I'm aware that posting exam questions is probably frowned upon, but this isn't homework, I think I'm genuinely misunderstanding some part of the algebra. The question is this:
Throughout this question, we shall denote by ¬$\neg$ the relation on a semigroup (A, * )$(A, * )$ defined so that elements x and y of the semigroup satisfy the relation x ¬ x$x \neg y$ if and only if there exists some element s$s$ of the semigroup A$A$ such that s * x = y * s$s * x = y * s$
Prove the relation ¬$\neg$ is a transitive relation on A$A$ for all semigroups (A, *)$(A, *)$.
Prove that the relation ¬$\neg$ is a reflexive relation on A$A$ for all semigroups (A, *)$(A, *)$.
Prove that if the semigroup (A, *)$(A, *)$ is a group, then the relation ¬$\neg$ on A$A$ is an equivalence relation.
Prove that if (A, *)$(A, *)$ is a group, and if the relation ¬$\neg$ is a partial order on A$A$, then the binary operation *$*$ of the group A$A$ is commutative.
I've proven it's transitive and reflexive without too much hassle, but can't get past 3.
My approach was such that I'd start with some s$s$ such that s * x = y * s$s * x = y * s$, and then play around with this equation until I'd get some new term, in terms of x^-1$x^{-1}$ and y^-1$y^{-1}$ such that (some term) * y = x * (some term), but I keep running around in circles. Then I thought s * x = y * s$s * x = y * s$ doesn't capture enough information about the problem in order to be molded into the solution, but I can't see what else I can add into it.
Thanks in advance, sorry about posting an exam question, but I figure this should expose something I'm missing in all similar problems.
