Definition:Semilattice
Definition
Let $\struct {S, \circ}$ be a semigroup.
Then $\struct {S, \circ}$ is called a semilattice if and only if $\circ$ is a commutative and idempotent operation.
Thus an algebraic structure is a semilattice if and only if it satisfies the semilattice axioms:
\((\text {SL} 0)\) | $:$ | Closure for $\circ$ | \(\ds \forall a, b \in S:\) | \(\ds a \circ b \in S \) | |||||
\((\text {SL} 1)\) | $:$ | Associativity of $\circ$ | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | |||||
\((\text {SL} 2)\) | $:$ | Commutativity of $\circ$ | \(\ds \forall a, b \in S:\) | \(\ds a \circ b = b \circ a \) | |||||
\((\text {SL} 3)\) | $:$ | Idempotence of $\circ$ | \(\ds \forall a \in S:\) | \(\ds a \circ a = a \) |
Join Semilattice
Let $\struct {S, \preceq}$ be an ordered set.
Suppose that for all $a, b \in S$:
- $a \vee b \in S$
where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$.
Then the ordered structure $\struct {S, \vee, \preceq}$ is called a join semilattice.
Meet Semilattice
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.
Also known as
Some sources hyphenate the word semilattice as semi-lattice.
Also see
- Results about semilattices can be found here.
Sources
- 1982: Peter T. Johnstone: Stone Spaces ... (previous) ... (next): Chapter $\text I$: Preliminaries, Definition $1.3$
- Semi-lattice. Encyclopedia of Mathematics. URL: https://www.encyclopediaofmath.org/index.php?title=Semi-lattice&oldid=39737