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)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 [1] on cs.se points to one possible application ie trinary valued circuit logic.

[1] building functionally complete boolean circuits out of trinary logic

I

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 as a group on wikipedia pages as some mathematically recognized or used object but couldnt find it(wikipedia doesnt seem to list the groups in terms of tables-- . does anyone have it show up elsewhere?

   a ref that does, pref online?). I guess this b ca  a c ab  c b bc  a b c
(prior comments point out it is not a group or semigroup). which one? where does it show up ie 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
   a b ca  a c ab  c b bc  a b c
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 [1] on cs.se points to one possible application ie trinary valued circuit logic.
[1] building functionally complete boolean circuits out of trinary logic
        

I looked for this multiplication table as a group on wikipedia pages but couldnt find it (wikipedia doesnt seem to list the groups in terms of tables-- does anyone have a ref that does, pref online?). I guess this is a group. which one? where does it show up ie applications of it? what can it be used for? "c" may or may not be the empty element. thx

   a b c

a  a c a
b  c b b
c  a b c