A. 382. S and T are disjoint sets, * is a binary operation on the elements of S and o is a binary operation on the elements of T. (That is, if a,bS and c,dT, then a*bS and codT). Each operation is associative. In other words, (S,*) and (T,o) are semigroups. It is also given that for every tT there are elements u,vT, such that uot=tov=t. Let denote an arbitrary mapping. Define the operation on the set ST as follows:
Show that the operation is associative if and only if f is a homomorphism, that is, f(a*b)=f(a)of(b) for all a,bS.
Czech competition problem
Deadline expired on 15 November 2005.