User:ArthurD8/sandbox

A third equivalent characterization is a Nowhere commutative semigroup that satisfies


 * $xy = yx$ implies $x = y$ for all $x, y ∈ S$,

This equation is sufficient to also ensure that the semigroup is a band since associativity requires that aa commutes with a so a = aa.

Then we have that in a Nowhere commutative band x commutes with xyx so they are equal which implies the first characterization.

Conversely the first characterization implies that if xy = yx then xy commutes with yx and so

x = xyx = xyyx = yxxy = yxy = y confirming equivalence with Nowhere commutative semigroup.