Algebraic Objects

Algebraic Objects are built using Sets and Axioms.

Admittedly, that is extremely vague. We don’t want to rewrite an 800 page textbook here. We defer to Wikipedia where someone wrote that it is typical for an algebraic object to be built using one or two sets and one or two binary operations.

Over time you will notice that usually when someone mentions two binary operations, one is Addition and one is Multiplication. We want to keep the very vague explanation of the first sentence, but possibly give you comfort that all the time you spent digging deeper into Sets, Addition and Multiplication was time well spent.

Our focus here is on more general views of Addition and Multiplication, and we explain them informally as follows:

• Addition is a binary operation that takes two elements from a set and the result of the calculation is also an element of that set.
• Multiplication is a binary operation where we take an element from a first set and an element from a second set and the result is an element in the first set.
  • It is acceptable for the second set to actually be the first set; if we didn’t say this someone could ask why it is legal to multiply a number by another number for which the result is a number.