### Semigroup Puzzles

#### Posted by John Baez

Suppose you have a semigroup: that is, a set with an associative product. Also suppose that

$x y x = x$

for all $x$ and all $y$.

**Puzzle 1.** Prove that

$x y z = x z$

for all $x,y$ and $z$.

**Puzzle 2.** Prove that

$x x = x$

for all $x$.

The proofs I know are not ‘deep’: they involve nothing more than simple equation-pushing. But the results were surprising to me, because they feel like you’re getting something for nothing.

Regarding Puzzle 2: of course $x y x = x$ gives $x x = x$ if you’re in a monoid, since you can take $y = 1$. But in a monoid, the law $x y x = x$ is deadly, since you can take $x = 1$ and conclude that $y = 1$ for all $y$. So these puzzles are only interesting for semigroups that *aren’t* monoids.

## Re: Semigroup Puzzles

First one: $xyz = xyzxz = xz$. Second one: $xx = xyxx = x$.

Really nothing, but what is something?