directed_partial_orders - MathStructures

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=====Directed partial orders=====

Abbreviation: **DPO**
====Definition====
A \emph{directed partial order} is a poset $\mathbf{P}=\langle P,\leq \rangle $ that is \emph{directed}, i.e. every finite subset
of $P$ has an upper bound in $P$, or equivalently, $P\ne\emptyset$, $\forall xy\exists z
(x\le z$ and $y\le z)$.
==Morphisms==
Let $\mathbf{P}$ and $\mathbf{Q}$ be directed partial orders. A morphism from $\mathbf{P}$ to 
$\mathbf{Q}$ is a function $f:Parrow Q$ that is order preserving: 

$x\le y\Longrightarrow f(x)\le f(y)$

====Examples====
Example 1: 

====Basic results====

====Properties====
^[[Classtype]]  |first-order |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &2\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
\end{array}$

====Subclasses====
[[Directed complete partial orders]] 

====Superclasses====
[[Partially ordered sets]] 


====References====

[(Ln19xx>
)]

Trace: • finite_ordered_semigroups_with_zero • distributive_lattices_with_operators • finite_binars_with_identity_and_zero • normal_valued_lattice-ordered_groups • weakly_associative_relation_algebras • residuated_lattice-ordered_semigroups • finite_commutative_binars_with_identity • commutative_partially_ordered_semigroups • pseudocomplemented_distributive_lattices • multiplicative_additive_linear_logic_algebras