Partial orders

As mentioned previously, a partial order is a relation which is reflexive, antisymmetric and transitive. If the $\leq$ relation on a set $S$ satisfies $\forall x,y \in S \quad x \leq y \vee y \leq x$, then it is a total order. If $x \in S: x \not\leq y \wedge y \not\leq x$, then $x$ and $y$ are incomparable.

Least/greatest elements, minima/maxima

For the $\leq$ relation on a set, the least element in a partial order is $x \in P$ such that $\forall y:\: x \leq y$.
The greatest element is the opposite.
The minimal element is $x$ if $\forall y:\: y \leq x \to y=x$.
The maximal element is $x$ if $\forall y:\: y \geq x \to y=x$.

Hasse diagram

The Hasse diagram of the partial order $P_\leq$ is the directed graph $G =(V,E)$ such that $V=P$ and $P = \{(x,y):x < y \wedge \nexists z \in P \text{ such that } x<y \wedge z<y\}$. This means $x$ and $y$ are adjacent. A dense partial order is where there is no $(x,y)$ which satisfies the conditions. Therefore there is no Hasse diagram.
All Hasse diagrams of partial orders are directed acyclic graphs because of the properties of partial orders.