I looked at here for getting an intuition about F-algebra, but I am still left with some questions.
Suppose I have a group signature as $\Sigma= (* : X \times X \rightarrow X, \thicksim: X \rightarrow X , e : \rightarrow X)$, with the following axioms in a unuiversal algebraic way:
- $x ∗ (y ∗ z) = (x ∗ y) ∗ z$ (Associativity)
- $e ∗ x = x = x ∗ e$ (Identity element)
- $x ∗ (\thicksim x) = e = (\thicksim x) ∗ x$ (Inverse element)
A model of the above signature is an assignment of two functions to its function symbols, and a constant to its constant symbol, such that the above three laws hold.
How the above structure with three axioms, can be encoded (represented) in an F-algebraic notion:
1) What is my endo-Functor F and why is that?
2) How these three laws are represented in F-algebra?
p.s.: I would appreciate if anybody refer to a textbook, or a document that I can read more examples to further understand the F-algebra concept.