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

The purposes of this paper are to introduce generalizations of quasi-prime ideals to the context of phi-quasi-prime ideals. Let phi:I(S) -> I(S) boolean OR {phi}? null be a function where I(S) is the set of all left ideals of an ordered LA-semigroup S. A proper left ideal A of an ordered LA-semigroup S is called a phi-quasi-prime ideal, if for each a,b is an element of S with ab is an element of A-phi(A), then a is an element of A or b is an element of 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 LA-semigroups. Finally, we obtain necessary and sufficient conditions of phi-quasi-prime ideal in order to be a quasi-prime ideal.