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, bin S with ab in A - phi (A), then a in A or bin 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.
Highlights
In 2010, Shah et al [35] studied ideals, M-systems, N -systems and I -systems of ordered LA-semigroups and provided that if A is a left ideal of an ordered LA-semigroup with left identity, 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
In 2012, Faisal et al [13] introduced the notion of fuzzy ordered Γ -LA-semigroups and studied (2, 2)-regular ordered Γ -LA**-semigroup in terms of fuzzy Γ -left ideals, fuzzy Γ -right ideals, fuzzy Γ -two-sided ideals, fuzzy Γ -generalized bi-ideals, fuzzy Γ -bi-ideals, fuzzy Γ -interior ideals and fuzzy Γ -(1; 2)-ideals. They proved that the set of all fuzzy Γ -two-sided ideals of a (2, 2)-regular ordered Γ -LA**-semigroup S forms a
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 φ-quasi-prime ideals
Summary
In 2010, Shah et al [35] studied ideals, M-systems, N -systems and I -systems of ordered LA-semigroups and provided that if A is a left ideal of an ordered LA-semigroup with left identity, 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. In 2012, Faisal et al [13] introduced the notion of fuzzy ordered Γ -LA-semigroups and studied (2, 2)-regular ordered Γ -LA**-semigroup in terms of fuzzy Γ -left ideals, fuzzy Γ -right ideals, fuzzy Γ -two-sided ideals, fuzzy Γ -generalized bi-ideals, fuzzy Γ -bi-ideals, fuzzy Γ -interior ideals and fuzzy Γ -(1; 2)-ideals. They proved that the set of all fuzzy Γ -two-sided ideals of a (2, 2)-regular ordered Γ -LA**-semigroup S forms a.
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