I am trying to model a problem. I have some elements $S=\{ a,b,c \}$ and a commutative operation $\cdot:S^2 \to S^2$ on them. The operation is defined as follows:
$ a^2 = (a,a)\\ b^2 = (b,b)\\ c^2 = (c,c)\\ ab = ba = (c,c)\\ ac = ca = (b,b)\\ bc = cb = (a,a) $
Is there any sort of nice algebraic structure hidden in this operation? A group structure or something like that would be nice. This isn't homework or anything. I am just trying to model a problem I found.