Definition talk:Semiring (Abstract Algebra)

Should a semiring really require commutativity? This would kind of defeat the point of Cancellable Semiring with Unity is Additive Semiring 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 is Additive Semiring (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 is Additive Semiring? 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)


 * There we are. I will start adapting the internal refs to match the new convention. --Lord_Farin 05:43, 15 June 2012 (EDT)
 * I think it is done. --Lord_Farin 06:16, 15 June 2012 (EDT)

General Overview
Had an idea.

These various descendants of the general ringoid all satisfy a subset of the Definition:Field Axioms. It occurred to be we could create a matrix with (across the top) all the field axioms: A0 to A4, M0 to M4, D, and down the side list all the various objects in an approximate order of complexity, with a cross (or tick, or whatever) in each box as relevant. This could be linked to from the "Also see" section of each of these, to save having to list every descendant of the ringoid on each page. (It would of course require that some of the axioms for some of the objects may need to be renumbered in order to remain consistent.) This is something which I've never seen anywhere and may or may not be useful, but it would be extremely pretty. --prime mover 08:30, 15 June 2012 (EDT)


 * I can see the merit of such pages; they can be of assistance when you are looking for a certain combination of axioms to be satisfied, easily finding the name you are looking for. Maybe also a tree-like structure with ringoid at the top and field at the bottom could be devised. A quick search yielded this WP; the table therein seems to adequately match your idea (but no worries - no similar structure is present on the ring page ;) ). Similarly, this figure describes a bit of my idea for a tree-like structure. --Lord_Farin 09:44, 15 June 2012 (EDT)


 * Okay, here we go (axioms indexed as on Definition:Field Axioms):


 * Definition:Ringoid: $D$
 * Definition:Semiring (Abstract Algebra): $A0,A1,M0,M1,D$
 * Definition:Additive Semiring: $A0,A1,A2,M0,M1,D$
 * Definition:Rig: $A0,A1,A2,A3,M0,M1,D$ (and that the $0$ is a zero for $\circ$)
 * Definition:Ring (Abstract Algebra): $A0,A1,A2,A3,A4,M0,M1,D$
 * Definition:Commutative Ring: $A0,A1,A2,A3,A4,M0,M1,M2,D$
 * Definition:Ring with Unity: $A0,A1,A2,A3,A4,M0,M1,M3,D$ (note that $M3$ imposes things not to be the null ring)
 * Definition:Commutative and Unitary Ring: $A0,A1,A2,A3,A4,M0,M1,M2,M3,D$ (note that $M3$ imposes things not to be the null ring)
 * Definition:Division Ring: $A0,A1,A2,A3,A4,M0,M1,M3,M4,D$
 * Definition:Field (Abstract Algebra): $A0,A1,A2,A3,A4,M0,M1,M2,M3,M4,D$


 * More can be created by putting 'commutative' and 'with unity' on semiring, add. semiring and rig. --Lord_Farin 10:14, 15 June 2012 (EDT)

Associativity ultimately hit us here
So we need to fix that. --Dfeuer (talk) 07:45, 9 January 2013 (UTC)
 * Definition:Non-Associative Semiring works. Note how, in general, non-associative means not necessarily associative. A good read is red herring in nLab. --barto (talk) (contribs) 14:06, 16 January 2018 (EST)
 * Ah I see, you probably want to allow non-associativity for only one operation. --barto (talk) (contribs) 14:19, 16 January 2018 (EST)

Commutativity of addition
Where does this definition come from? The 3 inline references at semiring, and those sources that do not require the existence of 0 and 1 (as we do, and indeed there are good arguments for that) all do require that $+$ (here $*$) is commutative. I understand we want a generalization of this where + need not be: Definition:Associative Ringoid works perfectly. --barto (talk) (contribs) 15:10, 16 January 2018 (EST)