## Equivalence relations

Abbreviation: EqRel

### Definition

An equivalence relation is a structure $\mathbf{X}=\langle X,\equiv\rangle$ such that $\equiv$ is a 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.

##### 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 equivalence relation is a qoset that is symmetric: $x\equiv y\Longrightarrow y\equiv x$

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 yes no no

### 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).

### Superclasses

[[Preordered sets]] supervariety