# What is the difference between a strict partial order and a strict weak order?

According to Wikipedia, strict partial orders and strict weak orders are transitive binary relations satisfying asymmetry, antisymmetry and irreflexivity. See screenshot below:

However, if strict partial orders and strict weak orders have different rows in the table, it must be because they are somehow different. Hence,

What is the difference between a strict partial order and a strict weak order?

Also, can you provide examples that illuminate the difference?

Without any qualifiers the term partial order typically means a relation on a set $$X$$ that is reflexive, antisymmetric, and transitive. An example is $$\subseteq$$ on the power set $$2^S$$ of a set $$S$$. The reflexive reduction of a partial order is called a strict partial order. The analogous example is $$\subsetneq$$ on $$2^S$$.

A partial order is a weak order if it contains no $${\mathbf 2}+{\mathbf 1}$$, in other words, if no distinct elements both incomparable to any third distinct element are comparable to one another. An example is the has-lesser-cardinality-than-or-is relation $$\preceq$$ on the power set $$2^S$$ of a finite set $$S$$. Its reflexive reduction is a strict weak order. The analogous example is has-lesser-cardinality-than $$\prec$$ on $$2^S$$. We have $$\{3,7\} \preceq \{1,2,4\}$$ and $$\{3,7\} \preceq \{3,7\}$$, but $$\{3,7\}$$ and $$\{10,20\}$$ are incomparable. By contrast, under $$\prec$$, $$\{3,7\}$$ is incomparable to itself.

Wikipedia's weak ordering page contains this quote:

A strict partial order $$<$$ is a strict weak ordering if and only if incomparability with respect to $$<$$ is an equivalence relation.

So it seems that every strict weak order is a strict partial order, but not conversely. For example, consider the relation on the set $$\{1,2,3,4,{\between}\}$$ where $$1<2<3<4$$ and $$1<{\between}<4$$ but neither $$2$$ nor $$3$$ is comparable to $$\between$$. This is a strict partial order, but it's not a strict weak order: $$2$$ is not comparable to $$\between$$, and $$\between$$ is not comparable to $$3$$, but $$2$$ is comparable to $$3$$. (Actually $$1$$ and $$4$$ aren't even needed for a more minimal example.) If this order were changed so that $$2$$ and $$3$$ were not comparable, then it would be a strict weak order.

• Thank you very much for your answer. How could one tweak your example for it to be a strict weak order? To be honest, I don’t quite understand the meaning of the Wikipedia quote you provided. Commented Jun 2 at 22:04
• It sounds like a strict weak order is a disjoint union of total orders. Commented Jun 2 at 22:06
• No, @RobertShore. See my answer to this question. Commented Jun 2 at 22:08