The sequence in which axioms of group present matter or not?

Question

I discussed this with my graduate teachers and some teachers say the sequence of axioms of group matter but some say does not matter.

So really sequence matter?

If matter then in which sense?

Correct me if i am wrong...

I think in some books before defining group they define

1. Groupoid

2. Semi group

3. Monoid

   And then they define group

Due to above mentioned order sequence matter or anything else?

Answer 1

Certainly the order in which a person or a book or any other entity defines these objects does not matter, assuming that one definition does not use the terminology of the other definition.

On the other hand, the logical relationships between these definitions is very important to understand. For instance, it is very important that you know why every group is a groupoid, and is a semigroup, and is a monoid.

Also, an author presenting this material has the option decide to present the definitions of these concepts in a certain order in order to make the logical relationships more transparent.

For example, Author#1 may decide to first give this definition:

A semigroup is a set equipped with an associative binary operation.

and then follow up with this definition:

A group is a semigroup in which there exists an identity element, and in which every element has an inverse.

But Author #2 might not wish to do bother defining semigroup, which means that their definition of a group will be a little longer:

A group is a set equipped with an associative binary operation in which there exists an identity element, and in which every element has an inverse.

There's nothing wrong with either of these approaches, and there are other ways to proceed as well.