Putnam 1978 A4

A bypass operation on a set $S$ is a mapping $B: S\times S \rightarrow S$ with the property $B(B(w, x), B(y,z)) = B(w,z)$ for all $w, x, y, z \in S$ . (a) Prove that $B(a,b)=c$ implies $B(c,c)=c$ when $B$ is a bypass. (b) Prove that $B(a,b)=c$ implies $B(a,x)=B(c,x)$ for all $x\in S$ when $B$ is a bypass. (c) Construct a bypass operation $B$ on a finite set S with the following three properties [list=i] [] $B(x,x)=x$ for all $x\in S$ . [] There exist $d$ and $e$ in $S$ with $B(d,e)=d \ne e.$ [*] There exist $f$ and $g$ in $S$ with $B(f,g)\ne f.$ [/list]