Definition:Join Semilattice
(Redirected from Definition:Upper Semilattice on Classical Set)
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Definition
Definition 1
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.
Definition 2
Let $\struct {S, \vee}$ be a semilattice.
Let $\preceq$ be the ordering on $S$ defined by:
- $a \preceq b \iff \paren {a \vee b} = b$
Then the ordered structure $\struct {S, \vee, \preceq}$ is called a join semilattice.
Also known as
A join semilattice is also known as an upper semilattice or a $\vee$-semilattice.
Some sources hyphenate: join semi-lattice.
Also see
- Results about join semilattices can be found here.
Sources
- Semi-lattice. Encyclopedia of Mathematics. URL: https://www.encyclopediaofmath.org/index.php?title=Semi-lattice&oldid=39737