Definition:Meet Semilattice

Definition

Definition 1

Let $\struct {S, \preceq}$ 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 $\struct {S, \wedge, \preceq}$ is called a meet semilattice.

Definition 2

Let $\struct {S, \wedge}$ be a semilattice.

Let $\preceq$ be the ordering on $S$ defined by:

$a \preceq b \iff \paren {a \wedge b} = a$

Then the ordered structure $\struct {S, \wedge, \preceq}$ is called a meet semilattice.

Also known as

A meet semilattice is also known as an lower semilattice or a $\wedge$-semilattice.

Some sources hyphenate: meet semi-lattice.

Also see

• Equivalence of Definitions of Meet Semilattice

• Meet Semilattice is Semilattice
• Meet Semilattice is Ordered Structure

• Definition:Join Semilattice
• Definition:Semilattice

• Results about meet semilattices can be found here.