As for your 2. Given a semigroup S you can embed it into a semigroup S^1 with identity in an obvious way and then treating S^1 as the basis of a vector space the mapping f defined by f(s)(x) = sx represents the semigroup as a semigroup of matrices. Much as any finite group can be represented by permutation matrices. On Tue, Aug 19, 2014 at 9:02 PM, Henry Baker <hbaker1@pipeline.com> wrote:
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.)
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