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< 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 under some assumptions it is possible to decompose $$\tilde{S}$$ into a direct sum of two subgroups: symmetric and asymmetric one.
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> 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.
Hubert PrzybycieńOn a direct sum decomposition of a group 
Given a commutative semigroup with the neutral element and the cancellation law 202424.png)
, denote by the set 
with the equivalence relation 
such that R(c2cd)202424.png)
if and only if 
. We will show that 202424.png)
is a group and under some assumptions it is possible to decompose into a direct sum of two subgroups: symmetric and asymmetric one.