11–20 of 59 posts
Most algebraic structures are best understood by which axioms it satisfies. For example basically every subset of axioms of an abelian group is useful enough to have a name. Wiki has a really nice table: Semigroupoid Small Category Groupoid Magma Quasigroup Unital Magma Loop Semigroup Inverse Semigroup Monoid Commutative monoid Group Abelian group https://en.wikipedia.org/wiki/Abelian_group
Most algebraic structures are best understood by which axioms it satisfies. For example basically every subset of axioms of an abelian group is useful enough to have a name. Wiki has a really nice table: Semigroupoid Small Category Groupoid Magma Quasigroup Unital Magma Loop Semigroup Inverse Semigroup Monoid Commutative monoid Group Abelian group https://en.wikipedia.org/wiki/Abelian_group
I'd like to seen an extension of this table with the negation of these axioms
For example, I can see the use of commutativity (ab = ba) and anticommutativity (ab = -ba), but I'm not sure what I'd do with the negation of commutativity (ab ≠ ba).
Also, the label "algebra" is vague here, and refers to an "algebra over a field", but sometimes it refers to an "algebra over a ring".
Earlier quoted context omitted.
I'd like to seen an extension of this table with the negation of these axioms
What would you do with that? For example, I can see the use of commutativity (ab = ba) and anticommutativity (ab = -ba), but I'm not sure what I'd do with the negation of commutativity (ab ≠ ba).
Earlier quoted context omitted.
What would you do with that? For example, I can see the use of commutativity (ab = ba) and anticommutativity (ab = -ba), but I'm not sure what I'd do with the negation of commutativity (ab ≠ ba).
Nope: the negation is "there is a couple a,b such that ab!=ba", which means just "strictly not commutative group": I do not think there is a relevant theory to be done about them (otherwise, I guess it would have been done).
Out of clarity this is an "algebra over a field" vs a more general concept of an algebra over a ring. More generally an algebra A, over a ring R, an R-algebra , is a ring A equipped with a map Hom(A,Z(R)). Algebra over a field is a special case. Here's a "fun" object for you to consider: https://en.wikipedia.org/wiki/Field_with_one_element
I don't think that's the usual definition of an algebra. For example, it would mean that there is no difference between an algebra over a non-commutative ring and over its centre, which seems weird; and it clashes with the usual habit to regard every non-0 commutative ring as a non-trivial ℤ-module, whereas, for example, the only homomorphism ℤ/2ℤ → ℤ is the trivial one.
I would expect rather the datum of an R-algebra structure on a ring A to be a ring homomorphism R → End_{gp}(A). EDIT: Now that I think of it, maybe got your A and R mixed up and meant the more restrictive definition, whereby the ring homomorphism I mention is supposed to factor through R → Z(A) → End_{gp}(A)? I'd call this more restricted notion, at least over a unital ring R, a unital algebra A (but often people want implicitly to assume unital-ness).