Simple Characterizations of Mathematical Structures By no means trivial, a simple characterization of a mathematical structure is a simply-stated one-liner in the following sense:
Some general structure is (surprisingly and substantially) more structured if and only if the former satisfies some (surprisingly and superficially weak) extra assumption.
For example, here are four simple characterizations in algebra:


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*A quasigroup is a group if and only if it is associative.

*A ring is an integral domain if and only if its spectrum is reduced and irreducible.

*A ring is a field if and only its ideals are $(0)$ and itself.

*A domain is a finite field if and only if it is finite.


I'm convinced that there are many beautiful simple characterizations in virtually all areas of mathematics, and I'm quite puzzled why they aren't utilized more frequently. What are some simple characterizations that you've learned in your mathematical studies?


The width (supremum of sizes of antichains) of the cardinals is $1$ if and only if it is $<k$ for some finite $k$.

Namely, every two cardinals are comparable, if and only if there is a fixed, finite $k$ such that given $k$ distinct cardinals there are two comparable.
One immediate corollary is that if the axiom of choice fails, then there are antichains of every finite length.
