Definition:Meet Semilattice
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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
- Results about meet semilattices can be found here.