directed_graphs - MathStructures

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=====Directed graphs=====

Abbreviation: **DiGraph**

====Definition====
A \emph{directed graph} (or \emph{digraph} for short) is a structure $\mathbf{G}=\langle G,E\rangle$ such that

$E$ is binary relation on $G$:  $E\subseteq G\times G$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

==Morphisms==
Let $\mathbf{G}$ and $\mathbf{H}$ be directed graphs. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:G\rightarrow H$ that preserves $E$: 
$\langle x,y\rangle\in E^{\mathbf G}\Longrightarrow \langle h(x), h(y)\rangle\in E^{\mathbf H}$

====Definition====
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$

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

====Basic results====


====Properties====
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

^[[Classtype]]                        |variety  |
^[[Equational theory]]                |decidable |
^[[Quasiequational theory]]           |decidable |
^[[First-order theory]]               |undecidable |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          |no |
^[[Congruence modular]]               |no |
^[[Congruence $n$-permutable]]        |no |
^[[Congruence regular]]               |no |
^[[Congruence uniform]]               |no |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            |yes |
^[[Strong amalgamation property]]     |yes |
^[[Epimorphisms are surjective]]      | |

====Finite members====

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &\\
  f(3)= &\\
  f(4)= &\\
  f(5)= &\\
\end{array}$     
$\begin{array}{lr}
  f(6)= &\\
  f(7)= &\\
  f(8)= &\\
  f(9)= &\\
  f(10)= &\\
\end{array}$


====Subclasses====
  [[...]] subvariety

  [[...]] expansion


====Superclasses====
  [[...]] supervariety

  [[...]] subreduct


====References====

[(Ln19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]

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