logoalt Hacker News

seanhunter • last Sunday at 5:05 PM • 1 reply • view on HN

Well if I check such a book that takes a category-theoretic approach to teaching abstract algebra (Aluffi “Algebra Chapter 0”), he says the following:

   > “ A semigroup is a set endowed with an associative operation; a monoid is a semigroup with an identity element. Thus a group is a monoid in which every element has an inverse”.
So according to Aluffi at least, the operation of a monoid is also associative. As you can see he does in fact also remove the associativity criterion from the description of a group by defining it in terms of a monoid. So he’s consistent with me at least.

Replies

MarkusQ • last Sunday at 6:09 PM

Right. And so is the article. When you are introducing an object you need to specify its properties, _including_those_it_inherits from objects you haven't defined.

If I haven't defined mammals, I say that bats are warm blooded animals that produce milk for their young, etc., but if I have (or expect my readers to know what a mammal is) I can just say they are mammals.