Commutativity is good in addition to associativity because then you can use Σ
Structure preservation in a morphism φ (aka φ(xy)=φ(x)φ(y)) is good because then if φ follows from the context, you can replace all of “φ(xyz)” and “φ(xy)φ(z)” and “φ(x)φ(yz)” with “xyz”
Bijectivity in a function f is good because then for any equation it is an equivalent transformation to apply f to both sides
Monotonicity in a function f is good because then you can apply it to both sides of an inequality
Huh, you’re right. That’s too bad, “well-definedness & injectivity” doesn’t flow so well, and I don’t see what comparable property surjectivity is good for.
In the same vein:
Commutativity is good in addition to associativity because then you can use Σ
Structure preservation in a morphism φ (aka φ(xy)=φ(x)φ(y)) is good because then if φ follows from the context, you can replace all of “φ(xyz)” and “φ(xy)φ(z)” and “φ(x)φ(yz)” with “xyz”
Bijectivity in a function f is good because then for any equation it is an equivalent transformation to apply f to both sides
Monotonicity in a function f is good because then you can apply it to both sides of an inequality
Don’t you just need injectivity here?
Huh, you’re right. That’s too bad, “well-definedness & injectivity” doesn’t flow so well, and I don’t see what comparable property surjectivity is good for.
Good stuff, thanks for your comment!