Talk:Identity Element is Idempotent

Merge request
I did try very hard to find this theorem before creating the page, but I missed that. I'd personally be inclined to turn that page into a master/transclusion one, and one way or another make sure we have Left Identity Element is Idempotent and Right Identity Element is Idempotent for anyone else looking for specific results. Despite the extreme triviality of it all, they are three different results requiring three different proofs. --Dfeuer (talk) 22:59, 13 March 2013 (UTC)


 * "they are three different results requiring three different proofs." But are they required at all? There is the seemingly trivial and the truly trivial. If you would like to post these then go for it. If you can put them to use though that's even better because it justifies the clutter. --Jshflynn (talk) 23:39, 13 March 2013 (UTC)


 * Used in two places. --Dfeuer (talk) 23:59, 13 March 2013 (UTC)


 * On Talk:Continuous Involution is Homeomorphism, Lord_Farin wrote: "On the trivial we build the intricate". That result is only slightly less trivial than this—in both cases, the main difficulty is in learning the required terminology. --Dfeuer (talk) 00:10, 14 March 2013 (UTC)


 * The page Identities are Idempotent is used in more than two places so I think it would be useful after all. Usually I go the other way by starting from the middle and figuring out what trivial proofs lower down I need. Its perfectly imaginable to me though that someone might build things from the ground up. To each their own. --Jshflynn (talk) 00:25, 14 March 2013 (UTC)


 * I went from the middle down here as well. I wanted to prove Idempotent Elements form Submonoid of Commutative Monoid.

rule of thumb
The usual rule of thumb I have been using for some time is: if there exists an explicit statement of a particular result (however trivial) then I make an effort to specify it as a separate page.

While the idempotence of the identity element (whether stated explicitly and proved rigorusly or not) is of fundamental importance in group theory (it allows one to state that the identity is the only idempotent element in a group), it is rarely explicitly stated as a result. Similarly for the left and right identities. The only work I have found it in is Dean, which provides the group axioms in the right-identity and right-inverses form (see Axiom:Right Group Axioms).

Consequently, as only Dean makes explicit mention of this idempotence, and then only for the right-identity form of this statement, I did not bother to separate out the statements, and as they all followed directly from the definition of the objects in question, no specific effort was made to do a blow-by-blow proof.

The true reason for this page existing is to provide an explanation for someone unfamiliar with the terminology what "the identity is idempotent" actually means. Yes, it follows directly from the definition of both "identity" and "idempotent" and as such is trivial - but the page still needs to exist so as to explain what is meant.

The same applies to many of these "trivial" proofs - they are specifically there so as to clarify certain fundamentally simple concepts which use words which may be unfamiliar to the reader. Also bear in mind that many use this site to whom English is not their native language. --prime mover (talk) 07:16, 14 March 2013 (UTC)

update
I have done what DFeuer suggested in the end. It was nagging at my OCD. --prime mover (talk) 11:07, 2 May 2015 (UTC)