Definition talk:Magma of Sets

Hi LF. I hope you don't mind these questions.

Question 1 (an obvious one)

Is this line:


 * $\phi_i: \mathcal P \left({X}\right)^{J_i} \to P \left({X}\right)$

supposed to be:


 * $\phi_i: \mathcal P \left({X}\right)^{J_i} \to \mathcal P \left({X}\right)$

Question 2

On lines 7 and 8 you present the defining conditions for a magma of sets. On line 9 you give a logically equivalent definition that on the surface seems miles easier to understand. But I've never come across this.

What do you mean by a set being closed under a mapping?

I am trying to understand this concept using finite $\sigma$ algebras. It's very difficult.

Please understand that I'm not making hard work for you. I just seek to see what you see. --Jshflynn (talk) 22:31, 2 October 2012 (UTC)


 * 1. Yes; good catch. 2. I mean with '$S$ is closed under $\phi_i$' that, upon feeding $\phi_i$ with only elements from $S$, it spits out an element of $S$. So if $S$ were to be a $\sigma$-algebra, it would need to be closed under the mappings "intersection", "countable union" and the constant mapping with image $X$.
 * I'd be glad if answering this questions opened at least one person's eyes for this concept (which is like an abstraction of algebraic structure). The condition that the $\phi_i$ are only partial mappings is to ensure that things like $\lim_{n \to \infty}$ could be included as well. --Lord_Farin (talk) 22:16, 3 October 2012 (UTC)


 * Technically, the concept of closure in this specific context has not been created yet. The more specialised ones of closed under an operation and closed for scalar product are just instances of the more general "closed under a mapping", but the latter concept still needs to be specified. Interestingly, although most texts are conscientious about making sure Definition:Closure (Abstract Algebra)/Algebraic Structure is defined as precisely as you like, the associated concept of $\phi \left({S}\right) \subseteq S$ is taken as understood. --prime mover (talk) 05:19, 4 October 2012 (UTC)