Let $A$ and $B$ be two sets. A relation from $A$ to $B$ is a subset $$R\subset A\times B.$$
This induces an inverse relation $$R^{-1}=\{(y, x)\in B\times A: (x, y)\in R\}$$ from $B$ to $A$. Also, given a third $C$ and relations $R\subset A\times B$ and $S\subset B\times C$ it is possible to define the composite relation $$R\circ S=\{(x, z)\in A\times C: \exists y\in B; (x, y)\in R\ \textrm{and}\ (y, z)\in S\}.$$
One last notation, for a set $A$ let $$\Delta_A=\{(a, a): a\in A\}$$ be the diagonal of $A$.
Let us fix a relation $R$ in a set $X$, that is, $R\subset X\times X$. This relation is called an order if:
($R$ is transitive) $R\circ R\subset R$;
($R$ is skew-symmetric) $R\cap R^{-1}=\Delta_X$.
The pair $(X, R)$ is called an ordered set.
Question 1. In the literature, an ordered set $(X, R)$ is called directed if for every $x, y\in X$ there exists $z\in X$ such that $$(x, z)\in R\ \textrm{and}\quad (y, z)\in R.$$ This makes me think $(X, R)$ is directed if and only if $$X\times X=R\circ R^{-1}.$$ Is that correct?
Question 2. How can one define using sets the notions of upper bound, lower bound, maximal element, minimal element, infimum and supremum?
Thanks.
To answer question 1, yes. What you have on the left is the relation that relates all elements, and what you have on the right is the relation that relates (x,y) if there’s some z such that (x,z) is related by R, and (z,y) is related by R$^{-1}$. And since R$^{-1}$ is the relation that relates (z,y) exactly when R relates (y,z), then that means R $\circ$R$^{-1}$ is the relation that relates (x,y) whenever there is a z such that R relates (x,z) and (y,z). Therefore a directed ordered set is exactly an ordered set where your equality:
$$X \times X = R \circ R^{-1}$$
Holds.
Regarding question 2, an order can be treated as a generalization of either the relation $ x \leq y$ or the relation $x \geq y$ Over the real numbers, because $x \leq y$ Is equivalent to $-x \geq -y$, which essentially means the two behave the same. If we want to treat our order like $ x \leq y$, then to define each of those terms, we simply take the familiar definitions in the real numbers, and substitute $(x,y) \in R$ wherever we see $x \leq y$ in our original definition, and similarly for other inequalities.
For example, to define supremum, we say that element $s$ is the supremum of set$ S$, when $\forall y \in S, (y,s)\in R$ and $\forall x$ such that $(y,x) \in R, (s,x) \in R$.