# Problem A. 382. (October 2005)

**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 pont)

**Deadline expired on November 15, 2005.**

**Solution.** (a) If *f* is a homomorphism then is trivially associative.

(b) Suppose that is associative and let *a*,*b* be two arbitrary elements of *S*. By the conditions there exist *u*,*v**T* such that *u*o*f*(*a*)=*f*(*a*) and *f*(*a***b*)o*v*=*f*(*a***b*). Then

### Statistics:

15 students sent a solution. 5 points: Erdélyi Márton, Estélyi István, Fischer Richárd, Gyenizse Gergő, Hujter Bálint, Jankó Zsuzsanna, Kónya 495 Gábor, Korándi Dániel, Nagy 224 Csaba, Paulin Roland. 4 points: Kisfaludi-Bak Sándor, Tomon István. 2 points: 2 students. 1 point: 1 student.

Problems in Mathematics of KöMaL, October 2005