# Definition:Meet Semilattice

From ProofWiki

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Suppose that for all $a, b \in S$:

- $a \wedge b \in S$,

where $a \wedge b$ is the meet of $a$ and $b$.

Then the ordered structure $\left({S, \wedge, \preceq}\right)$ is called a **meet semilattice**.

That it is indeed an ordered structure is proved on Meet Semilattice is Ordered Structure.

That a **meet semilattice** is a semilattice is proved on Meet Semilattice is Semilattice.