## what is this small 3 element quasigroup & what is it used for [closed]

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

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Here's a hint: you can't have two 'a' symbols in one row or column. Also, this is not appropriate for this site. – Steve D Jan 22 2012 at 18:11
This is not even associative (aa)b=c and a(ab)= a. – Benjamin Steinberg Jan 22 2012 at 18:37
Sorry, but your question isn't suitable for this site. If you look at the FAQ, you'll find some other sites that might be more suitable. Good luck. – Tom Leinster Jan 22 2012 at 18:39
It is not a semigroup. After you have done more research, I invite you to come back and edit this question, telling us what you have found and what (from a mathematically mature viewpoint) you wish to know. Read the FAQ and try for a more focussed and motivated question. Gerhard "Ask Me About System Design" Paseman, 2012.01.22 – Gerhard Paseman Jan 22 2012 at 21:36
I will flag this for reopening. I found Berman's catalog (regarding free spectra from 3 element algebras) and some references to the algebra above. I am willing to edit the question to make it more professional, but I find the content appropriate for MathOverflow, and not off topic. I think the question deserves a proper answer and I am willing to furnish one. Gerhard "Ask Me About General Algebra" Paseman, 2012.07.21 – Gerhard Paseman Jul 22 at 0:08