## question embedding sublattice into product-lattice

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• Hi all, I have a question which looks easy, but I haven t been able to get an answer. Suppose that $L$ is a finite lattice, and that $omega$ is the first
Message 1 of 3 , Jan 29, 2008
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Hi all,

I have a question which looks easy, but I haven't been able to get an

Suppose that $L$ is a finite lattice, and that $\omega$ is the first
infinite cardinal. Is it true that for every natural number $n$ there
is a (finite) natural number $m$ such that:
Whenever $a_1,..., a_n$ belong to $L^\omega$, there is a sublattice
$A$ of $L^\omega$, which is isomorphic to $L^m$, such that
$a_1,...,a_n$ belong to $A$?

Has anyone seen a proof or counterexample, results that look similar,
or papers that might prove useful?

Thanks!

Peter Ouwehand
• Dear Peter, It seems to me that this is essentially the same argument that says that the free object on $n$ generators in the variety generated by $L$ is the
Message 2 of 3 , Jan 29, 2008
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Dear Peter,

It seems to me that this is essentially the same argument that says that the free object on $n$ generators in the variety generated by $L$ is the sublattice of $L^{L^n}$ generated by the projection maps. Hence $m=|L|^n$ works! Here is an outline of the argument for your problem. Let $a: \omega\to L^n$ be the map defined by
$a(k)=(a_1(k),\dots,a_n(k))$
(for each $k\in\omega$), and denote by $S$ the range of $a$ (thus a subset of $L^n$). Put $X_s=a^{-1}\{s\}$, for each $s\in S$. Then the sets $X_s$, for $s\in S$, form a partition of $\omega$, and hence the lattice $A$ of all elements of $L^\omega$ which are constant on each $X_s$ is a sublattice of $L^\omega$, isomorphic to $L^{|S]}$. Now observe that each $a_i$ belongs to $A$.

Incidentally, there is nothing special about lattices in that argument (Jonsson's Lemma is not used), it would work just as well in any variety of algebras.

I hope that this is what you had in mind!

Cheers,
Fred

Le 29 janv. 08 à 11:04, p_ouwehand a écrit :

Hi all,

I have a question which looks easy, but I haven't been able to get an

Suppose that $L$ is a finite lattice, and that $\omega$ is the first
infinite cardinal. Is it true that for every natural number $n$ there
is a (finite) natural number $m$ such that:
Whenever $a_1,..., a_n$ belong to $L^\omega$, there is a sublattice
$A$ of $L^\omega$, which is isomorphic to $L^m$, such that
$a_1,...,a_n$ belong to $A$?

Has anyone seen a proof or counterexample, results that look similar,
or papers that might prove useful?

Thanks!

Peter Ouwehand

--
Friedrich Wehrung
LMNO, CNRS UMR 6139
Universit\'e de Caen, Campus 2
D\'epartement de Math\'ematiques, BP 5186
14032 Caen cedex
FRANCE

e-mail: wehrung@...
alternate e-mail: fwehrung@...

• I think this is easy. Let k = |L|. For f_1,...,f_n in L^{omega}, let p be the function defined on omega with p(i)=(f_a(i),...,f_n(i)). There are at most k^n
Message 3 of 3 , Jan 30, 2008
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I think this is easy. Let k = |L|. For f_1,...,f_n in L^{omega},
let p be the function defined on omega with
p(i)=(f_a(i),...,f_n(i)). There are at most k^n possible values of
p. Let M be the set
of all functions f in L^{omega} such that whenever p(i)=p(j) then
f(i)=f(j). Now M is isomorphic to L^r where r=|p(omega)| is no
greater than k^n. By dividing several of the equivalence classes
of elements of omega on which functions in M are constant, we
get a larger lattice M' with M' isomorphic to L^{r'}, r'=k^n.
Of course f_1,...,f_n all belong to M and to M'.

Ralph

--On Tuesday, January 29, 2008 10:04 AM +0000 p_ouwehand
<peter_ouwehand@...> wrote:

>
>
>
>
> Hi all,
>
> I have a question which looks easy, but I haven't been able to
>
> Suppose that $L$ is a finite lattice, and that $\omega$ is the
> first infinite cardinal. Is it true that for every natural
> number $n$ there is a (finite) natural number $m$ such that:
> Whenever $a_1,..., a_n$ belong to $L^\omega$, there is a
> sublattice $A$ of $L^\omega$, which is isomorphic to $L^m$, such
> that $a_1,...,a_n$ belong to $A$?
>
> Has anyone seen a proof or counterexample, results that look
> similar, or papers that might prove useful?
>
> Thanks!
>
> Peter Ouwehand
>
>

-----------------------------------------------------------------
Mckenzie, Ralph N
Vanderbilt University
Email: ralph.n.mckenzie@...
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