Another hint: AXIOM/FriCAS already has some of these new categories; in the non-associativity world we once provided
Monad ( 1) for *) MonadWithUnit ( 2) for *) However, it was never unified with the associative world, which really should be done to be consistent. Am 17.08.15 um 02:14 schrieb Waldek Hebisch: > Axiom core algebraic categories are taken from classis mostly > commutative algebra. I am thinking about extending them > to allow weaker assumpion. We have two basic operations, > '+' and '*'. Most domains assume that '+' is an operation > from an abelian group, and that '*' is from a monoid and > plays nicely with '+'. I consider the following categories: > > 1) Just operation '+' (named AdditiveOperation or possible > AdditiveMagma) > 2) '+' and 0 (neutral element) I am tempted to call it > AdditiveOperation0 > 3) Associativity: AdditiveSemigroup > 4) Associativity and 0: AdditiveMonoid > 5) Associativity and Inverse: AdditiveGroup > 6) Associativity and cancellation property (left/right) > > 3, 4, 5 and commutativity give existing categories: AbelianSemigroup, > AbelianMonoid, AbelianGroup > > Then similar categories for '*', starting from say > MultiplicativeOperation). > > Then we get categories combining the two operations: > - just operation > - 0 which is identity for '+' and satisfies 0*x = x*0 = 0 > (supposedly this is called Shell) > - near-semiring: two associative operations with right > distributive law > - near-ring: group with respect to addition > > I do not expect to be able to perform a lot of computations with > domains of such general categories. However, some exaples are > easy to create and I think it would be nice to have them > incorporated into our category hierarchy. > -- > Mit freundlichen Grüßen > > Johannes Grabmeier > > Prof. Dr. Johannes Grabmeier > Köckstraße 1, D-94469 Deggendorf > Tel. +49-(0)-991-2979584, Tel. +49-(0)-151-681-70756 > Tel. +49-(0)-991-3615-141 (d), Fax: +49-(0)-3224-192688 -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/fricas-devel. For more options, visit https://groups.google.com/d/optout.
