Ordered semilattices

Abbreviation: OSlat

Definition

An \emph{ordered semilattice} is a ordered semigroup $\mathbf{A}=\langle A,\cdot,\le\rangle$ that is

\emph{commutative}: $x\cdot y = y\cdot x$ and

\emph{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


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