A Generalization of Order Continuous Operators

Let E be a sublattice of a vector lattice F. A net {xα}α∈A ⊆ E is said to be F-order convergent to a vector x ∈ E (in symbols xα F o → x in E), whenever there exists a net {yβ}β∈B in F satisfying yβ ↓ 0 in F and for each β, there exists α0 such that |xα − x| ≤ yβ whenever α ≥ α0. In this manuscript, first we study some properties of F-order convergence nets and we extend some results to the general cases. Let E and G be sublattices of vector lattices F and H, respectively. We introduce F H-order continuous operators, that is, an operator T between two vector lattices E and G is said to be F H-order continuous, if xα F o → 0 in E implies T xα Ho → 0 in G. We will study some properties of this new classification of operators and its relationships with order continuous operators.