ordered_semilattices - MathStructures

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=====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====

^[[Classtype]]                        |universal  |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====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====
[[Commutative ordered semigroups]]


====References====

[(Lastname19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]

Trace: • partial_monoids • boolean_algebras • graphs • orthomodular_lattices • lattice-ordered_rings • directed_partial_orders • finite_binars • finite_commutative_binars • abelian_ordered_groups • commutative_monoids