Definition talk:Closed under Mapping

Isn't $S \subseteq X$ required? --abcxyz (talk) 19:38, 6 October 2012 (UTC)

Is it possible to make a simpler version of this which does not need to use the language of families? Subsets of a power set for example? --prime mover (talk) 20:53, 6 October 2012 (UTC)

@abcxyz: That would in general restrict $S$ too much, I think; although $X$ could always be extended, and then $\phi$ be made only a partial mapping. As PM noted before, the notion appears to be glossed over in the literature, so we're free to discuss and decide on the best shape ourselves.

@PM: Probably a similar approach to Definition:Pointwise Operation on Real-Valued Functions could be taken, first defining the notion for unary and binary functions, then for $I$-ary. Maybe a version using sets of sets could be crafted, but I haven't considered that yet. Would the first idea work for you? --Lord_Farin (talk) 21:43, 6 October 2012 (UTC)

I think so. The basic concept is simple: "a mapping is closed on $S$ if its image is a subset of $S$." While we have discussed multiary mappings (a binary op is a binary mapping), we haven't gone anywhere near the level of an arbitrarily-indexed-arity mapping which is pretty far out in this context. I understand you are trying to put something together which is as general as possible (and possibly at the fringes of mathematics) but I think it's important to bring it down to the basics first.

And I don't think you need to intersect with the domain of $\phi$ because image of $S$ under $\phi$ is the same thing as the image of $S \cap \phi$ under $\phi$. The elements of $S$ not in the domain of $\phi$ do not contribute anything to the image. Correct me if I'm missing a subtlety. --prime mover (talk) 22:26, 6 October 2012 (UTC)

I will get to the more basic cases tomorrow, I hope. As for the intersecion with $\Dom \phi$, I did it just to explicate that it is expressly *not* assumed that $S \subseteq \Dom \phi$. It seems pedagogically wrong to leave this out; it could confuse some readers. --Lord_Farin (talk) 21:17, 7 October 2012 (UTC)

@LF: Sorry, I didn't realize that. I guess it doesn't really matter, then. --abcxyz (talk) 02:23, 7 October 2012 (UTC)

No problem there; certainly, I value all your input since this notion is still under development. --Lord_Farin (talk) 21:17, 7 October 2012 (UTC)