Re: lattice theory

From: William Elliot (marsh_at_privacy.net)
Date: 09/26/04 

Date: Sun, 26 Sep 2004 14:00:09 +0000 (UTC)

From: Philip reny <preny@uchicago.edu>
Newsgroups: sci.math.research
Subject: Re: lattice theory

>William Elliot wrote:
>>From: Philip Reny <preny@uchicago.edu>
>
>>X = [0,1]^N; 0 = 0^N
>
>>||x|| = sup_i x_i
>
>>x <= y when for all i in N, x_i <= y_i

>>>Assume that:
>>>(1) (X, ||.||) is compact.

>>>(2) Every pair x,y in X possesses a l.u.b., xVy, in X.
>>lub x,y = prod_i max x_i,y_i = f:N -> [0,1], i -> max x_i,y_i
>
>Not necessarily true, and this is the main sticking point. The lub
>of {x,y} is, by definition, the smallest member of the set of all
>upper bounds of {x,y} contained in X. There is no reason for this to
>be the componentwise max, since X need not containing the
>componentwise max of the two vectors x,y.

As [0,1] is linear order, max x_i,y_i is defined.
Let z_i = max x_i,y_i
Thus x,y <= z. z is upper bound.

Now if x,y <= a, ie if a is an upper bound,
then for all i, x_i <= a_i and y_i <= a_i
Thus I conclude for all i, z_i = max x_i,y_i <= a_i.
Namely z <= a and subsequently z is the smallest upper bound.

Have you any dispute with this proof?

>>>(3) x_{n} --> x and y_{n} --> y imply x_{n} V y_{n} --> xVy.

>Technically, this means that the semilattice is "topological" in the
>sense of, say, Birkhoff's Lattice Theory text. That is, the join
>operator is continuous.

>>That x_n isn't a component of x but a sequence of points of X ?
>True.

>>>Prove or disprove:
>>>x_{n} --> 0 implies x_{n} V x_{n+1} V x_{n+2} V ... --> 0.
>
>>As x_n -> 0
>>for all eps, some n with for all k > n, ||x_k|| < eps.
>True.

>>X is a complete lattice.
>True, by (1).
I'm using 'complete' to mean having all sup's and inf's.
Are you using 'complete' to mean every Cauchy sequence converges?

>>Again by associativity of sup, don't I have
>> ||sup C|| = sup { ||x|| : x in C } ?
>No. Once again this is because lub is not necessarily the
>componentwise max.

for C subset X,
        sup C = f:N -> [0,1], k -> sup { x_k | x in C }
                = prod_k sup{ x_k | x in A }
is component wise and is proven like lub x,y was proven component wise.
Shall I detail proof or have you found flaw in the above proof?

>>For all eps, some n with for all k > n, ||x_k|| < eps. Thus
>> || sup { x_(n+k) | k in N } || <= eps
>>and with an iota of juggling eps, establish the limit you wish to
>>show?

>No, for the reasons stated above.

Email: privacy.org = agora.rdrop.com 

----

Relevant Pages

  • Re: lattice theory
    ... componentwise max, since X need not containing the componentwise max ... The reason is as stated above. ... sense of, say, Birkhoff's Lattice Theory text. ... This does not follow because lub is not necessarily the componentwise ...
    (sci.math.research)
  • Re: lattice theory
    ... >>componentwise max of the two vectors x,y. ... >>sense of, say, Birkhoff's Lattice Theory text. ... Once again this is because lub is not necessarily the ... >is component wise and is proven like lub x,y was proven component ...
    (sci.math.research)
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