Loading ...
Sorry, an error occurred while loading the content.

Re: [univalg] Epimorphisms of finite modular lattices

Expand Messages
  • Ralph Freese
    First let me apologize for being so slow in responding---I ve been traveling. The paper is The variety of modular lattices is not generated by its finite
    Message 1 of 5 , Jul 28, 2012
    • 0 Attachment
      First let me apologize for being so slow in responding---I've been
      traveling.

      The paper is "The variety of modular lattices is not generated by its
      finite members," Trans AMS 255, 1979. The lattice is easy to describe: let
      F and F' be countable fields of characteristics p and q, and let L and L'
      be the lattice of subspaces of a 4 dimensional vectors space over F (resp.
      F'). The two dimensional intervals of both lattices are M_\omega and so we
      can glue (via Hall-Dilworth) an upper interval of L to a lower interval
      of L'. The result is a 5-generated, simple modular lattice M of length 6.
      It has the property that prime (covering) intervals "stay distributive" in
      extensions. "Stays distributive" is the property defined in the message of
      2012/6/17 below. The proof is somewhat technical.

      When a lattice L has a covering with this "stays distributive" property,
      then the embedding of L(\leq) = {(a,b) in L^2 : a \leq b} into L^2 is an
      epimorphism that is not onto.

      Talking with Keith Kearnes we realized that a nondesarguean
      projective plane (as a modular lattice) also has this "stays distributive"
      property, showing epimorphisms need not be onto in the category of finite
      modular lattices as well.

      The advantage of the lattice M above is that, while it is not a projective
      modular lattice, much of its structure can be pulled back through
      homomorphisms. Using this it is possible to show that there are elements s
      < t in FM(5) such that the interval [s,t] is distributive (and "stays
      distributive"). If you now take a larger generating set X and for each x
      not in the original 5 generators, let x' = (x \join s) \meet t, then these
      generate a sublattice that is the free distributive lattice. Thus every
      sublattice of a free distributive lattice can be embedded into a free
      modular lattice. (On he other hand distributive sublattices of free
      lattices are quite restricted; in particular at most countable.)

      The lattice M is finite dimensional but not in the variety generated by
      all finite modular lattices. Christian Herrmann has shown that the variety
      of modular lattices is not generated by its finite dimension members. So
      the varieties of modular lattices generated by the finite members, the
      finite dimension members, and all members are distinct.

      Ralph


      On Mon, 18 Jun 2012, Gejza Jenca wrote:

      > 2012/6/17 Ralph Freese <ralph@...>
      >>
      >>
      >>
      >>
      >> There is a countable, simple modular lattice L of dimension 4 such that if a
      >> is covered by b in L and f : L -> M is an embedding into a modular lattice
      >> M, then the interval [f(a), f(b)] is distributive. Using this one can show
      >> that epimorphisms are not onto for modular lattices (and for finite
      >> dimensional modular lattices). But I'm not sure about finite modular
      >> lattices.
      >>
      >> Ralph Freese
      >>
      >
      > Thank you.
      >
      > Could you please provide a reference for the fact you mentioned?
      >
      > --
      > Gejza Jenca
      >
      >
      > ------------------------------------
      >
      > Yahoo! Groups Links
      >
      >
      >
      >
    • Ralph Freese
      PS. I should have mentioned that this ability to control the equations within an entire interval in the embedding s target began with Bjarni Jonsson, who
      Message 2 of 5 , Jul 28, 2012
      • 0 Attachment
        PS. I should have mentioned that this ability to control the equations
        within an entire interval in the embedding's target began with Bjarni
        Jonsson, who showed that if a and b are elements of M_3, a covered by b,
        and f: M_3 to L (L modular), the [f(a),f(b)] is arguesian.

        Bjarni asked the question which distributive lattices can be embedded into
        a free modular lattice and this is the genesis of the results of the paper
        described below. And despite the results below, that question is still
        open.

        Ralph


        On Sat, 28 Jul 2012, Ralph Freese wrote:

        >
        > First let me apologize for being so slow in responding---I've been
        > traveling.
        >
        > The paper is "The variety of modular lattices is not generated by its
        > finite members," Trans AMS 255, 1979. The lattice is easy to describe: let
        > F and F' be countable fields of characteristics p and q, and let L and L'
        > be the lattice of subspaces of a 4 dimensional vectors space over F (resp.
        > F'). The two dimensional intervals of both lattices are M_\omega and so we
        > can glue (via Hall-Dilworth) an upper interval of L to a lower interval
        > of L'. The result is a 5-generated, simple modular lattice M of length 6.
        > It has the property that prime (covering) intervals "stay distributive" in
        > extensions. "Stays distributive" is the property defined in the message of
        > 2012/6/17 below. The proof is somewhat technical.
        >
        > When a lattice L has a covering with this "stays distributive" property,
        > then the embedding of L(\leq) = {(a,b) in L^2 : a \leq b} into L^2 is an
        > epimorphism that is not onto.
        >
        > Talking with Keith Kearnes we realized that a nondesarguean
        > projective plane (as a modular lattice) also has this "stays distributive"
        > property, showing epimorphisms need not be onto in the category of finite
        > modular lattices as well.
        >
        > The advantage of the lattice M above is that, while it is not a projective
        > modular lattice, much of its structure can be pulled back through
        > homomorphisms. Using this it is possible to show that there are elements s
        > < t in FM(5) such that the interval [s,t] is distributive (and "stays
        > distributive"). If you now take a larger generating set X and for each x
        > not in the original 5 generators, let x' = (x \join s) \meet t, then these
        > generate a sublattice that is the free distributive lattice. Thus every
        > sublattice of a free distributive lattice can be embedded into a free
        > modular lattice. (On he other hand distributive sublattices of free
        > lattices are quite restricted; in particular at most countable.)
        >
        > The lattice M is finite dimensional but not in the variety generated by
        > all finite modular lattices. Christian Herrmann has shown that the variety
        > of modular lattices is not generated by its finite dimension members. So
        > the varieties of modular lattices generated by the finite members, the
        > finite dimension members, and all members are distinct.
        >
        > Ralph
        >
        >
        > On Mon, 18 Jun 2012, Gejza Jenca wrote:
        >
        >> 2012/6/17 Ralph Freese <ralph@...>
        >>>
        >>>
        >>>
        >>>
        >>> There is a countable, simple modular lattice L of dimension 4 such that if a
        >>> is covered by b in L and f : L -> M is an embedding into a modular lattice
        >>> M, then the interval [f(a), f(b)] is distributive. Using this one can show
        >>> that epimorphisms are not onto for modular lattices (and for finite
        >>> dimensional modular lattices). But I'm not sure about finite modular
        >>> lattices.
        >>>
        >>> Ralph Freese
        >>>
        >>
        >> Thank you.
        >>
        >> Could you please provide a reference for the fact you mentioned?
        >>
        >> --
        >> Gejza Jenca
        >>
        >>
        >> ------------------------------------
        >>
        >> Yahoo! Groups Links
        >>
        >>
        >>
        >>
        >
        >
        > ------------------------------------
        >
        > Yahoo! Groups Links
        >
        >
        >
        >
      Your message has been successfully submitted and would be delivered to recipients shortly.