# Differences

This shows you the differences between two versions of the page.

equivalence_relations [2010/07/29 15:46] (current)
Line 1: Line 1:
+=====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).
+
+
+The number of (nonisomorphic) equivalence relations is the number of partition patterns (= number of integer partitions).
+
+
+====Subclasses====
+
+====Superclasses====
+  [[Preordered sets]] supervariety
+
+
+====References====
+
+[(Lastname19xx>
+F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
+)]