Talk:Order Isomorphism Preserves Strictly Minimal Elements

Haven't $\prec_1$ and $\prec_2$ got to be orderings, not just general relations?

And also, the parenthetical "(for strict orderings)" needs to be explained.

I believe it really ought to say something like this:

Let $A_1$ and $A_2$ be classes.

Let $\prec_1 \subseteq A_1 \times A_1$ and $\prec_2 \subseteq A_2 \times A_2$ be strict orderings on $A_1$ and $A_2$ respectively.

(Alternatively you could write "Let $\left({A_1, \prec_1}\right)$ and $\left({A_1, \prec_2}\right)$ be relational structures where $\prec_1$ and $\prec_2$ are strict orderings."

Let $\phi : \left({A_1, \prec_1}\right) \to \left({A_1, \prec_2}\right)$ be an order isomorphism.

Let $B \subseteq A_1$.

Then $\phi$ maps $\prec_1$-minimal elements of $B$ to $\prec_2$-minimal elements of $\phi \left({B}\right)$.

Note I removed the words "the" from "minimal elements" in the last sentence in case there are no minimal elements in $B$ - as I don't believe it has been proved that there has to be any.

Thoughts? --prime mover 08:01, 11 August 2012 (UTC)

Agreed on all points, except for the alternative formulation (rel.struc. is in set theory, at least atm). For the last one, $\varnothing$ does the job. --Lord_Farin 08:58, 11 August 2012 (UTC)

Sorry, don't understand "For the last one, $\varnothing$ does the job." Does what job? I'm missing something. --prime mover 10:13, 11 August 2012 (UTC)

"[...] as I don't believe it has been proved that there has to be any". --Lord_Farin 10:14, 11 August 2012 (UTC)

Not sure the distinction is necessary. The point is that saying: "Then $\phi$ maps the $\prec_1$-minimal elements of $B$ to the $\prec_2$-minimal elements of $\phi \left({B}\right)$" presupposes that there exist minimal elements, whereas leaving out the "the" just means that "if there are (any) minimal elements, they will be mapped to elements in the image which can also be shown to be minimal." If there are none, then the statement still stands, and I'm not sure I see why it is necessary to invoke a statement including $\varnothing$ in it. How did you envisage such a statement to be crafted? --prime mover 10:27, 11 August 2012 (UTC)

I disagree, on second thought. If there are none, leaving 'the' in just gives a vacuous truth (it maps all zero minimal elements to min.elts). I didn't envisage it to be included, but simply pointed out that there were sets not containing minimal elements. --Lord_Farin 10:31, 11 August 2012 (UTC)

... incidentally "Definition:Relational Structure" has been written from the class-theory standpoint, it's here: Definition:Structure (Set Theory) and simply needs a merge and rewrite. --prime mover 10:29, 11 August 2012 (UTC)

I still need to sit down for a few moments and have a thorough thinking session about how to structure the class vs. set setup on PW. I'll get back to it. --Lord_Farin 10:31, 11 August 2012 (UTC)