Let our signature be that of a single binary operation symbol $*$. Consider the set of equations in that signature. I define a preorder on that set by saying $E \geq E'$ if and only if the equation $E$ implies the equation $E'$. By quotienting out by the equivalence relation of the preorder, we get a partial order. The top element of the partial order is the equivalence class of $x=y$, and the bottom element is the equivalence class of $x=x$. I want to know, what are the atoms and coatoms of this partial order? I believe that it has no atoms, and only three coatoms: $x+y=x$, $x+y=y$, and $x+y=z+w$. Is this true? If not, what are the atoms and coatoms of this partial order, if there are any?
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Tentative suggestion for proving no atoms: show (not sure this is true) that "$t=s$" is always strictly stronger than "$t+t=s+s$" for any terms $t,s$. – Noah Schweber Aug 04 '23 at 16:04