---
tag: 74X7JRG2J7
kind: concept
name: Semilattice
msc: [06A12]
sources:
  - url: https://en.wikipedia.org/wiki/Semilattice
same_as:
  wikidata: Q834585
  wikipedia: Semilattice
---

## Summary

A join-semilattice is a [[partially ordered set|0QV2JS9RJ3]] in which every two elements have a [[join|J2T75ATRV5]].
A meet-semilattice is one in which every two elements have a meet.
Algebraically, a semilattice is a [[semigroup|0XMEM406JM]] whose operation is commutative and idempotent.
The two views are equivalent, and a lattice is a set that is both a join- and a meet-semilattice for the same order.
Semilattices appear in the theory of semigroups, in domain theory and in the analysis of programs.

## Explanations

### L1 human

A join-semilattice is an ordering in which any two things have a smallest common upper bound, but not always a largest common lower bound.
Picture a rooted tree drawn with its root at the top: any two nodes have a closest common ancestor above them, and this is their join.
For example, take two branches $a$ and $b$ that meet at a top element $1$.
Then $a \vee b = 1$, but $a$ and $b$ have no common lower bound at all.
So this is a join-semilattice but not a meet-semilattice.