Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
@ZhenLin: I think rhenskyyy means that she know that right identity and left inverse cannot ensure the conditions being a group, but I think she want a concrete example to show that there exists a non-group set with multiplication satisfying associative law, right identity and left inverse. –  Yuchen Liu Sep 11 '12 at 12:16
1  
@jerrysciencemath You don't need to put your name in the post if you edit it. I think your edit is good, so leave it all, but take out "Edit(by jerrysciencemath):" –  Graphth Sep 11 '12 at 12:39
add comment

1 Answer

up vote 6 down vote accepted

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

share|improve this answer
    
This is a nifty and easy to remember example! –  rschwieb Sep 11 '12 at 12:51
    
You didn't tell which one is $e$, so I have difficulty finding the left inverse of (say) $b$. I know any one will do, but I think you need to be specific. –  Marc van Leeuwen Sep 11 '12 at 14:16
    
OP didn't posit a unique left inverse, and I don't see why I need to be specific. Isn't it enough that, as you point out, any one will do? –  Gerry Myerson Sep 11 '12 at 22:41
    
@GerryMyerson: Is there any example satisfying unique left inverse and unique right identity besides the conditions in the problem? –  Yuchen Liu Sep 12 '12 at 15:06
    
@jerry, I don't know. –  Gerry Myerson Sep 13 '12 at 2:53
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.