}_1{b}_2\cdots b_n$. Thus $\begin{array}{rl}& f\left(y\right)\ne 0,f\left(a_1\right)\ne 0,f\left(a_2\right)\ne 0,\dots ,f\left(a_m\right)\ne 0,\\ & f\left(b_1\right)\ne 0,f\left(b_2\right)\ne 0,\dots ,f\left(b_n\right)\ne 0.\end{array}$ Therefore, $(f^m\circ (x{)}_{\alpha }\circ f^n\left)\right(y)\ne 0.$ This implies that $\left[\right(f^m\circ (x{)}_{\alpha }\circ f^n)\cap f\left]\right(y)\ne 0.$ Consequently, f is a fuzzy almost $(m,n)$-ideal of S.

3.1. Minimal almost $(m,n)$-ideals and minimal fuzzy almost $(m,n)$-ideals

In this subsection, we give relationship between minimal almost $(m,n)$-ideals and minimal fuzzy almost $(m,n)$-ideals.

Definition 3.9

A fuzzy almost $(m,n)$-ideal f is called minimal if for all nonzero fuzzy almost $(m,n)$-ideals g of S such that $g\subseteq f$, we have $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(g\right).$

Theorem 3.10

Let S be a non-empty subset of a semigroup S. Then A is a minimal almost $(m,n)$-ideal of S if and only if $C_A$ is a minimal fuzzy almost $(m,n)$-ideal of S.

Proof.

Assume that A is a minimal almost $(m,n)$-ideal of S. By Theorem 3.7, $C_A$ is a fuzzy almost $(m,n)$-ideal of S. Let f be a fuzzy almost $(m,n)$-ideal of S such that $f\subseteq C_A.$ Then $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)\subseteq \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(C_A\right)=A.$ By Theorem 3.8, $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)$ is an almost $(m,n)$-ideal of S. Since A is minimal, $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)=A=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(C_A\right).$ Therefore, $C_A$ is minimal.

Conversely, assume that $C_A$ is a minimal fuzzy almost $(m,n)$-ideal of S. Let B be an almost $(m,n)$-ideal of S such that $B\subseteq A.$ Then $C_B$ is a fuzzy almost $(m,n)$-ideal of S such that $C_B\subseteq C_A.$ Hence, $B=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(C_B\right)=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(C_A\right)=A.$ Therefore, A is minimal.

Corollary 3.11

S has no proper almost $(m,n)$-ideal if and only if for all fuzzy almost $(m,n)$-ideals f of S, $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(f\right)=S$.

Proof.

This follows by Theorem 3.10.

3.2. Prime almost $(m,n)$-ideals and prime fuzzy almost $(m,n)$-ideals

In this subsection, we give relationship between prime almost $(m,n)$-ideals and prime fuzzy almost $(m,n)$-ideals.

Definition 3.12

Let S be a semigroup.

1. An almost $(m,n)$-ideal A of S is called prime if for all $x,y\in S,xy\in A$ implies $x\in A$ or $y\in A$.

2. A fuzzy almost $(m,n)$-ideal g of S is called prime if for all $x,y\in S,f\left(xy\right)\le max\left\{f\right(x),f(y\left)\right\}$.

Theorem 3.13

Let A be a non-empty subset of S. Then A is a prime almost $(m,n)$-ideal of S if and only if $C_A$ is a prime fuzzy almost $(m,n)$-ideal of S.

Proof.

Assume that A is a prime almost $(m,n)$-ideal of S. By Theorem 3.7, $C_A$ is a fuzzy almost $(m,n)$-ideal of S. Let $x,y\in S$. We consider two cases:

Case 1: $xy\in A$. So, $x\in A$ or $y\in A$. Then $max\left\{C_A\right(x),C_A(y\left)\right\}=1\ge C_A\left(xy\right)$.

Case 2: $xy\notin A$. Then $C_A\left(xy\right)=0\le max\left\{C_A\right(x),C_A(y\left)\right\}$.

Thus, $C_A$ is a prime fuzzy almost $(m,n)$-ideal of S. Conversely, assume that $C_A$ is a prime fuzzy almost $(m,n)$-ideal of S. By Theorem 3.7, A is an almost $(m,n)$-ideal of S. Let $x,y\in S$ such that $xy\in A$. Then $C_A\left(xy\right)=1$. By assumption, $C_A\left(xy\right)\le max\left\{C_A\right(x),C_A(y\left)\right\}$. Therefore, $max\left\{C_A\right(x),C_A(y\left)\right\}=1.$ Hence, $x\in A$ or $y\in A$. Thus, A is a prime almost $(m,n)$-ideal of S.

3.3. Semiprime almost $(m,n)$-ideals and semiprime fuzzy almost $(m,n)$-ideals

In this subsection, we give relationship between semiprime almost $(m,n)$-ideals and semiprime fuzzy almost $(m,n)$-ideals.

Definition 3.14

Let S be a semigroup.

1. An almost $(m,n)$-ideal A of S is called semiprime if for all $x\in S,x^2\in A$ implies $x\in A$.

2. A fuzzy almost $(m,n)$-ideal f is called semiprime if for all $x\in S,f\left(x^2\right)\le f\left(x\right)$.

Theorem 3.15

Let A be a non-empty subset of S. Then A is a semiprime almost $(m,n)$-ideal of S if and only if $C_A$ is a semiprime fuzzy almost $(m,n)$-ideal of S.

Proof.

Assume that A is a semiprime almost $(m,n)$-ideal of S. By Theorem 3.7, $C_A$ is a fuzzy almost $(m,n)$-ideal of S. Let $x\in S$. We consider two cases:

Case 1: $x^2\in A$. Then $x\in A$. So, $C_A\left(x\right)=1$. Hence, $C_A\left(x\right)\ge C_A\left(x^2\right)$.

Case 2: $x^2\notin A$. Then $C_A\left(x^2\right)=0\le C_A\left(x\right)$.

Thus, $C_A$ is a semiprime fuzzy almost $(m,n)$-ideal of S. Conversely, assume that $C_A$ is a semiprime fuzzy almost $(m,n)$-ideal of S. By Theorem 3.7, A is an almost $(m,n)$-ideal of S. Let $x\in S$ such that $x^2\in A$. Then $C_A\left(x^2\right)=1$. By assumption, $C_A\left(x^2\right)\le C_A\left(x\right)$. Since $C_A\left(x^2\right)=1$, $C_A\left(x\right)=1$. Hence, $x\in A$. Thus, A is a semiprime almost $(m,n)$-ideal of S.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ronnason Chinram  http://orcid.org/0000-0002-6113-3689

Additional information

Funding

This paper was supported by Algebra and Applications Research Unit, Prince of Songkla University.