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In an abstract algebra textbook, they define groups first and then abelian as a property that some groups have. Here, the author is defining abelian groups "fr
by MarkusQ 10mo ago
In an abstract algebra textbook, they define groups first and then abelian as a property that some groups have. Here, the author is defining abelian groups "from scratch" and doesn't have an earlier definition of groups to lean on.
In more advanced texts, they could simply say that a group is a moniod with inverses and could (by your reasoning, should) avoid specifying that groups are associative since this is a property of all monoids.
- seanhunter 10mo agoWell 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.
- MarkusQ 10mo agoRight. 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.