Table of Contents
Ordered semilattices
Abbreviation: OSlat
Definition
An ordered semilattice is a ordered semigroup $\mathbf{A}=\langle A,\cdot,\le\rangle$ that is
commutative: $x\cdot y = y\cdot x$ and
idempotent: $x\cdot x = x$
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $x\le y\Longrightarrow h(x)\le h(y)$.
Examples
Example 1:
Basic results
Properties
Finite members
$\begin{array}{rr} f(1)=&1\\ f(2)=&2\\ f(3)=&5\\ f(4)=&14\\ f(5)=&42\\ f(6)=&132\\ f(7)=&\\ f(8)=&\\ \end{array}$
This sequence is the Catalan numbers http://oeis.org/A000108
Subclasses
Superclasses
References
Trace: » ordered_semilattices