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.