Implication semilattice of 990 quasigroup equational laws

In his quest to disprove a claim by Peirce that all lattices are distributive, Ernst Schr"oder considered 135 years ago a list of 990 equational laws on quasigroups, analogous to associativity, such as $(x\mathrm{/}\mathrm{/}y)\ast z=(y\mathrm{/}\mathrm{/}x)\mathrm{\backslash }\mathrm{\backslash }z$. A quasigroup is a non-associative analogue of groups, specifically a set equipped with multiplication and right/left conjugate-division operations that are compatible. Each equation of interest identifies two three-variable expressions built from these operations. I determine all $114$ equivalence classes of their conjunctions, and all implications between them. This includes as a small corner the five-element non-distributive lattice identified by Schr"oder.