## Symmetric relations

Abbreviation: SymRel

### Definition

A symmetric relation is a structure $\mathbf{X}=\langle X,R\rangle$ such that $R$ is a binary relation on $X$ (i.e. $R\subseteq X\times X$) that is

symmetric: $xRy\Longrightarrow yRx$

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

Example 1:

### 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

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

### Superclasses

[[Directed graphs]] supervariety