---
tag: 57G2DV78JW
kind: concept
name: Infimum and supremum
aliases: [inf-sup, infima, infima-and-suprema, lub, lowest-upper-bound-axiom, smallest-upper-bound, suprema, supremum-and-infimum]
sources:
  - wikipedia: en:39382@r1374158516
    locator: Infimum and supremum
    note: adopted
    mention: wikipedia://en/39382
    sections: [Summary]
same_as:
  wikidata: Q17502105
  wikipedia: Infimum and supremum
inventory: [BD97KMBV7A]
---

## Summary

In mathematics, the **infimum** (abbreviated **inf**; : **infima**) of a [[subset|0BJR9QR08E]] $S$ of a [[partially ordered set|0QV2JS9RJ3]] $P$ is the greatest element in $P$ that is less than or equal to each element of $S,$ if such an element exists. If the infimum of $S$ exists, it is unique, and if *b* is a lower bound of $S$, then *b* is less than or equal to the infimum of $S$. Consequently, the term *greatest lower bound* (abbreviated as GLB) is also commonly used. The **supremum** (abbreviated **sup**; : **suprema**) of a subset $S$ of a partially ordered set $P$ is the least element in $P$ that is greater than or equal to each element of $S,$ if such an element exists. If the supremum of $S$ exists, it is unique, and if *b* is an upper bound of $S$, then the supremum of $S$ is less than or equal to *b*. Consequently, the supremum is also referred to as the *least upper bound* (or LUB).