Groupoid

A Groupoid is like a Group in that it is a Set with a Binary Operation and the Binary Operation has Closure.

A Groupoid differs from a Group in that there are no requirements for Associativity, Existence of an Identity Element or the Existence of Inverse Elements.

Appendix A

We found what is written below–we’re playing around with it, right now we kind of don’t like it.

Take a Group and replace the Binary Operation with a Partial Function.

A Partial Function f from X to Y is a function from S to Y where S is a subset of X. For the case where the subset S is equal to X, the Partial Function is a Total Function.

Subtraction

f(x,y) = x – y

of Counting Numbers (1, 2, 3, …) is a Partial Function because it is defined only when x > y.

Appendix B

This issue has come up elsewhere. Someone asks “to be a Groupoid, is it required that the operation isn’t associative and that no identity element exists and inverse elements don’t exist? The answer given, elsewhere, was, it’s allowable for things not required to still be present.

We at Mockingbird Academy have agreed that we can agree with this, but that we believe it is advantageous to have examples showing it both ways.