Order convergence

In mathematics, specifically in order theory and functional analysis, a filter $$\mathcal{F}$$ in an order complete vector lattice $$X$$ is order convergent if it contains an order bounded subset (that is, a subset contained in an interval of the form $$[a, b] := \{ x \in X : a \leq x \text{ and } x \leq b \}$$) and if $$\sup \left\{ \inf S : S \in \operatorname{OBound}(X) \cap \mathcal{F} \right\} = \inf \left\{ \sup S : S \in \operatorname{OBound}(X) \cap \mathcal{F} \right\},$$ where $$\operatorname{OBound}(X)$$ is the set of all order bounded subsets of X, in which case this common value is called the order limit of $$\mathcal{F}$$ in $$X.$$

Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology.

Definition
A net $$\left(x_{\alpha}\right)_{\alpha \in A}$$ in a vector lattice $$X$$ is said to decrease to $$x_0 \in X$$ if $$\alpha \leq \beta$$ implies $$x_{\beta} \leq x_{\alpha}$$ and $$x_0 = inf \left\{ x_{\alpha} : \alpha \in A \right\}$$ in $$X.$$ A net $$\left(x_{\alpha}\right)_{\alpha \in A}$$ in a vector lattice $$X$$ is said to order-converge to $$x_0 \in X$$ if there is a net $$\left(y_{\alpha}\right)_{\alpha \in A}$$ in $$X$$ that decreases to $$0$$ and satisfies $$\left|x_{\alpha} - x_0\right| \leq y_{\alpha}$$ for all $$\alpha \in A$$.

Order continuity
A linear map $$T : X \to Y$$ between vector lattices is said to be order continuous if whenever $$\left(x_{\alpha}\right)_{\alpha \in A}$$ is a net in $$X$$ that order-converges to $$x_0$$ in $$X,$$ then the net $$\left(T\left(x_{\alpha}\right)\right)_{\alpha \in A}$$ order-converges to $$T\left(x_0\right)$$ in $$Y.$$ $$T$$ is said to be sequentially order continuous if whenever $$\left(x_n\right)_{n \in \N}$$ is a sequence in $$X$$ that order-converges to $$x_0$$ in $$X,$$then the sequence $$\left(T\left(x_n\right)\right)_{n \in \N}$$ order-converges to $$T\left(x_0\right)$$ in $$Y.$$

Related results
In an order complete vector lattice $$X$$ whose order is regular, $$X$$ is of minimal type if and only if every order convergent filter in $$X$$ converges when $$X$$ is endowed with the order topology.