Convexity Conditions on f-Rings

Let n be a positive integer. An f-ring A is said to satisfy the left nth-convexity property if for any u, v, ∈ A such that v ≥ 0 and 0 ≤ u ≤ v^n , there exists a w e A such that u = ,wv. The right nth - convexity property is defined similarly and an f-ring is said to satisfy the nth -convexity property if it satisfies both the left and the right nth -covexity property. In this paper, we study arbitrary f-rings which satisfy one of the convexity properties.