---
tag: TV1V4JDRDD
kind: concept
name: Locally finite poset
msc: [06A07, 06A11]
specializes: [0QV2JS9RJ3]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_finite_poset
same_as:
  wikidata: Q6664687
  wikipedia: Locally finite poset
---

## Summary

A locally finite poset is a [[partially ordered set|0QV2JS9RJ3]] in which every closed interval $[x, y] = \set{z : x \le z \le y}$ is finite.
This is the condition under which the [[incidence algebra|1H9SFWHYEB]] and the Möbius function of the poset are defined.
Most posets of enumerative combinatorics are locally finite, even when they are infinite.

## Explanations

### L1 human

A poset may be infinite, but you can ask that between any two comparable elements there are only finitely many others.
The integers satisfy this: between $3$ and $10$ there are eight integers.
The positive integers ordered by divisibility satisfy it too: between $2$ and $12$ lie only the divisors of $12$ that are multiples of $2$.
The rationals fail: between $0$ and $1$ there are infinitely many.