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In the abstract algebra class, we have proved the fact that right identity and right inverse imply a group, while right identity and left inverse do not.

My question: Are there any good examples of sets (with operations on) with right identity and left inverse, not being a group?

To be specific, suppose $(X,\cdot)$ is a set with a binary operation satisfies the following conditions:

(i) $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ for any $a,b,c\in X$;

(ii) There exists $e\in X$ such that for every $a\in X$, $a\cdot e=a$;

(iii) For any $a\in X$, there exists $b\in X$ such that $b\cdot a=e$.

I want an example of $(X,\cdot)$ which is not a group.

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    See also: [Is a semigroup G with left identity and right inverses a group?](http://math.stackexchange.com/questions/433546/is-a-semigroup-g-with-left-identity-and-right-inverses-a-group)2015-03-09

1 Answers 1

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$\matrix{a&a&a\cr b&b&b\cr c&c&c\cr}$ That is, $xy=x$ for all $x,y$.

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    @jerry, I don't know.2012-09-13