Can you give me few examples of binary operation that it is not associative, not commutative but has an identity element?
Non-associative, non-commutative binary operation with a identity
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$\begingroup$
abstract-algebra
examples-counterexamples
binary-operations
associativity
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1Does it have to be both a left- and right- identity? – 2011-03-14
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0Vafa, maybe you should hand in your assignment before you post it here. – 2011-03-15
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2@Gerry Myerson: If I have posted one question from the assignment here, it does not mean that I am going to copy verbatim solutions from here into my solution of the assignment. In fact, as you will see in my solution, I have given a different example. I posted the question here because I was interested to see other binary operations that I could not think of. Next if I ever wanted to copy solutions from here into my solution manual, I would not be that stupid to ask the question with my real name and identity. (if this is ever what you meant.) – 2011-03-15
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0@Mitch: Yes, so if e is your identity and a is an element of your binary operation, then you should get $ae=ea=a$. – 2011-03-15
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0@Vafa Khalighi: If there is an identity in a associative, commutative groupoid, then it must be both left and right, but a left identity in an nonassociative noncommutative groupoid is not guaranteed to be a right identity. Thats why I asked (also, in all the excellent examples given by Jacob Schlater, the identity is only one sided). – 2011-03-15
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0@Mitch: You might be right. I do not know much about group theory to comment on this. This is Gerry Myerson's question for us. And I believe by identity he means that it is both right and left identity. – 2011-03-15
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2Vafa, I have read the comment you wrote to me. I can't answer there, so I'll answer here. **FIRST**, hand in your assignment; **THEN**, post questions from it online. Better: wait until all your classmates have handed in their assignments before you post assignment questions to a public website. I'm sure if you think about it, you'll see that this is the only sensible way to go about it. If your curiosity must be satisfied and you can't wait until everyone else has handed the assignment in, then come ask me f – 2011-03-15
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0What do you want me to do now? delete this? close this? – 2011-03-15
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0No, just be sure to credit anything you've seen here on this and other questions when you get your assignment in, and in the future, please do as I've suggested. – 2011-03-15
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0No problem. I will not post any more question of any assignment before its due date. – 2011-03-15