Equivalence relations

Abbreviation: EqRel


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

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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.


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: $x\equiv^{\mathbf X} y\Longrightarrow h(x)\equiv^{\mathbf Y}h(y)$


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


Example 1:

Basic results

Equivalence relations are in 1-1 correspondence with partitions.


Finite members


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\\


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



[[Preordered sets]] supervariety


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