Does a set and a transitive reflexive relation on that set form a category?

Question

If you have a set $S$ with a relationship on it $R$. With $R$ being transitive and reflexive can it be considered a category with elements of the set being the objects and the relationship being the morphisms?

Answer 1

Yes, this is a standard example called a preorder category. ( A preorder is a reflexive transitive relation.)

We take the objects of $\mathcal C$ to be elements of $S$.

There is an arrow $(a,b)$ from $a$ to $b$ if and only if $a≤_R b$; if it exists the arrow is unique. Then the identity arrow for $a$ is simply $(a,a)$.

The composition of arrows $(a,b)$ and $(b,c)$ is the arrow $(a,c)$. The composition obviously has the required associativity and identity properties.