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.
--
Waldek Hebisch
[email protected]
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