I asked this question on math stack exchange, here: https://math.stackexchange.com/questions/4985313/is-the-partial-order-of-all-equations-in-the-signature-of-magmas-a-lattice, but there were no comments or answers. So, I am asking it here on mathoverflow. Consider the signature of a single binary operation $*$, and consider the set $Eq$ of all equations in that signature. I can define a preorder $\geq$ on $Eq$ by stipulating that $E \geq E'$ precisely when $E$ implies $E'$. I can now consider the partial order obtained by quotienting out the preorder by its standard equivalence relation. My question is, is this partial order a lattice? If not, is it at least a join semilattice or a meet semilattice? If not even that, is it still at least the case that for every pair of elements, either the pair has a least upper bound or a greatest lower bound? I would be very interested to be given a pair of equations such that they have neither a least upper bound nor a greatest lower bound.

$\newcommand\Eq{\mathrm{Eq}}$I asked this question on math stack exchange, here, but there were no comments or answers. So, I am asking it here on mathoverflow. Consider the signature of a single binary operation $*$, and consider the set $\Eq$ of all equations in that signature. I can define a preorder $\geq$ on $\Eq$ by stipulating that $E \geq E'$ precisely when $E$ implies $E'$. I can now consider the partial order obtained by quotienting out the preorder by its standard equivalence relation. My question is, is this partial order a lattice? If not, is it at least a join semilattice or a meet semilattice? If not even that, is it still at least the case that for every pair of elements, either the pair has a least upper bound or a greatest lower bound? I would be very interested to be given a pair of equations such that they have neither a least upper bound nor a greatest lower bound.