reflexive_relations - MathStructures

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=====Reflexive relations=====

Abbreviation: **RefRel**

====Definition====
A \emph{reflexive 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

reflexive:  $xRx$

==Morphisms==
Let $\mathbf{X}$ and $\mathbf{Y}$ be reflexive 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]]                        |variety  |
^[[Equational theory]]           |decidable |
^[[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: • commutative_semigroups • relativized_relation_algebras • abelian_partially_ordered_groups • abelian_lattice-ordered_groups • bounded_distributive_lattices • finite_semilattices • principal_ideal_domains • quasi-mv-algebra • brouwerian_semilattices • representable_cylindric_algebras