equivalence_relations - MathStructures

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=====Equivalence relations=====

Abbreviation: **EqRel**

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

reflexive:  $x\equiv x$

symmetric:  $x\equiv y\Longrightarrow y\equiv x$

transitive: $x\equiv y\text{ and }y\equiv z\Longrightarrow x\equiv z$

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

====Definition====
An \emph{equivalence relation} is a [[qoset]] that is \emph{symmetric}: $x\equiv y\Longrightarrow y\equiv x$

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

====Basic results====
Equivalence relations are in 1-1 correspondence with [[partitions]].


====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====

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &2\\
  f(3)= &3\\
  f(4)= &5\\
  f(5)= &7\\
\end{array}$     
$\begin{array}{lr}
  f(6)= &11\\
  f(7)= &15\\
  f(8)= &22\\
  f(9)= &30\\
  f(10)= &42\\
\end{array}$

The number of (labelled) equivalance relations on an $n$ element set given by a sum of Stirlings formula (of the second kind). 

see also http://www.research.att.com/projects/OEIS?Anum=A000110

The number of (nonisomorphic) equivalence relations is the number of partition patterns (= number of integer partitions).

see also http://www.research.att.com/projects/OEIS?Anum=A000041

====Subclasses====

====Superclasses====
  [[Preordered sets]] supervariety


====References====

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

Trace: • commutative_bck-algebras • function_rings • quasitrivial_groupoids • hilbert_algebras • division_algebras • moufang_quasigroups • order_algebras • action_algebras • bounded_residuated_lattices • ordered_groups