Abstract
The purposes of this paper are to introduce generalizations of quasiprime ideals to the context of \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 quasiprime and \(\phi \)quasiprime ideals are obtained. Moreover, we investigate relationships between weakly quasiprime, almost quasiprime, \(\omega \)quasiprime, mquasiprime and \(\phi \)quasiprime ideals of ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroups. Finally, we obtain necessary and sufficient conditions of \(\phi \)quasiprime ideal in order to be a quasiprime ideal.
Introduction
In 2010, Shah et al. [35] studied ideals, Msystems, Nsystems and Isystems 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 quasiprime if and only if \(S A\) is an Msystem; A is quasisemiprime if and only if \(S  A\) is an Nsystem and A is quasiirreducible if and only if \(S  A\) is an Isystem. 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 \(\varGamma \)twosided ideals, fuzzy \(\varGamma \)generalized biideals, fuzzy \(\varGamma \)biideals, fuzzy \(\varGamma \)interior ideals and fuzzy \(\varGamma \)(1; 2)ideals. They proved that the set of all fuzzy \(\varGamma \)twosided 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 intraregular ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup in terms of interval valued fuzzy left (right, twosided) ideals. In 2014, Yousafzai et al. [46] introduced the notion of fully regular (Vregular) class of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup and characterized fully regular (Vregular) class of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup in terms of fuzzy (left, right, twosided, 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 0simple. 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 intraregular ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroups in terms of these generalized fuzzy ideals. In 2017, Yiarayong [42] have also studied prime, semiprime, quasiprime and quasisemiprime ideals of ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroups. In 2018, Amjid et al. [6] introduced the notion of smallest onesided ideals in an \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup.
In 2004, Stevanovic and Proti [37] introduced the notion of a 3potent element of an \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup and of AG3band. They studied several properties of AG3bands and AGbands. 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 (20062007) initiated the study of AGbands and AG*groupoids. They proved that an ideal A of an AGband 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 intervalvalued \((\in ,\in \vee q)\)fuzzy ideals, intervalvalued \((\in ,\in \vee q)\)fuzzy biideals and intervalvalued \((\in ,\in \vee q)\)fuzzy quasiideals 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 quasiideals and biideals 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 quasiregular \(\varGamma \)\({{\mathcal {L}}}{{\mathcal {A}}}\)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 0minimal (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 biideals, \((\alpha ,\beta )\)fuzzy interior ideals, \(({\bar{\beta }}, {\bar{\alpha }})\)fuzzy biideals 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 interiorhyperideals in \({{\mathcal {L}}}{{\mathcal {A}}}\)semihypergroups. They studied several properties of soft interiorhyperideals of LAsemihypergroups.
Motivated and inspired by the above works, the aim of this paper is to extend the concept of quasiprime ideals in multiplicative hyperrings given by Yiarayong [42] to the context of \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 quasiprime and \(\phi \)quasiprime ideals are obtained. Moreover, we investigate relationships between weakly quasiprime, almost quasiprime, \(\omega \)quasiprime, mquasiprime and \(\phi \)quasiprime ideals of ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroups. Finally, we obtain necessary and sufficient conditions of \(\phi \)quasiprime ideal in order to be a \(\phi \)quasiprime ideal.
\(\phi \)quasiprime ideals of ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroups
In this section, we give some basic properties of \(\phi \)quasiprime ideals and investigate \(\phi \)quasiprime ideals in several classes of ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroups and give its characterizations corresponding to \(\phi \)quasiprime 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 \(\phi \)quasiprime 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
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 \(\phi \)quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup S.
Remark 1
It is easy to see that every quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup S is a \(\phi \)quasiprime 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
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 \(\phi \)quasiprime ideal of S. Notice that \(b\cdot b = c \in P\), but \(b\not \in P\). Therefore P is not a quasiprime 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 \(\phi \)quasiprime 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 quasiprime ideal of S, then (A : s] is a \(\phi \)quasiprime 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 \(\phi \)quasiprime ideal of S. \(\square \)
Remark 2
Let A be a \(\phi \)quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup S and let \(\phi (A)\) be a quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime ideal of S. \(\square \)
The following theorem characterize that quasiprime ideals in terms of \(\phi \)quasiprime 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 quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup S. Then A is a \(\phi \)quasiprime ideal of S if and only if A is a quasiprime ideal of S.
Proof
First assume that A is a quasiprime ideal of S. Obviously, A is a \(\phi \)quasiprime ideal of S.
Conversely, assume that A is a \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime ideal of S that is not a quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup S that is not a quasiprime ideal, then \(A^{2} = \phi (A)\).
Proof
Since A is a \(\phi \)quasiprime ideal of S that is not a quasiprime 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, \(A^{2} = \phi (A)\). \(\square \)
\(\phi _{\alpha }\)quasiprime ideals
In this section, we introduce the concept of \(\phi \)quasiprime, \(\phi _{\emptyset }\)quasiprime, \(\phi _{n\ge 1}\)quasiprime and \(\phi _{\omega }\)quasiprime ideals of ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroups and study some basic properties of \(\phi \)quasiprime, \(\phi _{\emptyset }\)quasiprime, \(\phi _{n\ge 1}\)quasiprime and \(\phi _{\omega }\)quasiprime 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 \(\phi _{\alpha }\)quasiprime 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 \(\phi _{\alpha }\)quasiprime 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 \(\phi _{\emptyset }\)quasiprime ideal of S and hence A is a quasiprime 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 \(\phi _{1}\)quasiprime 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 \(\phi _{n}\)quasiprime ideal of S, and hence A is an almost quasiprime 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 \(\phi _{m}\)quasiprime ideal of S, and hence A is a mquasiprime 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 \(\phi _{\omega }\)quasiprime ideal of S, and hence A is an \(\omega \)quasiprime 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 \(\phi _{\emptyset }\)quasiprime ideal of S if and only if A is a quasiprime ideal of S.

2.
A left ideal A of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup S is a \(\phi _{1}\)quasiprime ideal of S if and only if A is a proper left ideal of S.

3.
If A is a quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup S, then A is a \(\phi _{\alpha }\)quasiprime ideal of S.
We start with our main result in which we give a characterization of \(\phi _{\alpha }\)quasiprime 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 \(\phi \)quasiprime ideal of S such that \(\phi \le \varphi \), then A is a \(\varphi \)quasiprime ideal of S.

2.
If A is a quasiprime ideal of S, then A is a \(\phi _{\omega }\)quasiprime ideal of S.

3.
If A is a \(\omega \)quasiprime ideal of S, then A is a mquasiprime ideal of S.

4.
If A is an almost quasiprime ideal of S, then A is a \(\phi _{1}\)quasiprime 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 \(\phi \)quasiprime ideal of S, we have \(a\in A\) or \(b\in A\). Hence A is a \(\varphi \)quasiprime 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, \(S_{1}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1} \times \ldots \times S_{k}\) is a \(\phi \)quasiprime ideal of \(S_{1}\times \ldots \times S_{k}\) if and only if \(A_{i}\) is a \(\psi _{i}\)quasiprime ideal of \(S_{i}\). First, we would like to show that, \(A_{1}\times S_{2}\) is a \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\) if and only if \(A_{1}\) is a \(\psi _{1}\)quasiprime 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 \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\).

2.

(a)
\(A_{1}\) is a \(\psi _{1}\)quasiprime 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 \(a_{2}\in A_{1}\).

(a)
Proof
First assume that \(A_{1}\times S_{2}\) is a \(\phi \)quasiprime 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 \(\psi _{1}\)quasiprime 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 \(\phi \)quasiprime 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 \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\).

2.

(a)
\(A_{2}\) is a \(\psi _{2}\)quasiprime 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 \(b_{2}\in A_{2}\).

(a)
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.
\(S_{1}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1} \times \ldots \times S_{k}\) is a \(\phi \)quasiprime ideal of \(S_{1}\times \ldots \times S_{k}\).

2.

(a)
\(A_{i}\) is a \(\psi _{i}\)quasiprime 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 \(a_{(2,i)}\in A_{i}\).

(a)
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 \(\phi _{0}\)quasiprime ideal of S, and hence A is a weakly quasiprime 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 quasiprime ideal. If A is a weakly quasiprime ideal of S, then \(A^{2} = \{0\}\).
Next we show that, if \(A_{i}\) is a \((\psi _{i})_{0}\)quasiprime ideal of \(S_{i}\), then \(S_{1}\times S_{2}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}\) is a \(\phi \)quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup \(S_{1}\times S_{2}\times \ldots \times S_{k}\) if \(S_{1}\times \ldots \times S_{i1} \times \left\{ 0\right\} \times S_{i+1}\times \ldots \times S_{k}\subseteq \phi ( S_{1}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k})\). First, we would like to show that, \(A_{1}\) is a \((\psi _{1})_{0}\)quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup \(S_{1}\), then \(A_{1} \times S_{2}\) is a \(\phi \)quasiprime 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 quasiprime 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 \(\phi \)quasiprime 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 quasiprime 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 \(\phi \)quasiprime 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 quasiprime 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 \(\phi \)quasiprime 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 \(S_{1}\times \ldots \times S_{i1} \times \left\{ 0\right\} \times S_{i+1}\times \ldots \times S_{k}\subseteq \phi ( S_{1}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k})\). Then \(A_{i}\) is a weakly quasiprime ideal of \(S_{i}\) if and only if \(S_{1}\times S_{2}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}\) is a \(\phi \)quasiprime 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 quasiprime ideal of S is \(\phi \)quasiprime ideal, but the converse does not necessarily hold. In Theorem 14 and Corollary 1 provide some conditions under which a \(\phi \)quasiprime ideal is a quasiprime 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 \(\phi \)quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup S. If \(\phi _{2} \not \le \phi \), then A is a quasiprime 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 \(\phi \)quasiprime 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 quasiprime ideal of S. \(\square \)
In the following theorem, we give a sort of consequences whose proof is similar to those of quasiprime 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 quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup S with left zero. If \(\phi _{2} \ne \phi _{0}\), then A is a quasiprime 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 quasiprime 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 \(\phi _{(2,2)}\)quasiprime ideal of \(S_{1} \times S_{2}\).
Proof
If \(A_{1}\) is a quasiprime ideal of \(S_{1}\), then \(A_{1} \times S_{2}\) is a quasiprime ideal of \(S_{1} \times S_{2}\). Obviously, \(A_{1} \times S_{2}\) is a \(\phi _{(2, 2)}\)quasiprime ideal of \(S_{1} \times S_{2}\). Assume that \(A_{1}\) is not a quasiprime 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,
It follows from Theorem 11 that \(A_{1} \times S_{2}\) is a \(\phi _{(2, 2)}\)quasiprime 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 quasiprime 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 \(\phi _{(2, 2)}\)quasiprime 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 quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup \(S_{i}\) with left zero such that \(({\psi _{j}})_{2}(S_{j}) = S_{j}\), then \(S_{1}\times S_{2}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}\) is a \(\phi _{(k, 2)}\)quasiprime ideal of \(S_{1}\times \ldots \times S_{k}\).
Next we show that, if \(A_{i}\) is a quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup \(S_{i}\), then \(S_{1}\times S_{2}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}\) is a \(\phi \)quasiprime 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 quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup \(S_{1}\), then \(A_{1}\times S_{2}\) is a \(\phi \)quasiprime 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 quasiprime ideal of \(S_{1}\).

2.
\(A_{1} \times S_{2}\) is a quasiprime ideal of \(S_{1}\times S_{2}\).

3.
\(A_{1}\times S_{2}\) is a \(\phi \)quasiprime 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 quasiprime 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 quasiprime 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 \(\phi \)quasiprime 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 quasiprime 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 quasiprime ideal of \(S_{2}\).

2.
\(S_{1} \times A_{2}\) is a quasiprime ideal of \(S_{1} \times S_{2}\).

3.
\(S_{1}\times A_{2}\) is a \(\phi \)quasiprime 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 quasiprime ideal of \(S_{i}\).

2.
\(S_{1}\times S_{2}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}\) is a quasiprime ideal of \(S_{1} \times \ldots \times S_{k}\).

3.
\(S_{1}\times S_{2}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}\) is a \(\phi \)quasiprime 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 \(\psi _{i}\)quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup \(S_{i}\), then \(S_{1}\times S_{2}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}\) is a \(\phi \)quasiprime 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 \(\psi _{1}\)quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup \(S_{1}\), then \(A_{1} \times S_{2}\) is a \(\phi \)quasiprime 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 \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\) if and only if \(A_{1}\) is a \(\psi _{1}\)quasiprime ideal of \(S_{1}\).
Proof
First assume that \(A_{1} \times S_{2}\) is a \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\). The proof is trivial and hence omitted.
Conversely, assume that \(A_{1}\) is a \(\psi _{1}\)quasiprime 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
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 \(\psi _{1}\)quasiprime 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 \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\) if and only if \(A_{2}\) is a \(\psi _{2}\)quasiprime 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 \(S_{1}\times S_{2}\times \ldots \times S_{i1} \times A_{i}\times S_{i+1}\times \ldots \times S_{k}\) is a \(\phi \)quasiprime ideal of \(S_{1} \times \ldots \times S_{k}\) if and only if \(A_{i}\) is a \(\psi _{i}\)quasiprime ideal of \(S_{i}\).
Next, we show that if \(A_{1} \times A_{2}\) is a \(\phi \)quasiprime ideal of an ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup \(S_{1} \times S_{2}\), then \(A_{i}\) is a \(\psi _{1}\)quasiprime 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 \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\) such that \(A_{2} \ne \psi _{2}(A_{2})\), then \(A_{1}\) is a \(\psi _{1}\)quasiprime ideal of \(S_{1}\).

2.
If \(A_{1} \times A_{2}\) is a \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\) such that \(A_{1} \ne \psi _{1}(A_{1})\), then \(A_{2}\) is a \(\psi _{2}\)quasiprime 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 \(\phi \)quasiprime 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 \(\psi _{1}\)quasiprime 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 \(\phi \)quasiprime 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 \(\psi _{i}\)quasiprime ideal of \(S_{i}\).
The next theorem gives conditions for a \(\phi \)quasiprime ideal to be quasiprime 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 \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\), then \(A = \phi (A)\) or A is a quasiprime ideal of \(S_{1} \times S_{2}\).
Proof
Suppose that A is a \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\) that is not a quasiprime 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,
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 \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\times \ldots \times S_{k}\), then \(A = \phi (A)\) or A is a quasiprime ideal of \(S_{1} \times S_{2}\times \ldots \times S_{k}\).
The above theorem shows the relationship between quasiprime ideals and \(\phi \)quasiprime 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 \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\) if and only if A is a quasiprime 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 \(\phi \)quasiprime ideal of \(S_{1} \times S_{2}\times \ldots \times S_{k}\) if and only if A is a quasiprime 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 quasiprime ideal. If A is a \(\phi \)quasiprime ideal of S such that \(\phi \le \phi _{2}\), then \(A^{2} = A^{n+1}\).
The next theorem gives conditions for a \(\phi \)quasiprime ideal to be \(\omega \)quasiprime 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 \(\phi \)quasiprime ideal of S if and only if A is an \(\omega \)quasiprime ideal of S.
Proof
First assume that A is a \(\phi \)quasiprime ideal of S. If A is a quasiprime ideal of S, then it is \(\omega \)quasiprime ideal. Now assume that A is not a quasiprime ideal of S. Then by Lemma 1, \(A^{2} = A^{n+1}\). By assumption, A is a \(\phi \)quasiprime ideal of S and \(\phi \le \phi _{n+1}\), which implies that A is a \(\phi _{n+1}\)quasiprime 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 \(\omega \)quasiprime ideal of S.
Conversely, assume that A is a \(\phi \)quasiprime ideal of S. The proof is trivial and hence omitted. \(\square \)
Conclusion
In study the structure of ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroups, we notice that the quasiprime ideals with special properties always play an important role. The purposes of this paper are to introduce generalizations of quasiprime ideals to the context of \(\phi \)quasiprime ideals. Some characterizations of quasiprime and \(\phi \)quasiprime ideals are obtained. Moreover, we investigate relationships between weakly quasiprime, almost quasiprime, \(\omega \)quasiprime, mquasiprime and \(\phi \)quasiprime ideals of ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroups. Finally, we obtain necessary and sufficient conditions of \(\phi \)quasiprime ideal in order to be a quasiprime ideal.
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Yiarayong, P. On generalizations of quasiprime ideals of an ordered left almost semigroups. Afr. Mat. (2021). https://doi.org/10.1007/s1337002100873x
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Keywords
 ordered \({{\mathcal {L}}}{{\mathcal {A}}}\)semigroup
 quasiprime ideal
 \(\phi \)quasiprime ideal
 \(\omega \)quasiprime
 \(\phi \)zero
Mathematics Subject Classification
 20M10
 16Y99