Name of algebraic structure (group-like) What is the name of an algebraic structure (group, quasi group, monoid) defined like the following:

*

*It is a set of three elements: $a,b,c$;

*Has an operation such that when applied to two elements of the group it returns the other element of the group (example: $ab=c$; $cb=a$);

*Has an "identity" structure such as $aa=a$, $bb=b$, $cc=c$;

*It is commutative: $ab=ba$; also $a(ba)=ac=b$.

This is no homework. I'm just trying to define a structure like this in order to learn, but as I'm reading about group-theory and semigroups it doesn't seem to really define this (for example, the "identity" here is ambiguous - so how should I call it?)
Thank you!
 A: This is a commutative quasigroup. However, personally I don't find this sort of naming to be particularly illuminating. To my mind the simplest way to think about this operation $\star$ is actually as a ternary relation $\{ (x, y, z) : x \star y = z \}$, where it has a very simple interpretation: if we set $a = 0, b = 1, c = 2$ then the ternary relation is precisely the relation $x + y + z \equiv 0 \bmod 3$. In other words, the operation is
$$x \star y = - x - y \bmod 3.$$
Writing it this way makes the quasigroup property clear and also exhibits an otherwise somewhat hidden $S_3$ symmetry: this condition is invariant under any permutation of $x, y, z$. It's also equivalent to the condition that $x, y, z$ are either all the same or all different, which you may recognize from the game Set.
A: The structure you are describing is called a Steiner quasigroup. The phrase Steiner quasigroup is sometimes abbreviated 'squag', so some folks call these objects squags.
In general, a (finite) Steiner triple system is an $(n,3,1)$-block design or a Steiner system of the form $S(2,3,n)$. This terminology means that a Steiner triple system is a finite geometry $\mathfrak S = (P,L)\;(=(\textrm{points},\textrm{lines}))$ which has $|P|=n$ points, $3$ points per line, and any $2$ points of the geometry determine a unique line. The 'algebraization' of such a geometry is called a 'Steiner quasigroup'. This algebraization $\mathfrak Q=(P,*)$ has the set $P$ of points as its underlying set equipped with a multiplication that encodes the lines. If $x\in P$, then one defines $x*x=x$, while if $x, y\in P$, $x\neq y$, then one defines $x*y=z$ iff $z$ is the third point on the unique line through $x$ and $y$. Your structure arises from a Steiner-type geometry $\mathfrak S = (P,L)$ with three points lying on one line ($|P|=3, |L|=1$). The next smallest Steiner quasigroup arises from a Steiner-type geometry with $|P|=7, |L|=7$, called the Fano plane.


