In the cross-section of math and computer science, lie one of the simplest and most powerful concepts. It’s called a monomorphism. This is the simplest example of a monomorphism that I’ve been able to construct.
While we may believe they’re complicated, after all, who talks frequently about words such as “Monomorphism?” To pure functions, they are crucial to understanding recursion, they’re simply functions that return the same type as the type supplied to the function at their most basic level.
The question is usually asked, what use are they? Being debatable, I wish to advert that topic and reveal a concrete example, helpful for me. While I don’t consider this a profound proof, I would consider calling it a lemma. To me a Lemma is a “less strong proof,” helpful in achieving a goal inside a proof.
I’ll start by calling it at Theorem. this is the Theorem of Absolute Value.
It’s the Cat group monomorphism of the absolute value function, please enjoy!
Symbolically it’s aMa’ . M is the Monomorphic Function. It takes an integer, increments it if it’s positive, decrements it if it’s negative, and returns 0, ie. the identity if it’s equal.
It should be clear the relation is symmetric about the a’ axis. Also, I prefer to bound it to some integer < infinity, thou its not required, it makes things easier in practice (especially in a graphic.
Since the Set( -Z, 0, +Z) is closed under addition and subtraction, it is an additive group, also while it has multiplication as well, it doesn’t remain closed under division, so it’s also a multiplicative ring.