A Partially Ordered Set is a set with a partial order.

Well, obviously. But what is a partial order?

The easiest way to see this is to start with a directed, acyclic graph. This is a web of nodes where each edge has a direction, and there are no loop, no way to follow the arrows and end up where you started.

No extend that with extra arrows so that if it's possible to get from A to B, put an arrow that gets there directly. This is called the transitive closure.

So a partial order is a way of saying that one thing is "less than" another thing, but where not everything is comparable. But there are consequences, because if A is "less than" B, and B is "less than" C, then A will certainly have to be "less than" C. It's transitive.

So a (strict) partial order is a relation which is:

• Irreflexive
• Transitive
 "Anti-symmetric" means that if "A thingy B" and "B thingy A," then A and B are equal.
We can extend this to non-strict partial order, where the concept is that of "less than or equal to," but then we need to add that the relation is "anti-symmetric"
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