## A small terminology question about lattice congruences

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• Dear members, Let L be a lattice, let Theta be a congruence such than L / Theta is a totally ordered set. Is there any name for such Thetas? Thank you, --
Message 1 of 11 , Jan 19, 2011
Dear members,

Let L be a lattice, let Theta
be a congruence such than L / Theta is
a totally ordered set.

Is there any name for such Thetas?

Thank you,

--
Gejza Jenca
• Not that I know of. GG Sent from my iPhonet
Message 2 of 11 , Jan 21, 2011
Not that I know of.

GG

Sent from my iPhonet

On 2011-01-19, at 9:53 AM, Gejza Jenca <gejza.jenca@...> wrote:

Dear members,

Let L be a lattice, let Theta
be a congruence such than L / Theta is
a totally ordered set.

Is there any name for such Thetas?

Thank you,

--
Gejza Jenca

• Not that I know of. GG Sent from my iPhonet
Message 3 of 11 , Jan 21, 2011
Not that I know of.

GG

Sent from my iPhonet

On 2011-01-19, at 9:53 AM, Gejza Jenca <gejza.jenca@...> wrote:

Dear members,

Let L be a lattice, let Theta
be a congruence such than L / Theta is
a totally ordered set.

Is there any name for such Thetas?

Thank you,

--
Gejza Jenca

• Perhaps ``linearizing’’ would be a good term to coin for these congruences? ``theta _linearizes_ L iff L/theta is a chain’’… Similarly, if K is a
Message 4 of 11 , Jan 21, 2011

Perhaps ``linearizing’’ would be a good term to coin for these congruences?  ``theta _linearizes_ L iff L/theta is a chain’’…

Similarly, if K is a class of lattices and L/theta is in K, then we might say ``theta _K-izes_ L’’:

1.        Theta _modularizes_ L if L/theta is modular

2.       Theta _distributizes_ L (klunky… hmm) if L/theta is distributive

3.       Theta _simplifies_ (no ``izes’’, but the same general idea…) L if L/theta is simple

4.       Theta _trivializes_ L if L/theta is trivial  (but here I would prefer ``annihilates’’, in concert with a similar concept in linear algebra…

This gets a bit too klunky in some cases, as in the case where L/theta is subdirectly irreducible…

But then we could say theta is a ``linearizer’’ of L if it ``linearizes’’ L, is a ``modularizer’’ of L if it modularizes L, etc.

Just a thought…

PS:  Happy New Year all!

Matt Insall, PhD
Associate Professor of Mathematics
Department of Mathematics and Statistics, suite 315
Missouri University of Science and Technology
(formerly University of Missouri - Rolla)

400 W. 12th St.

Rolla MO 65409-0020

(573)341-4901
insall@...

http://web.mst.edu/~insall/

Essays:

http://web.mst.edu/~insall/Essays/Bernoulli%20Functions%20versus%20modern%20functions/

http://web.mst.edu/~insall/Essays/Consistency%20in%20Mathematics/

http://web.mst.edu/~insall/Essays/First%20Lines/

http://web.mst.edu/~insall/Essays/Set%20Theory/

Effective Jan. 1, 2008, UMR became Missouri University of Science and Technology (Missouri S&T)

PS:  visit lulu.com!

viaR03

From: univalg@yahoogroups.com [mailto:univalg@yahoogroups.com] On Behalf Of George Gratzer
Sent: Friday, January 21, 2011 2:26 PM
To: univalg@yahoogroups.com
Subject: Re: [univalg] A small terminology question about lattice congruences

Not that I know of.

GG

Sent from my iPhonet

On 2011-01-19, at 9:53 AM, Gejza Jenca <gejza.jenca@...> wrote:

Dear members,

Let L be a lattice, let Theta
be a congruence such than L / Theta is
a totally ordered set.

Is there any name for such Thetas?

Thank you,

--
Gejza Jenca

• 2011/1/21 Insall, Matt ... Thank you, a good suggestion. I was thinking about the name cochain congruence -- since this notion is dual to
Message 5 of 11 , Jan 21, 2011
2011/1/21 Insall, Matt <insall@...>
>
>
>
> Perhaps ``linearizing’’ would be a good term to coin for these congruences?  ``theta _linearizes_ L iff L/theta is a chain’’…
>
>
>
> Similarly, if K is a class of lattices and L/theta is in K, then we might say ``theta _K-izes_ L’’:

Thank you, a good suggestion.

I was thinking about the name "cochain congruence" -- since this
notion is dual to
(embedding of a) chain.

Anyway, what I really wanted to know is whether someone in the past
worked with this type
of coungruences. Obviously, a google search for "lattice" "congruence"
"linear" "chain"
would give me about 50% of the papers on lattice theory in history. So
without knowing
the exact name, there is no chance to find something relevant.

--
GJ
• Ahhh, a slightly different question. It will perhaps take a few days to get to someone who has worked very extensively with them (if anyone has), considering
Message 6 of 11 , Jan 21, 2011
Ahhh, a slightly different question. It will perhaps take a few days to get to someone who has worked very extensively with them (if anyone has), considering that (a) you got little in response immediately, and (b) George Gratzer responded but did not tell you he had seen some work on such things.... :-) Have you got some nice results using this notion?

Matt Insall, PhD
Associate Professor of Mathematics
Department of Mathematics and Statistics, suite 315
Missouri University of Science and Technology
(formerly University of Missouri - Rolla)
400 W. 12th St.
Rolla MO 65409-0020

(573)341-4901
insall@...
http://web.mst.edu/~insall/

Essays:
http://web.mst.edu/~insall/Essays/Bernoulli%20Functions%20versus%20modern%20functions/
http://web.mst.edu/~insall/Essays/Consistency%20in%20Mathematics/
http://web.mst.edu/~insall/Essays/First%20Lines/
http://web.mst.edu/~insall/Essays/Set%20Theory/

Effective Jan. 1, 2008, UMR became Missouri University of Science and Technology (Missouri S&T)

PS:  visit lulu.com!

viaR03

-----Original Message-----
From: univalg@yahoogroups.com [mailto:univalg@yahoogroups.com] On Behalf Of Gejza Jenca
Sent: Friday, January 21, 2011 3:46 PM
To: univalg@yahoogroups.com
Subject: Re: [univalg] A small terminology question about lattice congruences

2011/1/21 Insall, Matt <insall@...>
>
>
>
> Perhaps ``linearizing'' would be a good term to coin for these congruences?  ``theta _linearizes_ L iff L/theta is a chain''...
>
>
>
> Similarly, if K is a class of lattices and L/theta is in K, then we might say ``theta _K-izes_ L'':

Thank you, a good suggestion.

I was thinking about the name "cochain congruence" -- since this
notion is dual to
(embedding of a) chain.

Anyway, what I really wanted to know is whether someone in the past
worked with this type
of coungruences. Obviously, a google search for "lattice" "congruence"
"linear" "chain"
would give me about 50% of the papers on lattice theory in history. So
without knowing
the exact name, there is no chance to find something relevant.

--
GJ

------------------------------------

• ... Hi, I believe that Dilworth s famous 1950 result that the number of chains needed to cover a poset is its width arose from the following problem: If D is a
Message 7 of 11 , Jan 21, 2011
> Anyway, what I really wanted to know is whether someone in the past
> worked with this type of coungruences. Obviously, a google search for

Hi,

I believe that Dilworth's famous 1950 result that the number of chains
needed to cover a poset is its width arose from the following problem:

If D is a finite distributive lattice it can be in embedded into the
boolean lattice 2^n, of course, and the smallest n is the length of D.
What is the least number of chains into which it can be embedded? In other
words, what is the smallest number of XXXX-congruences that meet to 0 in
Con(D)? Answer: the width of J(D) (the poset of join-irreducible elements)
and this equals the maximum of the number of covers of elements of D.

Ralph

On Fri, 21 Jan 2011, Gejza Jenca wrote:

> 2011/1/21 Insall, Matt <insall@...>
>>
>>
>>
>> Perhaps ``linearizing’’ would be a good term to coin for these congruences?  ``theta _linearizes_ L iff L/theta is a chain’’…
>>
>>
>>
>> Similarly, if K is a class of lattices and L/theta is in K, then we might say ``theta _K-izes_ L’’:
>
> Thank you, a good suggestion.
>
> I was thinking about the name "cochain congruence" -- since this
> notion is dual to
> (embedding of a) chain.
>
> Anyway, what I really wanted to know is whether someone in the past
> worked with this type
> of coungruences. Obviously, a google search for "lattice" "congruence"
> "linear" "chain"
> would give me about 50% of the papers on lattice theory in history. So
> without knowing
> the exact name, there is no chance to find something relevant.
>
> --
> GJ
>
>
> ------------------------------------
>
>
>
>
>
• I would also mention the paper F. W. Anderson and R. L. Blair, Representations of distributive lattices as lattices of functions. Math. Ann. 143 (1961),
Message 8 of 11 , Jan 22, 2011
I would also mention the paper

F. W. Anderson and R. L. Blair,
Representations of distributive lattices as lattices of functions.
Math. Ann. 143 (1961), 187--211.

GG

On 2011-01-21, at 9:53 PM, Ralph Freese wrote:

> Anyway, what I really wanted to know is whether someone in the past
> worked with this type of coungruences. Obviously, a google search for

Hi,

I believe that Dilworth's famous 1950 result that the number of chains
needed to cover a poset is its width arose from the following problem:

If D is a finite distributive lattice it can be in embedded into the
boolean lattice 2^n, of course, and the smallest n is the length of D.
What is the least number of chains into which it can be embedded? In other
words, what is the smallest number of XXXX-congruences that meet to 0 in
Con(D)? Answer: the width of J(D) (the poset of join-irreducible elements)
and this equals the maximum of the number of covers of elements of D.

Ralph

On Fri, 21 Jan 2011, Gejza Jenca wrote:

> 2011/1/21 Insall, Matt <insall@...>
>>
>>
>>
>> Perhaps ``linearizing’’ would be a good term to coin for these congruences?  ``theta _linearizes_ L iff L/theta is a chain’’…
>>
>>
>>
>> Similarly, if K is a class of lattices and L/theta is in K, then we might say ``theta _K-izes_ L’’:
>
> Thank you, a good suggestion.
>
> I was thinking about the name "cochain congruence" -- since this
> notion is dual to
> (embedding of a) chain.
>
> Anyway, what I really wanted to know is whether someone in the past
> worked with this type
> of coungruences. Obviously, a google search for "lattice" "congruence"
> "linear" "chain"
> would give me about 50% of the papers on lattice theory in history. So
> without knowing
> the exact name, there is no chance to find something relevant.
>
> --
> GJ
>
>
> ------------------------------------
>
>
>
>
>

• 2011/1/22 Ralph Freese ... Oh, thank you, now I see why they do not have an established name! In the case of a finite lattice L, they
Message 9 of 11 , Jan 22, 2011
2011/1/22 Ralph Freese <ralph@...>
>
>
>
> > Anyway, what I really wanted to know is whether someone in the past
> > worked with this type of coungruences. Obviously, a google search for
>
> Hi,
>
> I believe that Dilworth's famous 1950 result that the number of chains
> needed to cover a poset is its width arose from the following problem:
>
> If D is a finite distributive lattice it can be in embedded into the
> boolean lattice 2^n, of course, and the smallest n is the length of D.
> What is the least number of chains into which it can be embedded? In other
> words, what is the smallest number of XXXX-congruences that meet to 0 in
> Con(D)? Answer: the width of J(D) (the poset of join-irreducible elements)
> and this equals the maximum of the number of covers of elements of D.
>
> Ralph

Oh, thank you, now I see why they do not have an established name!

In the case of a finite lattice L, they are just the (duals of) chains in the
J(D), where D is the maximal distributive image of L and it is now clear to me
that they belong to the realm of finite poset theory rather than to lattice
theory.

--
Gejza
• Yesterday evening I looked at the ceiling and began to reason about the wooden panels, which reminded me of a chess board. Soon I came to these equivalences
Message 10 of 11 , Jan 23, 2011
Yesterday evening I looked at the ceiling and began to reason about the wooden panels, which reminded me of a chess board. Soon I came to these equivalences for fields on a (finite) dimensional chess board:

For field x with row/column (r,c), color(x) and #neighbour=(count of neighbours) you can have

a) equivalence by color(x)
b) equivalence by #neighbours
c) equivalence by integer function r+c
d) equivalence by integer r-1/c
e) ...

In relational calculus
a) to c) could be used as an index with multiples, cpo
d) produces unique numbering and could serve as a key

Happy reasoning,
Jens
• Yesterday evening I looked at the ceiling and began to reason about the wooden panels, which reminded me of a chess board. Soon I came to these equivalences
Message 11 of 11 , Jan 23, 2011
Yesterday evening I looked at the ceiling and began to reason about the wooden panels, which reminded me of a chess board. Soon I came to these equivalences for fields on a (finite) dimensional chess board:

For field x with row/column (r,c), color(x) and #neighbour=(count of neighbours) you can have

a) equivalence by color(x)
b) equivalence by #neighbours
c) equivalence by integer function r+c
d) equivalence by rational r-1/c
e) ...

In relational calculus
a) to c) could be used as an index with multiples, cpo
d) produces unique numbering and could serve as a key

Happy reasoning,
Jens
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