symmetric_relations - MathStructures

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=====Symmetric relations=====

Abbreviation: **SymRel**

====Definition====
A \emph{symmetric relation} is a structure $\mathbf{X}=\langle X,R\rangle$ such that $R$ is a \emph{binary relation on $X$} 
(i.e. $R\subseteq X\times X$) that
is

symmetric:  $xRy\Longrightarrow yRx$

Remark: This is a template.
If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.

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 $\mathbf{X}$ and $\mathbf{Y}$ be symmetric relations. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $h:A\rightarrow B$ that is a homomorphism: 
$xR^{\mathbf X} y\Longrightarrow h(x)R^{\mathbf Y}h(y)$

====Definition====

====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]]                        |quasivariety  |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   |yes |
^[[Residual size]]                    | |
^[[Congruence distributive]]          |no |
^[[Congruence modular]]               |no |
^[[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====

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


====Subclasses====

====Superclasses====
  [[Directed graphs]] supervariety


====References====

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

Trace: • abelian_p-groups • completely_regular_hausdorff_spaces • abelian_partially_ordered_groups

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