---
tag: WMR73J0S26
kind: concept
name: Pseudocomplement
msc: ["06D15"]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudocomplement
same_as:
  wikidata: Q18345290
  wikipedia: Pseudocomplement
---

## Summary

Let $L$ be a [[lattice|WH4SFAACAS]] with a least element $0$, and let $a \in L$.
The pseudocomplement of $a$ is the largest element whose meet with $a$ is $0$, when such an element exists.
It is written $a^*$.
In a [[Boolean algebra|0XADJKM699]] it is the usual complement, and in a [[Heyting algebra|NG943XNQWH]] it is $a \to 0$.
Pseudocomplements give a weak, intuitionistic form of negation.

## Explanations

### L1 human

The pseudocomplement of $a$ is the biggest thing that does not overlap $a$ at all.
For the open sets of the real line, the pseudocomplement of an open set $U$ is the interior of its complement.
If $U = (0, 1)$, this is $(-\infty, 0) \cup (1, \infty)$.
The points $0$ and $1$ belong neither to $U$ nor to $U^*$, so $U \cup U^*$ is not the whole line, unlike a true complement.