In my previous post, I said when a subset of the domain of a Group forms a SubGroup of that Group. In this blog post, I continue showing how Abstract Algebra can be done in a more usable and logically correct way.
So let's start by defining the term Left Coset of a subset of the domain of a Group by an element of the Group.
Let 𝒢 = <G, e, op, inv> be a Group and let H be a subset of G. …
In Abstract Algebra we usually care only that a subset of the domain/carrier of a Group forms a subgroup of that group and not so much for the actual subgroup.
In this post, I’ll use the following definition for a Group.
Let G be a set. Let e be an element of G. Let op be a binary operation over G. Let inv be a unary operation over G. We say that <G, e, op, inv> is a Group if:
(∀ a ∊ G)(∀ b ∊ G)(∀ c ∊ G)[op(a, op(b, c)) = op(op(a, b), c)];
(∀ a ∊ G)[op(e…
Maybe I should have used the word property instead of "axiom". The quotes are on purpose. I have used "axiom" only because I have seen the term "Group axioms" being used more than "Group properties". And yes the use of "axiom" is a problem. It should be property. But the real problem for me is with the usual definition. Precisely the fact that when the usual definition is translated from natural language to the language of modern mathematics (the language of Set Theory which is a first order language) the translation is not trivial. The last two properties have to be combined together. And for me this is just a Math smell and it should ring a bell that there is indeed a problem.
This statement is the root cause of your concerns and your basic mistake. Nowhere in any math book you read this, because it’s simply wrong.
Have you read any book about Set Theory or Mathematical Logic? And saying that it's "simply wrong" is wrong :D
I don't know (have not touched/worked with) the Peano axioms. But from what I have found here https://mathworld.wolfram.com/PeanosAxioms.html is that the theory they imply (because they make a statement about every subset of the domain) is from second order and I have not worked yet in the domain of second order logic. But just give me a month and I will give you a well defined set of axioms in a second order language.
For now all I can do…
I totally agree that those are properties and not axioms. This is why I put them in quotations. The reason why I did this is because usually they are called Group axioms or so I have most seen them being called.
I agree we do no't have do ask the identity to be unique it follows from the associativity of operation and it's existence. Actually in non of the definitions I put in mine blog post I have required/stated that it should be unique, even in the last one.
Yes I totally agree that if there isn't an identity the…
Also there is no way to have 2 different identity elements (both left and right identities). And this follows straight from the usual prove that the identity element is unique (that prove depends only on the existence of at least one neutral element and the fact that the binary operation is associative).
Actually, I did skip the remarks on purpose. Because I wanted to first give a clear logical explanation of the problem. And then to make my point that in order for this definition to work it should be proven that the element stated to exist is unique... I mean it should ring a bell that the "axioms" that are usually given depend on each other in a nontrivial way. I mean heaving to prove something that gives you the right to write the definition the way you did is an obvious math smell (the equivalent of the term code smell from programming). I also believe that definitions should be well thought out and easy to use the way we have given them.
There are real problems with the usual definition. But the biggest one is not that this definition does not define only Groups but the fact that it does not reflect the way Groups are used.
I’ve decided to take the usual definition straight from Wikipedia only because I do not what to point fingers at anyone! …
From time to time I see pictures like the one I drew on WEB pages which try to explain SQL JOINs…
Those pictures “try to show the difference between the different SQL JOINs” with Venn diagrams. And here is the thing SQL JOINs are for joining data…
All those pictures out there on WEB are trying to demonstrate the difference by showing which records are used in the most general case for creating the result of the join operation. The main problem is that joins are all about joining records from two tables. …
So well that it makes me enjoy writing imperative OOP code in it as much as I enjoy writing pure functional code in Haskell.
For the last 4 years of my life, I have really dedicated myself to mathematics. I was planning to become a hardcore logician but at the last moment, I decided not to pursue a professional math career, because of my vision of how math should really be done. And instead to learn how to design and implement large-scale distributed systems in a good way and seek professional realization as a Software engineer.
This is my first…
Passionate about Programming. Interested in Highly Distributed Systems and the Microservice Architecture. In love with Math and proving things.