**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*,*b**S* and *c*,*d**T*, then *a***b**S* and *c*o*d**T*). Each operation is associative. In other words, (*S*,*) and (*T*,o) are semigroups. It is also given that for every *t**T* there are elements *u*,*v**T*, such that *u*o*t*=*t*o*v*=*t*. Let denote an arbitrary mapping. Define the operation on the set *S**T* as follows:

Show that the operation is associative if and only if *f* is a homomorphism, that is, *f*(*a***b*)=*f*(*a*)o*f*(*b*) for all *a*,*b**S*.

Czech competition problem

(5 points)

**Deadline expired on 15 November 2005.**