The tag boolean-rings is rigorously equivalent to the more used boolean-algebras. I suggest to create a synonym.
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I want to start out by giving two examples:
1) Graham's problem is to decide whether a given edge-coloring (with two colors) of the complete graph on vertices $\lbrace-1,+1\rbrace^n$ contains a planar $K_4$ colored with just one color. Graham's result is that such a $K_4$ exists provided $n$ is ...
The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below.
Let $k$ be a field. The following are equivalent for a $k$-algebra $A$:
$A$ is pseudocompact,
$A$ is profinite,
...
