Optimization and Control Theory Department: 2008-04-01 Seminar

Hubert Przybycień

On a direct sum decomposition of a group $\tilde{S}$

Given a commutative semigroup with the neutral element and the cancellation law $(S,+)$ , denote by $\tilde{S}$ the set $S^{2}$ with the equivalence relation $R$ such that $(a,b)R(c,d)$ if and only if $a+d=b+c$ . We will show that $(\tilde{S},+)$ is a group and under some assumptions it is possible to decompose $\tilde{S}$ into a direct sum of two subgroups: symmetric and asymmetric one.