It seems like there would be utility in a search engine or database through which the user inputs the operation table of a magma (I think that's the right level of algebraic structural generality) to output a compilation of all known phenomena modeled by that operation table. For example, entering the Caley Table of the Klein-Four Group would return several results from music theory and geometry, component-wise addition mod $2$ of ordered pairs, etc. This could serve as a sort of search capability to find isomorphisms, which might be an aid to analogical thinking.
Does such a tool exist? Would it be useful? Are there few enough of such isomorphisms that algebraists just learn them all, or are there enough that entering most reasonably sized operation tables would return matches?
