unbounded order-to-order continuous operators and order-to-unbounded order continuous operators on riesz spaces unbounded order-to-order continuous operators on riesz spaces

Let e and f be two riesz spaces. an operator t : e→ f between two riesz spaces is said to be unbounded order-to-order continuous whenever x∝→ 0 in e implies tx∝ → 0 in f for each net (x∝)⊆ e. this paper aims to investigate several properties of a novel class of operators and their connections to established operator classifications. furthermore, we introduce a new class of operators, which we refer to as order-to-unbounded order continuous operators. an operator t : e→ f rightarrow f between two riesz spaces is said to beorder-to-unbounded order continuous (for short, ouo-continuous), if x∝→ 0 in e implies tx∝ → 0 in f for each net (x∝)⊆ e.in this manuscript, we investigate the lattice properties of a certain class of objects and demonstrate that, under certain conditions, order continuity is equivalent to unbounded order-to-order continuity of operators on riesz spaces. additionally, we establish that the set of all unbounded order-to-order continuous linear functionals on a riesz space e forms a band of e∼.