Rel is a concrete category over Sets, but how to concretize that?

Question

The traditional denotation of a structured set object is something like

$(1)\qquad(X,\mathcal S_1,\dots\mathcal S_n)$

for some "structures" $\mathcal S_1,\dots\mathcal S_n$ on a set X.

The modern definition of a category with sets as objects and functions as morphisms includes a faithful functor $F:\mathbf C\to\mathbf{Set}$, that is, a functor such that $\alpha\ne \beta\implies F(\alpha)\ne F(\beta)$ for all morphisms $\alpha,\beta\in Mor(\mathbf C)$.

Now there is a faithful functor from Rel to Set, $\mathcal P(\cdot)$ the covariant power set functor, and my question is:

is it possible to express this category as a category of structured sets as in $(1)$ with structure preserving functions as morphisms?

Answer 1

Well, $\mathbf{Rel}$ is a full subcategory of $\mathbf{CJS}$, the category of complete join-semilattices (and join-preserving maps). The embedding sends a set $X$ to the powerset $\mathscr{P} X$ and a relation $R : X \not\to Y$ to the unique join-preserving map $r : \mathscr{P} X \to \mathscr{P} Y$ where $r \{ x \} = \{ y \in Y : x \mathrel{R} y \}$. Thus you can realise $\mathbf{Rel}$ as a full subcategory of a category of (infinitary) algebraic structures.