A formalization of the notion of Hasse diagrams

Question

I have read many textbooks on order theory, but none of them give a formal definition of Hasse diagrams. I would like someone to give me a formalization of that notion. Just to be clear, we are talking about Hasse diagrams on the real plane, $\mathbb{R}^2$.

Answer 1

A Hasse diagram for the partial order $\preceq$ on the finite set $\{a_1,\ldots,a_n\}$ consists of

• a set of points $P_j=(x_j,y_j)$ for $j=1,\ldots,n$, all different, and having the property that $y_j\le y_k$ whenever $a_j\preceq a_k$;
• a line segment $P_jP_k$ whenever $a_j\prec a_k$ and there is no $m$ such that $a_j\prec a_m\prec a_k$. Here $x\prec y$ means $x\preceq y$ and $x\ne y$.