What's probably worst is that this is a repost, as I was stuck on 3. before. The question is this:
Throughout this question, we shall denote by $\neg$ the relation on a semigroup $(A, * )$ defined so that elements x and y of the semigroup satisfy the relation $x \neg y$ if and only if there exists some element $s$ of the semigroup $A$ such that $s * x = y * s$
Prove the relation $\neg$ is a transitive relation on $A$ for all semigroups $(A, *)$.
Prove that the relation $\neg$ is a reflexive relation on $A$ for all semigroups $(A, *)$.
Prove that if the semigroup $(A, *)$ is a group, then the relation $\neg$ on $A$ is an equivalence relation.
Prove that if $(A, *)$ is a group, and if the relation $\neg$ is a partial order on $A$, then the binary operation $*$ of the group $A$ is commutative.
You guys helped me through 3, so now I feel like a total jerk asking for more help, but there's not much more I can do.
If anyone could give me a hint that'd be great. Basically, never proven that anything is commutative before, so bear with me. The only real equation I have to work with is $s * x = y * s$, and it's inverse $s^-1 * y = x * s^-1 $, which, one of my numerous issues is that equation exists only if $y=x$ as the thing is a partial order, which means that if I go and show that $s^-1 * s = s * s^-1$, that's meaningless, as of course something times the inverse itself is equal both ways (as the thing is a group), and if I show $x*y = y*x$ this is equally as meaningless, as I'm not showing it for any $x$ or $y$, but for $x$ and $y$ which are equal, as per the whole partial order deal.
Basically here for guidance, any help is tremendously appreciated.