Can every partially ordered set (POSET) take the form of a directed acyclic graph (DAG)? A POSET (Partially ordered set) is a set on the elements of which we have established a partial order relation ($\leq$), i.e. a relation which is:


*

*reflexive: $x\leq x,$ for every x in S

*anti-symmetric: $x \leq y \wedge y \leq x \Rightarrow x=y $

*transitive: $x\leq y, y\leq x \Rightarrow x\leq z$


My question is if every POSET can take the form of a DAG (Directed Acyclic Graph) if we view its elements as the nodes and the relation itself as the edge set.

YES. Every POSET can take the form of a DAG. 
However, in order to obtain a less cluttered graph, you’d better avoid drawing every single edge. (see figure) 
You can omit the edges that can be inferred from the reflexive and transitive properties. 
Moreover you can arrange nodes in order to orient every edge upward and omit vector tips.
In this way you obtain a Hasse diagram: it is just a stripped down DAG.

