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# semi group

A semi group stasfies cancelation laws is a group how you prove that?

Note by Sai Venkata Raju Nanduri
3 years, 10 months ago

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The result is not true if your semigroup does not have an identity. Consider the semigroup $$\{n \in \mathbb{N} : n \ge 2\}$$ under ordinary multiplication, which satisfies the cancellation laws but does not have an identity.

The result is not true for infinite unital semigroups. Consider the semigroup $$\mathbb{N}$$ under ordinary multiplication.

The result is true for finite unital semigroups. The cancellation laws guarantee that (for example) left multiplication by $$x \in G$$ is injective. Why does that guarantee the existence of a right inverse for $$x$$? · 3 years, 10 months ago