Definition talk:Semiring (Abstract Algebra)

Should a semiring really require commutativity? This would kind of defeat the point of Cancellable Semiring with Unity has Commutative Distributand that made us rewrite this stuff in the first place. I don't have a good source book at home to check with offhand - I can try to remember to see what Dummit and Foote do tomorrow, or other people can jump in with some authorative source. Note that any change here would have to be reflected in Definition:Rig, and leaving it as is requires minor reworking to Natural Numbers form Commutative Semiring and Cancellable Semiring with Unity has Commutative Distributand (i.e. renaming the second one to something that will probably be unwieldy). --Alec (talk) 21:24, 5 June 2012 (EDT)
 * There are so many definitions and variants for all these objects, I suppose it depends on what you want to do with them when you've defined them. Since (at the moment on this site) we haven't really dug out any use for these things yet (apart from as building blocks for the more compound mainstream objects from rings on up) this is probably a question that we can leave as is till either we find some sources which discuss the matters in depth, or explore it ourselves for our Ph.D. theses. :-) It may have been touched upon in via category theory, but I haven't got round to studying that properly yet. --prime mover 01:17, 6 June 2012 (EDT)
 * ...hang on, I see where you're coming from now: if a semiring is defined with additive commutativity, then do we really need Cancellable Semiring with Unity has Commutative Distributand? Good question, but whatever the definition of semiring I suggest the latter result does have use. We just need to establish the context. --prime mover 02:17, 6 June 2012 (EDT)
 * Maybe we could call a 'semiring with commutative distributand' an 'additive semiring'? --Lord_Farin 03:15, 6 June 2012 (EDT)
 * Sounds good to me. --Alec (talk) 21:06, 13 June 2012 (EDT)

PM agreed? This brings with it some work as the definition of semiring will be broadened and consequently refs need to be adapted to refer to 'additive semiring' when this is required; therefore I would like an approval before getting started. --Lord_Farin 13:35, 14 June 2012 (EDT)
 * Hmm ... I think so. The concept may not have been defined before in this form anywhere else, in which case it seems the right thing to do to innovate (as long as we indicate that the usage is newly-coined here, as far as we can tell). We're not Wikipedia, thank goodness, so we are not dependent upon citations from tertiary sources. Okay, let's go for it. --prime mover 15:10, 14 June 2012 (EDT)