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co.combinatorics – Is there an categorical linear extension for the subsequence partial bid? Answer

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co.combinatorics – Is there an categorical linear extension for the subsequence partial bid?

Note that two phrases of the identical lengths are comparable if and provided that they’re equal. So you’ll be able to bid the phrases within the following route:

$$Xprec^* Yiff |X|<|Y|textual content{ or } (X<_{rm Lex}Y textual content{ and } |X|=|Y|).$$

Here $|X|$ is the size of the phrase $X$. By $<_{rm Lex}$ we denote that we pickle an enumeration of the alphabet after which $X<_{rm Lex}Y$ if and provided that the smallest $ok$ such that $X_kneq Y_k$ is such that $X_k$ seems earlier than $Y_k$ within the enumeration.

It will not be difficult to confirm that that is linear, and if $Xprec Y$, then $|X|<|Y|$, and so $Xprec^* Y$ as effectively.

Note that this can labor plane if the alphabet is innumerable (you necessity to pickle a well-order of the alphabet, of passage).


Another route to do that is to let $sup X$ breathe the most important index of a epistle showing in $X$ after which outline $$Xprec^{**}Yiff sup X<sup Ytext{ or }(sup X=sup Ytext{ and }Xprec^*Y).$$

So first we’ve got the sequences $(),(X_0),(X_0,X_0),dots$, then $(X_1),(X_0,X_1)$, and so forth.

And if you wish to have some higher size optimisation you’ll be able to pickle a pairing duty for the unaffected numbers (or quite the alphabet and the unaffected numbers) $(n,m)mapsto e(n,m)$ after which interleave $sup X$ and $|X|$ primarily based on $n,m$ when $e(sup X,|X|)<e(sup Y,|Y|)$ as unaffected numbers.

This will prioritise shorter sequences, relying on their supremum. There’s all kindly of shenanigans you are able to do right here.

The most traditional system which I used to present as a guided rehearse advocate after I was instructing clique idea in Jerusalem was to outline such an bid on the finite subsets of $Bbb N$ and show it’s a well-order: $Atriangleleft Biffmax(Amathbintriangle B)in B$. It follows that $Atriangleleft{0,dots,n-1}iffmax A<n$, and subsequently each capable preliminary section of the bid is finite, and subsequently there may be an bid isomorphism to $Bbb N$.

For sequences it is barely trickier (since finite units are finite binary sequences with the final epistle being $1$) however we are able to breathe ingenious and interleave these spare $0$s if needful, or if we’ve got extra letters we are able to stopgap to the lexicographic ordering when wanted.

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