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A small terminology question about lattice congruences

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  • Gejza Jenca
    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
    • George Gratzer
      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

      • George Gratzer
        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

        • Insall, Matt
          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

          • 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
            • Insall, Matt
              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


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

                Yahoo! Groups Links
              • Ralph Freese
                ... 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
                  >
                  >
                  > ------------------------------------
                  >
                  > Yahoo! Groups Links
                  >
                  >
                  >
                  >
                • George Gratzer
                  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
                    >
                    >
                    > ------------------------------------
                    >
                    > Yahoo! Groups Links
                    >
                    >
                    >
                    >


                  • Gejza Jenca
                    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
                    • 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 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
                      • 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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