# Some properties of decompositions of a commutative group

## Full text

(1)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COM MENT ATIONES MATHEMATICAE XXVIII (1989) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Serin I: PRACE MATEMATYCZNE XXVIII (1989)

Ko m in e k

## Some properties of decompositions of a commutative group

Abstract. Considering decompositions of a commutative group, we study some relations between the sets: A + A , А л - А ' , А 'Л - А , A — A , A — A ' and A — A , where A ' denotes the complement of A . Some examples which illustrate our results are also presented.

g

g

Le m m a.

x g

x g

Th e o r e m 1.

### уфВх + B x we have y — B x a A x. It follows from symmetry of the sets A x and

(2)

250 Z. K o m in ek

Th e o r e m 2.

### such that A —\ x = ? x — A which ends the proof.

(3)

Properties o f decompositions o f a commutative group 2 5 1

Ф

e

n (J

neZ

e

Theorem 3.

Ф

Ф 0

Ф

Ф

Ф

Ф

Ф

Theorem 4.

Ф

3,

### From the Theorem 4 we get immediately the following

— Commentationes Math. 28.2

(4)

252 Z. K o m in ek

Co r o l l a r y 1.

Ф

Ф

+

Ф

Th e o r e m 5.

Ф

y

^

a y —

Ф

Co r o l l a r y

### 2. Let V be a linear vector space. For every subset A of V either A or A' contains a basis of the space V.

References

 F. B a g e m ih l, S o m e s e ts o f s u m s a n d d iffe r e n c e s , Michigan Math. J. 4 (1957), 289-290.

 Z. K o m in e k , M e a s u r e , c a te g o r y , a n d th e s u m o f s e ts , Amer. Math. Monthly 90 (October (1983), 561-562.

 M. K u c z m a , A n I n t r o d u c t i o n to th e T h e o r y o f F u n c tio n a l E q u a tio n s a n d I n e q u a litie s ,

P.W.N. Uniwersytet Slqski, Warszawa-Krakow-Katowice 1985.

 H. I. M ille r , S o m e d e c o m p o s itio n th r o r e m s f o r th e r e a l lin e, Radovi Matematicki 1 (1985), 31-37.

 S. P ic c a r d , S u r les e n s e m b le s p a r fa its , Mem. Univ. Neuchâtel 16 (1942).

 H. S t e in h a u s , S u r les d is ta n c e s d e s p o in ts d e s e n s e m b le s d e m e s u r e p o s itiv e , Fund. Math. 1 (1920), 99-104.

Updating...

## References

Related subjects :