Sigma-Algebra as Magma of Sets

Theorem
The concept of $\sigma$-algebra is an instance of a magma of sets.

Proof
It will suffice to define partial mappings such that the axiom for a magma of sets crystallises into the axioms for a $\sigma$-algebra.

Let $X$ be any set, and let $\mathcal P \left({X}\right)$ be its power set.

Define:


 * $\phi_1: \mathcal P \left({X}\right) \to \mathcal P \left({X}\right): \phi_1 \left({S}\right) := X$


 * $\phi_2: \mathcal P \left({X}\right) \to \mathcal P \left({X}\right): \phi_1 \left({S}\right) := X \setminus S$


 * $\phi_3: \mathcal P \left({X}\right)^\N \to \mathcal P \left({X}\right): \phi_3 \left({\left({S_n}\right)_{n \in \N}}\right) := \displaystyle \bigcup_{n \mathop \in \N} S_n$

It is blatantly obvious that $\phi_1, \phi_2$ and $\phi_3$ capture the axioms for a $\sigma$-algebra.