(Pedantic notes: A group doesn’t have two operations, you’re thinking of rings or fields or modules or algebras or other such things; one of them distributes over the other, but not the other way around.)
I agree that it would be annoying if we had different names for all the different things we currently call multiplication.
It sounds as if you might prefer a system where instead of saying “R is a ring” we have some concise way of saying what operations R has and what axioms it satisfies. Something like “R is an (01+-*,ACD) algebra”, meaning it has addition & subtraction (= addition and additive inverses), multiplication but not necessarily division (so not necessarily multiplicative inverses), multplication is associative and commutative, and there’s a distributive law. And you’d say that instead of “commutative ring with unity” or “commutative ring” or “ring”, so there’d be a bit more verbosity and a bit more scope for not reading carefully exactly what’s being assumed, but less ambiguity, and the terminology wouldn’t so strongly favour a smallish set of particular types of structure that have their own names. Instead of “monoid” we’d say “0+ algebra”, instead of “semigroup” we’d say “+ algebra”, instead of “abelian group” we’d say “0+- algebra”, instead of “group” we’d say “1*/ algebra”, etc.
(My notation there assumes that it’s understood that something called addition is always commutative and associative.)
There’s definitely something to be said for that. On the other hand, those structures with special names have those names because those types of structures keep coming up. “Group” is only five letters and one syllable, and a lot of things are groups.
I was thinking of fields. I am doing a pretty bad math sin (atleast for the context) on not being precise and referring to the right target with “distributes over the other”. It is too foggy in my mind, I faintly recall something about being able to project + structures into * via something, but is wasn’t as straighforward and trivial than I thought.
While there migth have been at various point paid various amounts of attention on which things should be named and which should not be I doubt that it is completely reconstructible from anonymising everything and building it back again. A thing being named is somewhat sticky and will probably push back to outstay its welcome partly because of terminological inertia. A significant reason why we call things we call them is that other know what we are referring to and that it is customary.
(Pedantic notes: A group doesn’t have two operations, you’re thinking of rings or fields or modules or algebras or other such things; one of them distributes over the other, but not the other way around.)
I agree that it would be annoying if we had different names for all the different things we currently call multiplication.
It sounds as if you might prefer a system where instead of saying “R is a ring” we have some concise way of saying what operations R has and what axioms it satisfies. Something like “R is an (01+-*,ACD) algebra”, meaning it has addition & subtraction (= addition and additive inverses), multiplication but not necessarily division (so not necessarily multiplicative inverses), multplication is associative and commutative, and there’s a distributive law. And you’d say that instead of “commutative ring with unity” or “commutative ring” or “ring”, so there’d be a bit more verbosity and a bit more scope for not reading carefully exactly what’s being assumed, but less ambiguity, and the terminology wouldn’t so strongly favour a smallish set of particular types of structure that have their own names. Instead of “monoid” we’d say “0+ algebra”, instead of “semigroup” we’d say “+ algebra”, instead of “abelian group” we’d say “0+- algebra”, instead of “group” we’d say “1*/ algebra”, etc.
(My notation there assumes that it’s understood that something called addition is always commutative and associative.)
There’s definitely something to be said for that. On the other hand, those structures with special names have those names because those types of structures keep coming up. “Group” is only five letters and one syllable, and a lot of things are groups.
I was thinking of fields. I am doing a pretty bad math sin (atleast for the context) on not being precise and referring to the right target with “distributes over the other”. It is too foggy in my mind, I faintly recall something about being able to project + structures into * via something, but is wasn’t as straighforward and trivial than I thought.
While there migth have been at various point paid various amounts of attention on which things should be named and which should not be I doubt that it is completely reconstructible from anonymising everything and building it back again. A thing being named is somewhat sticky and will probably push back to outstay its welcome partly because of terminological inertia. A significant reason why we call things we call them is that other know what we are referring to and that it is customary.