Intervals in a poset; locally finite, lower-finite and upper-finite posets

Let $(P,\le)$ be a poset (Partial order and partially ordered set). For comparable elements $x\le y$, the closed interval from $x$ to $y$ is

$[x,y]:=\{z\in P:x\le z\le y\}.$

The principal ideal below $y$ and the principal filter above $x$ are

$P_{\le y}:=\{z\in P:z\le y\},\qquad P_{\ge x}:=\{z\in P:x\le z\}.$

The poset $P$ is

• locally finite when $[x,y]$ is finite for every $x\le y$;
• lower-finite when $P_{\le y}$ is finite for every $y\in P$;
• upper-finite when $P_{\ge x}$ is finite for every $x\in P$.

Here finite has the meaning of Finite, countably infinite, countable, uncountable, and finite cardinalities are those of The cardinality $\lvert A\rvert$ of a finite set. Every lower-finite poset is locally finite because $[x,y]\subseteq P_{\le y}$, and every upper-finite poset is locally finite because $[x,y]\subseteq P_{\ge x}$; both conclusions use that a subset of a finite set is finite (A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$, and equality holds if and only if $B = A$).

Local finiteness controls sums over one interval $[x,y]$. It does not imply that a whole principal ideal or principal filter is finite. The one-sided hypotheses are therefore stated separately because global inversion sums range over those larger sets.