On generalizations of quasi-prime ideals of an ordered left almost semigroups

Abstract

The purposes of this paper are to introduce generalizations of quasi-prime ideals to the context of $\phi$-quasi-prime ideals. Let $\phi : {\mathcal {I}}(S) \rightarrow {\mathcal {I}}(S) \cup \left\{ \emptyset \right\}$ be a function where ${\mathcal {I}}(S)$ is the set of all left ideals of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S. A proper left ideal A of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S is called a $\phi$-quasi-prime ideal, if for each $a, b\in S$ with $ab \in A - \phi (A)$, then $a \in A$ or $b\in A$. Some characterizations of quasi-prime and $\phi$-quasi-prime ideals are obtained. Moreover, we investigate relationships between weakly quasi-prime, almost quasi-prime, $\omega$-quasi-prime, m-quasi-prime and $\phi$-quasi-prime ideals of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. Finally, we obtain necessary and sufficient conditions of $\phi$-quasi-prime ideal in order to be a quasi-prime ideal.

1 Introduction

In 2010, Shah et al. [35] studied ideals, M-systems, N-systems and I-systems of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups and provided that if A is a left ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left identity, then A is quasi-prime if and only if $S- A$ is an M-system; A is quasi-semiprime if and only if $S - A$ is an N-system and A is quasi-irreducible if and only if $S - A$ is an I-system. Nowadays many scholars have studied different aspects of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups see [4, 7, 14, 33, 44, 47,48,49]. In 2012, Faisal et al. [13] introduced the notion of fuzzy ordered $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups and studied (2, 2)-regular ordered $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$**-semigroup in terms of fuzzy $\varGamma$-left ideals, fuzzy $\varGamma$-right ideals, fuzzy $\varGamma$-two-sided ideals, fuzzy $\varGamma$-generalized bi-ideals, fuzzy $\varGamma$-bi-ideals, fuzzy $\varGamma$-interior ideals and fuzzy $\varGamma$-(1; 2)-ideals. They proved that the set of all fuzzy $\varGamma$-two-sided ideals of a (2, 2)-regular ordered $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$**-semigroup S forms a semilattice structure with identity S. In 2013, Khan et al. [18] characterized an intra-regular ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup in terms of interval valued fuzzy left (right, two-sided) ideals. In 2014, Yousafzai et al. [46] introduced the notion of fully regular (V-regular) class of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup and characterized fully regular (V-regular) class of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup in terms of fuzzy (left, right, two-sided, interior, bi-, generalized bi- and quasi) ideals. In 2015, Khan et al. [24] defined (0, 2)-ideals and (1, 2)-ideals of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups and proved that the ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S is 0-(0, 2)-bisimple if and only if S is right 0-simple. In 2016, Yousafzai et al. [51] introduced the notion of $(\in _{\gamma }, \in _{\gamma }\vee q_{\delta })$-fuzzy (left, right, bi-) ideals of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup and characterized intra-regular ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups in terms of these generalized fuzzy ideals. In 2017, Yiarayong [42] have also studied prime, semiprime, quasi-prime and quasi-semiprime ideals of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. In 2018, Amjid et al. [6] introduced the notion of smallest one-sided ideals in an ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup.

In 2004, Stevanovic and Proti [37] introduced the notion of a 3-potent element of an ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup and of AG-3-band. They studied several properties of AG-3-bands and AG-bands. Nowadays many scholars have studied different aspects of ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups and ${{\mathcal {L}}}{{\mathcal {A}}}$-semihypergroups see [5, 10, 16, 19, 22, 23, 26, 29, 32, 39, 40]. In [27, 28] Mushtaq and Khan (2006-2007) initiated the study of AG-bands and AG*-groupoids. They proved that an ideal A of an AG-band is prime iff if ideal (S) is totally ordered; it is prime iff it is strongly irreducible. In 2012, Khan and Anis [17] proved that $S/ \gamma$ is a maximal separative semilattice homomorphic image of an ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S. In 2013, Shah and Rehman [17] studied several properties of locally associative $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. They proved that for a locally associative $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S with a left identity, $S/\rho$ is a maximal weakly separative homomorphic image of S, where $\rho$ is a relation on S defined by: $a\rho b$ if and only if $a\gamma b^{n} = b^{n+1}$ and $b\gamma a^{n} = a^{n+1}$ for some positive integer n and for all $\gamma \in \varGamma$, where $a, b \in S$. In 2014, Abdullah et al. [3] introduced the concept of interval-valued $(\in ,\in \vee q)$-fuzzy ideals, interval-valued $(\in ,\in \vee q)$-fuzzy bi-ideals and interval-valued $(\in ,\in \vee q)$-fuzzy quasi-ideals of an ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup. Nowadays many scholars have studied different aspects of fuzzy subsets on ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups and $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups see [9, 11, 12, 15, 20, 30, 31, 36, 38, 41, 45, 50]. In 2015, Abbasi and Basar [1] studied quasi-ideals and bi-ideals of locally associative $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups, (m, n) simple $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups, minimal bi-$\varGamma$-ideal, semiprime $\varGamma$-ideal and quasi-regular $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. They proved that the product of two (m, n)-$\varGamma$-ideals of a locally associative $\varGamma$-${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S with left identity is an (m, n)-$\varGamma$-ideal of S. In 2016, Khan et al. [25] have also studied (m, n)-ideals and 0-minimal (m, n)-ideals of ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups and proved that A is a (0, 2)-ideal of S if and only if A is a left ideal of some left ideal of S. In 2017, Khan et al. [21] defined $(\alpha , \beta )$-fuzzy bi-ideals, $(\alpha ,\beta )$-fuzzy interior ideals, $({\bar{\beta }}, {\bar{\alpha }})$-fuzzy bi-ideals and $({\bar{\beta }}, {\bar{\alpha }})$-fuzzy interior ideals in ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. In 2019, Younas and Mushtaq [43] proved that the set of idempotent elements in a left permutable inverse ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup is an order ideal. In 2020, Abbasi et al. [2] introduced the notion of soft interior-hyperideals in ${{\mathcal {L}}}{{\mathcal {A}}}$-semihypergroups. They studied several properties of soft interior-hyperideals of LA-semihypergroups.

Motivated and inspired by the above works, the aim of this paper is to extend the concept of quasi-prime ideals in multiplicative hyperrings given by Yiarayong [42] to the context of $\phi$-quasi-prime ideals. Let $\phi : {\mathcal {I}}(S) \rightarrow {\mathcal {I}}(S) \cup \left\{ \emptyset \right\}$ be a function where ${\mathcal {I}}(S)$ is the set of all left ideals of M. A proper left ideal A of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S is called a $\phi$-quasi-prime ideal, if for each $a, b\in S$ with $ab \subseteq A - \phi (A)$, then $a\in A$ or $b\in A$. Some characterizations of quasi-prime and $\phi$-quasi-prime ideals are obtained. Moreover, we investigate relationships between weakly quasi-prime, almost quasi-prime, $\omega$-quasi-prime, m-quasi-prime and $\phi$-quasi-prime ideals of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. Finally, we obtain necessary and sufficient conditions of $\phi$-quasi-prime ideal in order to be a $\phi$-quasi-prime ideal.

2 $\phi$-quasi-prime ideals of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups

In this section, we give some basic properties of $\phi$-quasi-prime ideals and investigate $\phi$-quasi-prime ideals in several classes of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups and give its characterizations corresponding to $\phi$-quasi-prime ideals in ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups.

For the sake of completeness, we state some definitions in the same fashion as found in [8] which are used throughout this paper.

Definition 1

Let S be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup and let $\phi : {\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function where ${\mathcal {I}}(S)$ be a set of all left ideals of S. A proper left ideal A of S is called a $\phi$-quasi-prime ideal if for each $a, b\in S$ with $ab \in A - \phi (A)$, then $a\in A$ or $b\in A$.

We now present the following example satisfying above definition.

Example 1

Let $S = \left\{ a, b, c \right\}$ be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with following multiplication given by

$$\begin{aligned} \begin{array}{l|lll} \cdot &{}a &{}b &{}c \\ \hline a &{}a &{}a &{}a \\ b &{}a &{}b &{}b \\ c &{}a &{}b &{}c \\ \end{array} \end{aligned}$$

We define a mapping $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ as follows: $\phi (A) = \emptyset$ for every $A\in {\mathcal {I}}(S)$. Clearly, $\left\{ a\right\}$ and $\{a, b\}$ are $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S.

Remark 1

It is easy to see that every quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S is a $\phi$-quasi-prime ideal of S.

The following example shows that the converse of Remark 1 is not true.

Example 2

Let $S = \left\{ a, b, c \right\}$ be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with following multiplication given by

$$\begin{aligned} \begin{array}{l|lll} \cdot &{}a &{}b &{}c \\ \hline a &{}a &{}a &{}a \\ b &{}c &{}c &{}c \\ c &{}a &{}a &{}c \\ \end{array} \end{aligned}$$

Consider the proper left ideal $P = \{a, c\}$ of the ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S. Define $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ by $\phi (A) = A$ for every $A\in {\mathcal {I}}(S)$. It is easy to see that P is a $\phi$-quasi-prime ideal of S. Notice that $b\cdot b = c \in P$, but $b\not \in P$. Therefore P is not a quasi-prime ideal of S.

Let A be a left ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S and let $\phi : {\mathcal {I}}(S) \rightarrow {\mathcal {I}}(S) \cup \left\{ \emptyset \right\}$ be a function. Since $A - \phi (A) = A - (A \cap \phi (A))$ for $A\in {\mathcal {I}}(S)$, without loss of generality, we will assume that $\phi (A) \subseteq A$. Throughout this paper, as it is noted earlier, if $\phi$ is a function, then we always assume that $\phi (A) \subseteq A$.

Theorem 1

Let A be a $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S with left identity. For each element s of $S - A$ if $\phi (A)$ is a quasi-prime ideal of S, then (A : s] is a $\phi$-quasi-prime ideal of S with $(\phi (A) : s]\subseteq \phi (A : s]$.

Proof

Obviously, (A : s] is a left ideal of S. Let a and b be any elements of S such that $ab \in (A : s] - \phi (A : s]$. Since $(\phi (A) : s]\subseteq \phi (A : s]$, we have $a(sb) = s(ab)\in A - \phi (A)$. By assumption, $a\in A$ or $bs\in A$. If $bs\in \phi (A)$, then $b\in \phi (A)$ or $s\in \phi (A)$. Now, if $bs\not \in \phi (A)$, then $bs\in A - \phi (A)$. Then from hypothesis, $b\in A$ or $s\in A$. In any case, we have $a\in A$ or $b\in A$, which implies that $sa\in (sA]\subseteq (A] = A$ or $sb\in (sA]\subseteq (A] = A$. Consequently, $a\in (A: s]$ or $b\in (A: s]$ and hence (A : s] is a $\phi$-quasi-prime ideal of S. $\square$

Remark 2

Let A be a $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S and let $\phi (A)$ be a quasi-prime ideal of S with $s_{1}\not \in A, s_{2}\not \in (A: s_{1}], s_{3}\not \in ((A : s_{1}] :s_{2}], \ldots$ and $(\phi (A): s_{1}] \subseteq \phi (A : s_{1}], ((\phi (A): s_{1}] : s_{2}] \subseteq \phi ((A : s_{1}] :s_{2}], \ldots$. Then $(A: s_{1}], ((A : s_{1}] :s_{2}], \ldots$ are $\phi$-quasi-prime ideals of S and $A\subseteq (A : s_{1}] \subseteq ((A : s_{1}] : s_{2}] \subseteq \ldots$.

In the following result, we give an equivalent definition of $\phi$-quasi-prime ideals in an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup.

Theorem 2

Let S be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup and let $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function. The following conditions are equivalent:

• 1. A is a $\phi$-quasi-prime ideal of S.

• 2. For each an element a of S if $a \in S - A$, then $(A : a] = (\phi (A) : a]\cup A$.

Proof

First assume that A is a $\phi$-quasi-prime ideal of S. It is easy to see that, $(\phi (A) : a)]\cup A \subseteq (A : a]$. Let b be an element of S such that $b \in (A : a]$. Then we have, $ab \in A$. If $ab \not \in \phi (A)$, then $ab \in A - \phi (A)$. Since A is a $\phi$-quasi-prime ideal of S, we have $a \in A$ or $b \in A$. By assumption, $b \in A$ that is, $b \in (\phi (A) : a] \cup A$. Now, if $ab \in \phi (A)$, then $b \in (\phi (A) : a] \subseteq (\phi (A) : a] \cup A$. In any case, we have $(A : a] \subseteq (\phi (A) : a] \cup A$ and hence $(A : a] = (\phi (A) : a]\cup A$.

Conversely, assume that 2 holds. Let a and b be any elements of S such that $ab \in A - \phi (A)$. Then we have, $b \in (A : a]$ and $b \not \in (\phi (A) : a]$. If $a\in A$, then there is nothing to prove. Now, if $a \not \in A$, then $(A : a] = (\phi (A) : a] \cup A$. Since $b \in (A : a]$ and $b \not \in (\phi (A) : a]$, we have $b \in A$. Therefore A is a $\phi$-quasi-prime ideal of S. $\square$

The following theorem characterize that quasi-prime ideals in terms of $\phi$-quasi-prime ideals of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S.

Theorem 3

Let $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function and let $\phi (A)$ be a quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S. Then A is a $\phi$-quasi-prime ideal of S if and only if A is a quasi-prime ideal of S.

Proof

First assume that A is a quasi-prime ideal of S. Obviously, A is a $\phi$-quasi-prime ideal of S.

Conversely, assume that A is a $\phi$-quasi-prime ideal of S. Let a and b be any elements of S such that $ab \in A$. If $ab \not \in \phi (A)$, then $ab \in A - \phi (A)$. By assumption, $a\in A$ or $b \in A$. Now if $ab \in \phi (A)$, then $a\in A$ or $b \in A$. In any case, we have A is a $\phi$-quasi-prime ideal of S. $\square$

Now we introduce the notion of a $\phi$-zero in an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup.

Definition 2

Let $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function and let A be a $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S. An order pair (a, b), where $a, b \in S$ is a $\phi$-zero if

• 1. $ab \in \phi (A)$,

• 2. $a\not \in A$ and $b \not \in A$.

Remark 3

Note that a proper left ideal A of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S is a $\phi$-quasi-prime ideal of S that is not a quasi-prime ideal of S if and only if A has a $\phi$-zero (a, b) for some $a, b \in S$.

Theorem 4

Let $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function and let A be a $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S. Suppose that B is a left ideal of S and $a\in S$ such that $aB \subseteq A$. If for every an element b of S such that (a, b) is not a $\phi$-zero of A, then $a\in A$ or $B\subseteq A$.

Proof

Assume, $a\not \in A$ and $B\not \subseteq A$. Then there exists an element $c \in B$ such that $c \not \in A$. If $ac\not \in \phi (A)$, then $ac \in A - \phi (A)$. Since A is a $\phi$-quasi-prime ideal of S, we have $a\in A$ or $c\in A$. Next, let $ac\in \phi (A)$. By hypothesis, $a\in A$ or $c\in A$. In any case, we have $a\in A$ or $c\in A$, which is a contradiction. Hence, $a\in A$ or $B\subseteq A$. $\square$

Theorem 5

Let $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function and let A be a $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S. For each elements $a, b \in S$ if (a, b) is a $\phi$-zero of A, then $aA \subseteq \phi (A)$.

Proof

Let a be an element of S such that $aA \not \subseteq \phi (A)$. Then there exists an element c of A such that $ac \not \in \phi (A)$. Thus we have, $a(b \cup c) = \left( ab\right) \cup \left( ac\right) \not \subseteq \phi (A)$, which implies that $a(b \cup c) \subseteq A - \phi (A)$. Since A is a $\phi$-quasi-prime ideal of S, we have $a \in A$ or $b \cup c \subseteq A$. Therefore, $a \in A$ or $b \in A$, which is a contradiction. Consequently, $aA \subseteq \phi (A)$. $\square$

Theorem 6

Let $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function. If A is a $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S that is not a quasi-prime ideal, then $A^{2} = \phi (A)$.

Proof

Since A is a $\phi$-quasi-prime ideal of S that is not a quasi-prime ideal, we have A has a $\phi$-zero (a, b) for some $a, b\in S$ by Remark 3. Assume, $cd \not \in \phi (A)$ for some $c, d \in A$. Then we have, $\left( a\cup c\right) \left( b\cup d\right) = ab \cup cb \cup ad\cup cd \not \subseteq \phi (A)$ by Theorem 5. This implies that, $\left( a\cup c\right) \left( b\cup d\right) \subseteq A - \phi (A)$. By assumption, $a\cup c\subseteq A$ or $b\cup d \subseteq A$. Therefore, $a\in A$ or $b\in A$, which is a contradiction. Hence, $A^{2} = \phi (A)$. $\square$

3 $\phi _{\alpha }$-quasi-prime ideals

In this section, we introduce the concept of $\phi$-quasi-prime, $\phi _{\emptyset }$-quasi-prime, $\phi _{n\ge 1}$-quasi-prime and $\phi _{\omega }$-quasi-prime ideals of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups and study some basic properties of $\phi$-quasi-prime, $\phi _{\emptyset }$-quasi-prime, $\phi _{n\ge 1}$-quasi-prime and $\phi _{\omega }$-quasi-prime ideals of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. Our starting points are the following definitions:

Definition 3

Let $\alpha \in \mathbf{Z} ^{+} \cup \left\{ \omega \right\} \cup \left\{ \emptyset \right\}$ and let $\phi _{\alpha } : {\mathcal {I}}(S) \rightarrow {\mathcal {I}}(S) \cup \left\{ \emptyset \right\}$ be a function where ${\mathcal {I}}(S)$ is a set of all left ideals of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S. A proper left ideal A of S is called a $\phi _{\alpha }$-quasi-prime ideal if for each $a, b\in S$ with $ab \in A - \phi _{\alpha }(A)$, then $a\in A$ or $b\in A$.

Let A be a $\phi _{\alpha }$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S.

• If $\phi _{\alpha }(A) = \emptyset$ for every $A\in {\mathcal {I}}(S)$, then we say that $\phi _{\alpha } = \phi _{\emptyset }$ and A is called a $\phi _{\emptyset }$-quasi-prime ideal of S and hence A is a quasi-prime ideal of S.

• If $\phi _{\alpha }(A) = A$ for every $A\in {\mathcal {I}}(S)$, then we say that $\phi _{\alpha } = \phi _{1}$ and A is called a $\phi _{1}$-quasi-prime ideal of S.

• If $\phi _{\alpha }(A) = A^{2}$ for every $A\in {\mathcal {I}}(S)$, then we say that $\phi _{\alpha } = \phi _{2}$ and A is called a $\phi _{n}$-quasi-prime ideal of S, and hence A is an almost quasi-prime ideal of S.

• If $\phi _{\alpha }(A) = A^{m}$ for every $A\in {\mathcal {I}}(S)$, then we say that $\phi _{\alpha } = \phi _{m\ge 3}$ and A is called a $\phi _{m}$-quasi-prime ideal of S, and hence A is a m-quasi-prime ideal of S.

• If $\phi _{\alpha }(A) = \displaystyle \bigcap \limits ^{\infty }_{i=1}A^{i}$ for every $A\in {\mathcal {I}}(S)$, then we say that $\phi _{\alpha } = \phi _{\omega }$ and A is called a $\phi _{\omega }$-quasi-prime ideal of S, and hence A is an $\omega$-quasi-prime ideal of S.

Remark 4

Let $\alpha \in \mathbf{Z} ^{+} \cup \left\{ \omega \right\} \cup \left\{ \emptyset \right\}$ and let $\phi _{\alpha } : {\mathcal {I}}(S) \rightarrow {\mathcal {I}}(S) \cup \left\{ \emptyset \right\}$ be a function.

• 1. A left ideal A of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S is a $\phi _{\emptyset }$-quasi-prime ideal of S if and only if A is a quasi-prime ideal of S.

• 2. A left ideal A of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S is a $\phi _{1}$-quasi-prime ideal of S if and only if A is a proper left ideal of S.

• 3. If A is a quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S, then A is a $\phi _{\alpha }$-quasi-prime ideal of S.

We start with our main result in which we give a characterization of $\phi _{\alpha }$-quasi-prime ideals in ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. For that, we need the following proposition.

Proposition 1

Let S be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup and let $\phi , \varphi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be two functions. Then the following properties hold:

• 1. If A is a $\phi$-quasi-prime ideal of S such that $\phi \le \varphi$, then A is a $\varphi$-quasi-prime ideal of S.

• 2. If A is a quasi-prime ideal of S, then A is a $\phi _{\omega }$-quasi-prime ideal of S.

• 3. If A is a $\omega$-quasi-prime ideal of S, then A is a m-quasi-prime ideal of S.

• 4. If A is an almost quasi-prime ideal of S, then A is a $\phi _{1}$-quasi-prime ideal of S.

Proof

1. Let a and b be any elements of S such that $ab \in A - \varphi (A)$. Since $\phi \le \varphi$, we have $\phi (A) \subseteq \varphi (A)$. Then we have, $ab \in A - \varphi (A)\subseteq A - \phi (A)$. Since A is a $\phi$-quasi-prime ideal of S, we have $a\in A$ or $b\in A$. Hence A is a $\varphi$-quasi-prime ideal of S.

2 - 4. It are obvious. $\square$

Remark 5

Let S be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup and let ${\mathcal {I}}(S)$ be a set of all left ideals of S. It is easy to see that, $\phi _{\emptyset } \le \phi _{\omega }\le \ldots \le \phi _{n+1}\le \phi _{n}\le \ldots \le \phi _{2}\le \phi _{1}$.

Let $S_{1}$ and $S_{2}$ be two ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. Then $S_{1} \times S_{2}$ is an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup and for each left ideal of $S_{1} \times S_{2}$ is of the form $A_{1} \times A_{2}$ for some left ideals $A_{1}$ and $A_{2}$ of $S_{1}$ and $S_{2}$, respectively.

Next we show that, $S_{1}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1} \times \ldots \times S_{k}$ is a $\phi$-quasi-prime ideal of $S_{1}\times \ldots \times S_{k}$ if and only if $A_{i}$ is a $\psi _{i}$-quasi-prime ideal of $S_{i}$. First, we would like to show that, $A_{1}\times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$ if and only if $A_{1}$ is a $\psi _{1}$-quasi-prime ideal of $S_{1}$.

Theorem 7

Let $S_{1}$ and $S_{2}$ be two ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups with left identities and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function with $\phi = \psi _{1}\times \psi _{2}$. Then the following conditions are equivalent:

• 1. $A_{1}\times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$.

• 2.

  • (a) $A_{1}$ is a $\psi _{1}$-quasi-prime ideal of $S_{1}$ where $\psi _{2}(S_{2}) \ne S_{2}$.

  • (b) For each elements $(a_{1}, b_{1}), (a_{2}, b_{2})$ of $S_{1}\times S_{2}$ such that $a_{1}a_{2}\in \psi _{1}(A_{1})$ if $b_{1}\in S_{2} - ( \psi _{2}(S_{2}) : S_{2}]$, then $a_{1}\in A_{1}$ or $a_{2}\in A_{1}$.

Proof

First assume that $A_{1}\times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$.

• (a) Let $a_{1}$ and $a_{2}$ be any elements of $S_{1}$ such that $a_{1}a_{2} \in A_{1} - \psi _{1}(A_{1})$. Then we have, $(a_{1}, e)(a_{2}, e) = (a_{1}a_{2}, e) \in A_{1}\times S_{2} - \psi _{1}(A_{1})\times \psi _{2}(S_{2}) = A_{1}\times S_{2} - \phi (A_{1}\times S_{2})$. By assumption, $(a_{1}, e)\in A_{1}\times S_{2}$ or $(a_{2}, e)\in A_{1}\times S_{2}$. Therefore, $a_{1}\in A_{1}$ or $a_{2}\in A_{1}$ and hence $A_{1}$ is a $\psi _{1}$-quasi-prime ideal of $S_{1}$.

• (b) Let $(a_{1}, b_{1}),$ and $(a_{2}, b_{2})$ be any elements of $S_{1} \times S_{2}$ be such that $a_{1}a_{2}\in \psi _{1}(A_{1})$ and $a_{1}, a_{2}\not \in A_{1}$. In fact, since $b_{1}\in S_{2} - ( \psi _{2}(S_{2}) : S_{2}]$, there exists an element $b_{2}$ of $S_{2}$ such that $b_{2}b_{1} \not \in \psi _{2}(S_{2})$. Thus, $(a_{1}, b_{2})(a_{2}, b_{1}) = (a_{1}a_{2}, b_{2}b_{1}) \in A_{1} \times S_{2} - \psi _{1}(A_{1}) \times \psi _{2}(S_{2}) = A_{1} \times S_{2} - \phi (A_{1} \times S_{2})$. Then by part (a), i.e., $(a_{1}, b_{2})\in A_{1}\times S_{2}$ or $(a_{2}, b_{1})\in A_{1}\times S_{2}$. Therefore, $a_{1} \in A_{1}$ or $a_{2} \in A_{1}$, which is a contradiction. Consequently, $b_{1}\in ( \psi _{2}(S_{2}) : S_{2}]$.

Assume that 2 holds. Let $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ be any elements of $S_{1} \times S_{2}$ be such that $(a_{1}a_{2}, b_{1}b_{2}) = (a_{1}, b_{1})(a_{2}, b_{2}) \in A_{1}\times S_{2} - \phi (A_{1}\times S_{2}) = A_{1}\times S_{2} - \psi _{1}(A_{1})\times \psi _{2}(S_{2})$. If $a_{1}a_{2}\not \in \psi _{1}(A_{1})$, then $a_{1}a_{2}\in A_{1} - \psi _{1}(S_{1})$. Then by part (a), $a_{1} \in A_{1}$ or $a_{2} \in A_{1}$. Thus, $(a_{1}, b_{1})\in A_{1}\times S_{2}$ or $(a_{2}, b_{2}) \in A_{1}\times S_{2}$ and thus we are done. If $a_{1}a_{2}\in \psi _{1}(A_{1})$, then $b_{1}b_{2}\not \in \psi _{2}(S_{2})$, which implies that $b_{2}\not \in (\psi _{2}(S_{2}): S_{2}]$. Hence by part (b), $a_{1}\in A_{1}$ or $a_{2}\in A_{1}$. Therefore, $(a_{1}, b_{1}) \in A_{1} \times S_{2}$ or $(a_{2}, b_{2}) \in A_{1} \times S_{2}$ and hence $A_{1}\times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$. $\square$

The following theorem can be seen in a similar way as in the proof of Theorem 7.

Theorem 8

Let $S_{1}$ and $S_{2}$ be two ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups with left identities and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function with $\phi = \psi _{1}\times \psi _{2}$. Then the following conditions are equivalent:

• 1. $S_{1}\times A_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$.

• 2.

  • (a) $A_{2}$ is a $\psi _{2}$-quasi-prime ideal of $S_{2}$ where $\psi _{1}(S_{1}) \ne S_{1}$.

  • (b) For each elements $(a_{1}, b_{1}), (a_{2}, b_{2})$ of $S_{1}\times S_{2}$ such that $b_{1}b_{2}\in \psi _{2}(A_{2})$ if $a_{1}\in S_{1} - ( \psi _{1}(S_{1}) : S_{1}]$, then $b_{1}\in A_{2}$ or $b_{2}\in A_{2}$.

The proof of the next result is similar to that of Theorem 7.

Theorem 9

Let $S_{i}$ be a ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left identity and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function with $\phi = \psi _{1}\times \ldots \times \psi _{k}$. Then the following conditions are equivalent:

• 1. $S_{1}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1} \times \ldots \times S_{k}$ is a $\phi$-quasi-prime ideal of $S_{1}\times \ldots \times S_{k}$.

• 2.

  • (a) $A_{i}$ is a $\psi _{i}$-quasi-prime deal of $S_{i}$ where $\psi _{j}(S_{j}) \ne S_{j}$.

  • (b) For each elements $(a_{(1,1)}, \ldots , a_{(k,1)}), (a_{(1,2)}, \ldots , a_{(k,2)})$ of $S_{1}\times \ldots \times S_{k}$ such that $a_{(1,i)}a_{(2,i)}\in \psi _{i}(A_{i})$ if $a_{(j,1)} \in S_{j} - ( \psi _{j}(S_{j}) : S_{j})$ for all $j\in \left\{ 1,\ldots , k\right\} - \{i\}$, then $a_{(1,i)}\in A_{i}$ or $a_{(2,i)}\in A_{i}$.

Recall that an element 0 of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S is called a left zero element of S if $0s \le 0$ for any $s\in S$.

Let S be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left zero. If $\phi _{\alpha }(A) = \left\{ 0 \right\}$ for every $A\in {\mathcal {I}}(S)$, then we say that $\phi _{\alpha } = \phi _{0}$ and A is called a $\phi _{0}$-quasi-prime ideal of S, and hence A is a weakly quasi-prime ideal of A.

As a simple consequence of Theorem 6, we give the following result.

Theorem 10

Let $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function and let A be a left ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S with left zero that is not a quasi-prime ideal. If A is a weakly quasi-prime ideal of S, then $A^{2} = \{0\}$.

Next we show that, if $A_{i}$ is a $(\psi _{i})_{0}$-quasi-prime ideal of $S_{i}$, then $S_{1}\times S_{2}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}$ is a $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{1}\times S_{2}\times \ldots \times S_{k}$ if $S_{1}\times \ldots \times S_{i-1} \times \left\{ 0\right\} \times S_{i+1}\times \ldots \times S_{k}\subseteq \phi ( S_{1}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k})$. First, we would like to show that, $A_{1}$ is a $(\psi _{1})_{0}$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{1}$, then $A_{1} \times S_{2}$ is a $\phi$-quasi-prime ideal if $\left\{ 0\right\} \times S_{2} \subseteq \phi (A_{1} \times S_{2})$.

Theorem 11

Let $S_{1}$ and $S_{2}$ be two ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups with left zeroes and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function with $\phi = \psi _{1}\times \psi _{2}$. If $A_{1}$ is a weakly quasi-prime ideal of $S_{1}$ such that $\left\{ 0 \right\} \times S_{2} \subseteq \phi (A_{1} \times S_{2})$, then $A_{1} \times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$.

Proof

Let $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ be any elements of $S_{1}\times S_{2}$ be such that

$(a_{1}, b_{1})(a_{2}, b_{2})\in A_{1}\times S_{2} - \phi (A_{1}\times S_{2})$.

In fact, since $\left\{ 0\right\} \times S_{2} \subseteq \phi (A_{1} \times S_{2})$, we have $(a_{1}a_{2}, b_{1}b_{2}) = (a_{1}, b_{1})(a_{2}, b_{2})\not \in \left\{ 0\right\} \times S_{2}$, which means that $a_{1}a_{2}\ne 0$. Then we have, $a_{1}a_{2} \in A_{1} - (\psi _{1})_{0}(A_{1})$. Since $A_{1}$ is a weakly quasi-prime ideal of $S_{1}$, we have $a_{1}\in A_{1}$ or $a_{2}\in A_{1}$. Therefore, $(a_{1}, b_{1})\in A_{1}\times S_{2}$ or $(a_{2}, b_{2})\in A_{1}\times S_{2}$ and hence $A_{1} \times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$. $\square$

From Theorem 11 we can easily obtain the following theorem.

Theorem 12

Let $S_{1}$ and $S_{2}$ be two ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups with left zeroes and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function with $\phi = \psi _{1}\times \psi _{2}$. If $A_{2}$ is a weakly quasi-prime ideal of $S_{2}$ such that $S_{1} \times \left\{ 0 \right\} \subseteq \phi (S_{1} \times A_{2})$, then $S_{1} \times A_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$.

From Theorems 11, 12 we can easily obtain the following theorem.

Theorem 13

Let $S_{i}$ be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left zero and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function with $\phi = \psi _{1}\times \ldots \times \psi _{k}$ and $S_{1}\times \ldots \times S_{i-1} \times \left\{ 0\right\} \times S_{i+1}\times \ldots \times S_{k}\subseteq \phi ( S_{1}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k})$. Then $A_{i}$ is a weakly quasi-prime ideal of $S_{i}$ if and only if $S_{1}\times S_{2}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}$ is a $\phi$-quasi-prime ideal of $S_{1}\times S_{2}\times \ldots \times S_{k}$.

Next, let S be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup. Clearly, every quasi-prime ideal of S is $\phi$-quasi-prime ideal, but the converse does not necessarily hold. In Theorem 14 and Corollary 1 provide some conditions under which a $\phi$-quasi-prime ideal is a quasi-prime ideal in an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup.

Theorem 14

Let $\phi , \phi _{3}:{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be two functions and let A be a $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S. If $\phi _{2} \not \le \phi$, then A is a quasi-prime ideal of S.

Proof

Let $a_{1}$ and $a_{2}$ be any elements of S such that $a_{1}a_{2} \in A$. If $a_{1}a_{2}\not \in \phi (A)$, then $a_{1}a_{2}\in A - \phi (A)$. Since A is a $\phi$-quasi-prime ideal of S, we have $a_{1}\in A$ or $a_{2}\in A$. Next, let $a_{1}a_{2}$ be an element of $\phi (A)$. Since $\phi _{2} \not \le \phi$, we have $A^{2} \not \subseteq \phi (A)$. Then there exist elements $b_{1}$ and $b_{2}$ of A such that $b_{1}b_{2}\not \in \phi (A)$, which means that $\left( a_{1}\cup b_{1}\right) \left( a_{2}\cup b_{2}\right) = a_{1}b_{1}\cup a_{2}b_{1}\cup a_{1}b_{2} \cup a_{2}b_{2} \subseteq A - \phi (A)$. By hypothesis, $a_{1}\cup b_{1} \subseteq A$ or $a_{2}\cup b_{2}\subseteq A$. Therefore, $a_{1} \in A$ or $a_{2}\in A$ and hence A is a quasi-prime ideal of S. $\square$

In the following theorem, we give a sort of consequences whose proof is similar to those of quasi-prime ideals in ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups.

Corollary 1

Let $\phi _{n}:{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function and let A be a weakly quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S with left zero. If $\phi _{2} \ne \phi _{0}$, then A is a quasi-prime ideal of S.

Proof

Similar to the proof of Theorem 14. $\square$

Let $S_{i}$ be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup. For each elements k, n of $\mathbf{Z} ^{+}$ such that $k\ge 2, n \ge 1, ({\psi _{i}})_{n}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ and let $\phi _{(k, n)} = (\psi _{1})_{n}\times (\psi _{2})_{n}\times \ldots \times (\psi _{k})_{n}$.

Theorem 15

If $A_{1}$ is a weakly quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{1}$ with left zero such that $(\psi _{2})_{2}(S_{2}) = S_{2}$, then $A_{1} \times S_{2}$ is a $\phi _{(2,2)}$-quasi-prime ideal of $S_{1} \times S_{2}$.

Proof

If $A_{1}$ is a quasi-prime ideal of $S_{1}$, then $A_{1} \times S_{2}$ is a quasi-prime ideal of $S_{1} \times S_{2}$. Obviously, $A_{1} \times S_{2}$ is a $\phi _{(2, 2)}$-quasi-prime ideal of $S_{1} \times S_{2}$. Assume that $A_{1}$ is not a quasi-prime ideal of $S_{1}$. Then by Corollary 1, $(\psi _{1})_{2} \le (\psi _{1})_{0}$, which implies that $\left( A_{1}\right) ^{2} = \left\{ 0\right\}$. By assumption,

$$\begin{aligned} \begin{array}{lcl} \phi _{(2,2)}(A_{1} \times S_{2}) &{}=&{} (\psi _{1})_{2}\times (\psi _{2})_{2}(A_{1} \times S_{2}) \\ &{}=&{} (\psi _{1})_{2}(A_{1}) \times (\psi _{2})_{2}(S_{2}) \\ &{}=&{} \left\{ 0\right\} \times S_{2}. \end{array} \end{aligned}$$

It follows from Theorem 11 that $A_{1} \times S_{2}$ is a $\phi _{(2, 2)}$-quasi-prime ideal of $S_{1} \times S_{2}$. $\square$

From Theorem 15 we can easily obtain the following theorem.

Theorem 16

If $A_{2}$ is a weakly quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{2}$ with left zero such that $({\psi _{1}})_{2}(S_{1}) = S_{1}$, then $S_{1} \times A_{2}$ is a $\phi _{(2, 2)}$-quasi-prime ideal of $S_{1} \times S_{2}$.

From Theorems 15, 16 we can easily obtain the following theorem.

Theorem 17

If $A_{i}$ is a weakly quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{i}$ with left zero such that $({\psi _{j}})_{2}(S_{j}) = S_{j}$, then $S_{1}\times S_{2}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}$ is a $\phi _{(k, 2)}$-quasi-prime ideal of $S_{1}\times \ldots \times S_{k}$.

Next we show that, if $A_{i}$ is a quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{i}$, then $S_{1}\times S_{2}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}$ is a $\phi$-quasi-prime ideal of $S_{1} \times \ldots \times S_{2}$ if $\psi _{j}(S_{j}) \ne S_{j}$. First, we would like to show that, $A_{1}$ is a quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{1}$, then $A_{1}\times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1}\times S_{2}$ if $\psi _{2}(S_{2}) \ne S_{2}$.

Theorem 18

Let $S_{1}$ and $S_{2}$ be two ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups with left identities and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\phi = \psi _{1}\times \psi _{2}$. Then the following conditions are equivalent:

• 1. $A_{1}$ is a quasi-prime ideal of $S_{1}$.

• 2. $A_{1} \times S_{2}$ is a quasi-prime ideal of $S_{1}\times S_{2}$.

• 3. $A_{1}\times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1}\times S_{2}$ where $\psi _{2}(S_{2}) \ne S_{2}$.

Proof

$(1 \Rightarrow 2)$. Assume that $A_{1}$ is a quasi-prime ideal of $S_{1}$. Let $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ be any elements of $S_{1} \times S_{2}$ be such that $(a_{1}a_{2}, b_{1}b_{2}) = (a_{1}, b_{1})(a_{2}, b_{2}) \in A_{1} \times S_{2}$, which implies that $a_{1}a_{2}\in A_{1}$. By assumption, $a_{1}\in A_{1}$ or $a_{2}\in A_{1}$. Therefore, $(a_{1}, b_{1}) \in A_{1}\times S_{2}$ or $(a_{2}, b_{2}) \in A_{1}\times S_{2}$. Consequently, $A_{1} \times S_{2}$ is a quasi-prime ideal of $S_{1}\times S_{2}$.

$(2 \Rightarrow 3)$. It is obvious.

$(3 \Rightarrow 1)$. Assume that 3 holds. Let $a_{1}$ and $a_{2}$ be any elements of $S_{1}$ be such that $a_{1}a_{2} \in A_{1}$. Since $\psi _{2}(S_{2}) \ne S_{2}$, there exists an element c of $S_{2}$ such that $c\not \in \psi _{2}(S_{2})$. In fact, since $(a_{1}, e)(a_{2}, c) = (a_{1}a_{2}, c) \not \in A_{1} \times \psi _{2}(S_{2})$ and $\phi (A_{1} \times S_{2}) = ( \psi _{1}\times \psi _{2})(A_{1} \times S_{2}) \subseteq A_{1} \times \psi _{2}(S_{2})$, we have $(a_{1}, e)(a_{2}, c) \not \in \phi (A_{1} \times S_{2})$, which means that $(a_{1}, e)(a_{2}, c) \in A_{1} \times S_{2} - \phi (A_{1} \times S_{2})$. Since $A_{1}\times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1}\times S_{2}$, we have $(a_{1}, e)\in A_{1}\times S_{2}$ or $(a_{2}, c)\in A_{1}\times S_{2}$. Therefore, $a_{1}\in A_{1}$ or $a_{2}\in A_{1}$ and hence $A_{1}$ is a quasi-prime ideal of $S_{1}$. $\square$

From Theorem 18 we can easily obtain the following theorem.

Theorem 19

Let $S_{1}$ and $S_{2}$ be two ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups with left identities and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\phi = \psi _{1}\times \psi _{2}$. Then the following conditions are

• 1. $A_{2}$ is a quasi-prime ideal of $S_{2}$.

• 2. $S_{1} \times A_{2}$ is a quasi-prime ideal of $S_{1} \times S_{2}$.

• 3. $S_{1}\times A_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$, where $\psi _{1}(S_{1}) \ne S_{1}$.

From Theorems 18, 19 we can easily obtain the following theorem.

Theorem 20

Let $S_{i}$ be a ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left identity and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\phi = \psi _{1}\times \ldots \times \psi _{k}$. Then the following conditions are equivalent:

• 1. $A_{i}$ is a quasi-prime ideal of $S_{i}$.

• 2. $S_{1}\times S_{2}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}$ is a quasi-prime ideal of $S_{1} \times \ldots \times S_{k}$.

• 3. $S_{1}\times S_{2}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}$ is a $\phi$-quasi-prime ideal of $S_{1} \times \ldots \times S_{2}$ with $\psi _{j}(S_{j}) \ne S_{j}$.

Next, we show that if $A_{i}$ is a $\psi _{i}$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{i}$, then $S_{1}\times S_{2}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}$ is a $\phi$-quasi-prime ideal of $S_{1} \times \ldots \times S_{k}$ if $\psi _{j}(S_{j}) = S_{j}$. First, we would like to show that, $A_{1}$ is a $\psi _{1}$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{1}$, then $A_{1} \times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$, if $\psi _{2}(S_{2}) = S_{2}$.

Theorem 21

Let $S_{1}$ and $S_{2}$ be two ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups with left identities and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\psi _{2}(S_{2}) = S_{2}$ and $\phi = \psi _{1}\times \psi _{2}$. Then $A_{1} \times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$ if and only if $A_{1}$ is a $\psi _{1}$-quasi-prime ideal of $S_{1}$.

Proof

First assume that $A_{1} \times S_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$. The proof is trivial and hence omitted.

Conversely, assume that $A_{1}$ is a $\psi _{1}$-quasi-prime ideal of $S_{1}$. Let $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ be any elements of $S_{1} \times S_{2}$ be such that

$$\begin{aligned} \begin{array}{lcl} (a_{1}a_{2}, b_{1} b_{2}) &{}=&{} (a_{1}, b_{1})(a_{2}, b_{2})\in A_{1} \times S_{2} - \phi (A_{1}\times S_{2}) \\ &{}=&{} A_{1} \times S_{2} - (\psi _{1}\times \psi _{2})(A_{1}\times S_{2}) \\ &{}=&{} A_{1} \times S_{2} - \psi _{1}(A_{1}) \times S_{2}. \end{array} \end{aligned}$$

Obviously, $a_{1} a_{2}\in A_{1} - \psi _{1}(A_{1})$. By assumption, $a_{1}\in A_{1}$ or $a_{2}\in A_{1}$. Consequently, $A_{1}$ is $\psi _{1}$-quasi-prime ideal of $S_{1}$. $\square$

From Theorem 21 we can easily obtain the following theorem.

Theorem 22

Let $S_{1}$ and $S_{2}$ be two ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups with left identities and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\psi _{1}(S_{1}) = S_{1}$ and $\phi = \psi _{1}\times \psi _{2}$. Then $S_{1} \times A_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$ if and only if $A_{2}$ is a $\psi _{2}$-quasi-prime ideal of $S_{2}$.

From Theorems 21,22 we can easily obtain the following theorem.

Theorem 23

Let $S_{i}$ be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left identity and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\psi _{j}(S_{j}) = S_{j}$ and $\phi = \psi _{1}\times \ldots \times \psi _{k}$. Then $S_{1}\times S_{2}\times \ldots \times S_{i-1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}$ is a $\phi$-quasi-prime ideal of $S_{1} \times \ldots \times S_{k}$ if and only if $A_{i}$ is a $\psi _{i}$-quasi-prime ideal of $S_{i}$.

Next, we show that if $A_{1} \times A_{2}$ is a $\phi$-quasi-prime ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{1} \times S_{2}$, then $A_{i}$ is a $\psi _{1}$-quasi-prime ideal of $S_{i}$ for all $i = 1, 2$.

Theorem 24

Let $A_{1}$and $A_{2}$ be any proper left ideals of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups with left identities $S_{1}$ and $S_{2}$, respectively and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\phi = \psi _{1}\times \psi _{2}$. Then the following properties hold:

• 1. If $A_{1} \times A_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$ such that $A_{2} \ne \psi _{2}(A_{2})$, then $A_{1}$ is a $\psi _{1}$-quasi-prime ideal of $S_{1}$.

• 2. If $A_{1} \times A_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$ such that $A_{1} \ne \psi _{1}(A_{1})$, then $A_{2}$ is a $\psi _{2}$-quasi-prime ideal of $S_{2}$.

Proof

1. Let $a_{1}$ and $a_{2}$ be any elements of $S_{1}$ be such that $a_{1}a_{2}\in A_{1} - \psi (A_{1})$. If $A_{2} \ne \psi _{2}(A_{2})$, then there exists an element c of $S_{2}$ such that $c \notin \psi _{2}(A_{2})$. This implies that, $(a_{1}, e)(a_{2}, c) = (a_{1}a_{2}, c) \in A_{1}\times A_{2} - \psi _{1}(A_{1})\times \psi _{2}(A_{2}) = A_{1}\times A_{2} - \phi (A_{1}\times A_{2})$. Since $A_{1} \times A_{2}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$, we have $(a_{1}, e)\in A_{1} \times S_{2}$ or $(a_{2}, c)\in A_{1} \times S_{2}$. Therefore, $a_{1}\in A_{1}$ or $a_{2}\in A_{1}$ and hence $A_{1}$ is a $\psi _{1}$-quasi-prime ideal of $S_{1}$.

2. This follows from part 1. $\square$

From Theorem 24 we can easily obtain the following theorem.

Theorem 25

Let $A_{i}$ be a proper left ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{i}$ with left identity and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\phi = \psi _{1}\times \psi _{2} \times \ldots \times \psi _{k}$. If $A_{1} \times A_{2}\times \ldots \times A_{k}$ is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2} \times \ldots \times S_{k}$ such that $A_{j} \ne \psi _{j}(A_{j})$, then $A_{i}$ is a $\psi _{i}$-quasi-prime ideal of $S_{i}$.

The next theorem gives conditions for a $\phi$-quasi-prime ideal to be quasi-prime ideal in an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup.

Theorem 26

Let $S_{i}$ be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left identity and left zero and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\psi _{i}(S_{i}) \ne S_{i}$ and $\phi = \psi _{1}\times \psi _{2}$. If A is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$, then $A = \phi (A)$ or A is a quasi-prime ideal of $S_{1} \times S_{2}$.

Proof

Suppose that A is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$ that is not a quasi-prime ideal of $S_{1} \times S_{2}$. To show that $A\ne \phi (A)$. First assume, $A_{1} \times A_{2} = A \ne \phi (A) = \phi (A_{1} \times A_{2}) = \psi _{1}(A_{1}) \times \psi _{2}(A_{2})$. Then there exists an element i of $\left\{ 1, 2\right\}$ such that $A_{i} \ne \psi _{i}(A_{i})$. We may assume that $A_{1} \ne \psi _{1}(A_{1})$, there exists an element $c_{1}$ of $A_{1}$ such that $c_{1} \notin \psi _{1}(A_{1})$. We will to show that $A_{2} = S_{2}$. Next, assume, $A_{2} \ne S_{2}$, it follows that there exists an element $c_{2}$ of $S_{2}$ such that $c_{2} \not \in A_{2}$. In fact, since $(e, c_{2})(c_{1}, e) = (c_{1}, c_{2}e)\not \in \psi _{1}(A_{1})\times \psi _{2}(A_{2}) = \phi (A)$, we have $(e, c_{2})(c_{1}, e) \in A - \phi (A)$. Thus, $(e, c_{2})) \in A$ or $(c_{1}, e) \in A$. Obviously, $c_{2} \in A_{2}$, which is a contradiction. Therefore, $A = A_{1} \times S_{2}$, which means that $(0, e)\in A$. By Theorem 14,

$$\begin{aligned} \begin{array}{lcl} (0,e) &{}=&{} (0, e)^{2} \in A^{2} \\ &{}=&{} \phi _{2}(A) \\ &{}\subseteq &{} \phi (A) \\ &{}=&{} \psi _{1}(A_{1}) \times \psi _{2}(A_{2}), \end{array} \end{aligned}$$

which is a contradiction. Hence, $A = \phi (A)$. $\square$

From Theorem 26 we can easily obtain the following theorem.

Theorem 27

Let $S_{i}$ be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left identity and left zero and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\psi _{i}(S_{i}) \ne S_{i}$ and $\phi = \psi _{1}\times \psi _{2}\times \ldots \times \psi _{k\ge 2}$. If A is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}\times \ldots \times S_{k}$, then $A = \phi (A)$ or A is a quasi-prime ideal of $S_{1} \times S_{2}\times \ldots \times S_{k}$.

The above theorem shows the relationship between quasi-prime ideals and $\phi$-quasi-prime ideals in an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup $S_{1}\times S_{2}$. From the above theorem, we have the following theorem.

Theorem 28

Let $S_{i}$ be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left identity and left zero and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\psi _{i}(S_{i}) \ne S_{i}, \phi = \psi _{1}\times \psi _{2}$ and $A \ne \phi (A)$. Then A is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}$ if and only if A is a quasi-prime ideal of $S_{1} \times S_{2}$.

Proof

This follows from Theorem 26. $\square$

From Theorem 27 we can easily obtain the following theorem.

Theorem 29

Let $S_{i}$ be an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup with left identity and left zero and let $\psi _{i}:{\mathcal {I}}(S_{i})\rightarrow {\mathcal {I}}(S_{i})\cup \left\{ \emptyset \right\}$ be a function such that $\psi _{i}(S_{i}) \ne S_{i}, \phi = \psi _{1}\times \psi _{2}\times \ldots \times \psi _{k\ge 2}$ and $A \ne \phi (A)$. Then A is a $\phi$-quasi-prime ideal of $S_{1} \times S_{2}\times \ldots \times S_{k}$ if and only if A is a quasi-prime ideal of $S_{1} \times S_{2}\times \ldots \times S_{k}$.

As a simple consequence of Theorem 6, we give the following result.

Lemma 1

Let $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$

be a function and let A be a left ideal of an ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroup S that is not a quasi-prime ideal. If A is a $\phi$-quasi-prime ideal of S such that $\phi \le \phi _{2}$, then $A^{2} = A^{n+1}$.

The next theorem gives conditions for a $\phi$-quasi-prime ideal to be $\omega$-quasi-prime ideal in a commutative semigroup.

Theorem 30

Let $\phi :{\mathcal {I}}(S)\rightarrow {\mathcal {I}}(S)\cup \left\{ \emptyset \right\}$ be a function where $\phi \le \phi _{n+1}$. Then A is a $\phi$-quasi-prime ideal of S if and only if A is an $\omega$-quasi-prime ideal of S.

Proof

First assume that A is a $\phi$-quasi-prime ideal of S. If A is a quasi-prime ideal of S, then it is $\omega$-quasi-prime ideal. Now assume that A is not a quasi-prime ideal of S. Then by Lemma 1, $A^{2} = A^{n+1}$. By assumption, A is a $\phi$-quasi-prime ideal of S and $\phi \le \phi _{n+1}$, which implies that A is a $\phi _{n+1}$-quasi-prime ideal of S. On the other hand, $\phi _{\omega }\left( A\right) = A^{n+1} = \phi _{n+1}\left( A\right)$. Therefore A is an $\omega$-quasi-prime ideal of S.

Conversely, assume that A is a $\phi$-quasi-prime ideal of S. The proof is trivial and hence omitted. $\square$

4 Conclusion

In study the structure of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups, we notice that the quasi-prime ideals with special properties always play an important role. The purposes of this paper are to introduce generalizations of quasi-prime ideals to the context of $\phi$-quasi-prime ideals. Some characterizations of quasi-prime and $\phi$-quasi-prime ideals are obtained. Moreover, we investigate relationships between weakly quasi-prime, almost quasi-prime, $\omega$-quasi-prime, m-quasi-prime and $\phi$-quasi-prime ideals of ordered ${{\mathcal {L}}}{{\mathcal {A}}}$-semigroups. Finally, we obtain necessary and sufficient conditions of $\phi$-quasi-prime ideal in order to be a quasi-prime ideal.