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chain

Defines: 
finite chain, infinite chain, dense chain, complete chain, join of chains, chain homomorphism
Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

03-00 no label found

Comments

As far as I remember "chain from A to B" is defined as a maximal chain having A and B as min. and max. elements correspondingly. Do I remember correctly? If yes, then the entry "Chain" needs correction to say about "chain from A to B".

Also we should state how existence of chains from A to B is related with axiom of choice and Kuratowski's lemma in particular.
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts

> As far as I remember "chain from A to B" is defined
> as a maximal chain having A and B as min. and
> max. elements correspondingly. Do I remember
> correctly?

This is not correct. There is nothing that requires
that a chain from A to B be maximal.

> Also we should state how existence of chains from A
> to B is related with axiom of choice and Kuratowski's
> lemma in particular.

If $A \le B$, then there always exists a chain from A
to B, namely $\{A, B\}$. Choice is not required here.
The axiom of choice become relevant when we ask for a
maximal chain.

>> As far as I remember "chain from A to B" is defined
>> as a maximal chain having A and B as min. and
>> max. elements correspondingly. Do I remember
>> correctly?

> This is not correct. There is nothing that requires
> that a chain from A to B be maximal.

Anyway we should define "a maximal chain from A to B" and just
"chain from A to B". (Should we now file a correction to "Chain"?)

Indeed my memory suggests that the word "maximal" can be omitted in this context accordingly definitions in some book. But my memory may be flaky.
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts

My experience is that most authors allow chains to be non-maximal, and when they want maximal they say so. The same goes for flags (a synonym for chains in some contexts). A flag is not generally maximal, unless you add the prefix "maximal flag" One could certainly define chain this way, but I don't think it is standard, in particular because there are many many uses of chains, including in PM articles, which are not intended only for maximal chains.

For example, the parabolic subgroups are stabilizers of certain chains/flags of subspaces. If we only allowed maximal then all parabolics would be Borel, and that is simply not the case. In the same vein, one can define a simplicial complex from a poset by taking the faces to be all chains in the poset. If all chains were maximal you wouldn't get a simplicial complex and you would not have subfaces (that is, you would not have edges of 2-simplex and vertices of 1-simplex, etc.) These are just some ways that chain is used in very well established ways and were the assumption of maximal would not be feasable.

So no, I don't think this is a correction. I think the standard solution to such problems on PM is to attach an entry that defines maximal chain, and if you wish, state WITH REFERENCE (not flaky memory) that some authors assume all chains are maximal. (I say with reference because the references at my fingertips don't do this so I think it is a rare assumption and worth supporting if it is true.)

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