Order‑Theoretic Limsup and Liminf

Definition (Complete lattice).
A partially ordered set $(L,\le )$ is called a complete lattice if for every $S\subseteq L$, the supremum $\sup S$ and infimum $\inf S$ exists in $L$.

Definition (Order‑theoretic limits).
Let $(L,\le )$ be a complete lattice and $\{x_n\}\subseteq L$ be a sequence in $L$. Define the upper‑tail suprema and lower‑tail infima of $\{x_n\}$ by

$$s_n:=\underset{m\ge n}{\sup }{x}_m,i_n:=\underset{m\ge n}{\inf }{x}_m.$$

Define the limit suprema and limit infima by

$$\underset{n\to \infty }{\mathrm{limsup}}{x}_n:=\underset{n}{\inf }{s}_n,\underset{n\to \infty }{\mathrm{liminf}}{x}_n:=\underset{n}{\sup }{i}_n.$$

Definition (limsup and liminf of sequence of numbers).
Taking $L=[-\infty ,\infty ]$ with the usual order recovers the classical limsup and liminf for sequences of real numbers.

Definition (limsup and liminf of sequence of sets).
Let $X$ be a set. Taking $L=\mathcal{P}(X)$ ordered by inclusion reproduces the limsup and liminf of sequence of sets

$$\underset{n\to \infty }{\mathrm{limsup}}{A}_n=\bigcap _{N=1}^{\infty }\bigcup _{n\ge N}{A}_n,\underset{n\to \infty }{\mathrm{liminf}}{A}_n=\bigcup _{N=1}^{\infty }\bigcap _{n\ge N}{A}_n.$$