To begin topology, we took a set and added some structure to it by designating certain subsets as *open sets*.

To begin algebra, we will start with a set and add some structure to it by defining a *binary operation*.

An operation is a rule that takes one or more objects from a set and results in another object. For example, the addition operation takes two numbers and results in another number.

If the result is always in the same set the inputs came from, the set is said to be **closed** under that operation.

If an operation takes two inputs, it is called a **binary operation**.

So addition is an example of a binary operation and the set of integers is closed under that operation.

String concatenation is another example where two strings are concatenated to form a third.

Note that as long as you can define the rule (if need be just by listing the result for each pair of inputs) you have an operation.

So there is nothing wrong with defining a set {A, B, C} and defining some rule # such that: A#A = A, A#B = C, A#C = B, B#A = A, B#B = B, B#C = C, C#A = A, C#B = A, C#C = A

In this example, don't try to look for any pattern. I just randomly picked some results. All we require to have a binary operation is that a result is defined for each pair of inputs.

**UPDATE**: next post

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The original post was in the category: poincare_project but I'm still in the process of migrating categories over.