semi group

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

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 $x$ ?

Mark Hennings - 7 years, 8 months ago

But same problem can we prove in this way by using this statemet that is in a group G a,b,x,y belongs to G ax=b and ya=b have unique solution

sai venkata raju nanduri - 7 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$