[math-fun] Stupid associativity question

I'm interested in binary operations on a finite set S that are associative, period; no other constraints. 1. Are there "fast" methods for checking associativity given the operation table? 2. Can all such associative operations be emulated "efficiently" using an isomorphism in which the operation is matrix multiplication (not just over standard rings, but extended like Knuth & the APL language do) ? (RCS knows why I'm asking this question.)