Definition talk:Idempotence

Probably no point in defining an "idempotent algebraic structure" as this has already been implicitly covered in "idempotent operation". What we have got is Definition:Idempotent Semigroup which is significantly more useful because it combines the concepts of closure, assoc and idemp in one definition, and this grouping is significant enough, and crops up frequently enough, to be treated as a type of object in its own right. It also makes definitions of objects which have these three properties somewhat shorter and (IMO) easier therefore to comprehend. (It is assumed that anyone making an investigation into e.g. semilattices will be sufficiently familiar with abstract algebra to be able to take semigroups and idempotence in their stride - and those who don't have a link to some easy pages to familiarise themselves with.) --prime mover 05:37, 28 January 2012 (EST)


 * And another philosophical point: is there any direct need to introduce a "set" as such? The definitions "mapping" and "binary operation" should be enough, as the fact that they themselves are defined with relation to "sets" already means that it is already implicit that $S$ is a set. However, I have introduced the concept at the top of the page. --prime mover 05:42, 28 January 2012 (EST)


 * I feel there is a need to introduce a set; if only for the fact that the definitions of mapping and binary operation may be generalised in the future to cover more than mere sets. Also, readers should not be bothered with even the remote question as to what $S$ is; for all they could know, there could be an omission stating that $S$ is some specific type of set. Clarity above everything.


 * Okay, I concur.


 * As to your first point, I will look through the various definitions of algebraic structures and link them together where I can. --Lord_Farin 06:16, 28 January 2012 (EST)


 * Lastly, it would have been epic if you would have closed that (still open) parenthesis in your second comment :) --Lord_Farin 06:37, 28 January 2012 (EST)


 * Parenthesis duly closed.


 * There are several instances of objects in abstract algebra whose meaning is obvious but which occur often enough for it to make sense to be defined as a single entity: "Commutative Semigroup", "Commutative and Unitary Ring", and so on. It makes it easier to provide a nexus for which results pertaining to such entities can be coordinated. It does not matter too much to have commutative semigroup in a page, so it's probably not worth trawling through all pages for such, but when I find them I generally go through and change them. Or, when I'm too stupid to do anything more complicated, I will take a morning to go search for them all methodically and change them one by one.