ON PRIME LEFT(RIGHT) IDEALS OF GROUPOIDS-ORDERED GROUPOIDS

Abstract. Recently, Kehayopulu and Tsingelis studied for prime ideals of groupoids-ordered groupoids. In this paper, we give some results on prime left(right) ideals of groupoid-ordered groupoid. These results are generalizations of their results. If (G, ·,≤) is an ordered groupoid, a non-empty subset A of G is called a left (resp. right) ideal of G if 1) GA ⊆ A (resp. AG ⊆ A) and 2) a ∈ A, b ≤ a for b ∈ G implies b ∈ A ([2-4]). If G is a groupoid, a left (resp. right) ideal of G is a non-empty subset A of G such that GA ⊆ A (resp. AG ⊆ A). A subset P of a groupoid is said to be prime if ab ∈ P implies a ∈ P or b ∈ P (see [5]). A prime left (resp. right) ideal of a groupoid (ordered groupoid) is prime as left (resp. right) ideals ([2, 3]). If (G, ·,≤) is an ordered groupoid, a zero of G is an element 0 of G such that 0x = x0 = 0 and 0 ≤ x for every x ∈ G ([1]). If G is a groupoid, a zero of G is an element 0 of G such that 0x = x0 = 0 for every x ∈ G. Recently, Kehayopulu and Tsingelis gave some results for prime ideals of groupoids. In this paper, we give analogous results for prime left(right) ideals of groupoids. These results are generalizations of the results of Kehay-opulu and Tsingelis. Proposition 1. Let G be a groupoid (resp. ordered groupoid) and C a chain (under set inclusion) of prime left ideals of G. If ⋂L∈C L is non-empty, then it is a prime left ideal of G.