this following multiplication table arises in a deep area of theoretical computer science proofs eg complexity theory & am looking at it more in isolation or outside of TCS. looked for this multiplication table on wikipedia as some mathematically recognized or used object but couldnt find it. does it show up elsewhere?
a b c a a c a b c b b c a b c
(prior comments point out it is not a group or semigroup but seem to confirm its at least a quasigroup). are there applications of it? what can it be used for? am esp interested in some way of implementing a boolean logic with it. (however if there was some way to link it to group theory that would be very interesting too). "c" may or may not be the empty element. thx
note— may edit this question later to describe at least one very specific case where it appears in theoretical computer science, but for now am leaving it open so as not to bias possible answers. question  on cs.se points to one possible application ie trinary valued circuit logic.