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

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