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class_operators

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  • Jens-Uwe Grabowski
    Hello all, class operators for algebras (e.g. H...all homomorphic images or S...all subalgebras) and how they are related (e.g HS =SH) are well known. My
    Message 1 of 4 , Sep 14, 2000
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      Hello all,

      class operators for algebras (e.g. H...all homomorphic images or
      S...all subalgebras) and how they are related (e.g HS>=SH) are well known.

      My question: Are there such considerations for relational systems
      (or even first-order structures) including operators like e.g.
      formation of retracts, products, reduced (wrt. a filter over the index set)
      products, etc. ?

      Jens.

      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
      Jens-Uwe Grabowski email: grabo@...-dresden.de
      WWW: www.math.tu-dresden.de/~grabo

      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    • Matt Insall
      There definitely are some class operators that are studied. A trivial example is the Galois connection established by mapping a model class to its theory, and
      Message 2 of 4 , Sep 14, 2000
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        There definitely are some class operators that are studied. A trivial
        example is the Galois connection established by mapping a model class to
        its theory, and then mapping the theory to the class of its models. See
        any of the following texts:


        Model Theory, by Chang and Keisler

        Models and Ultraproducts by Bell and Slomson

        Handbook of Mathematical Logic by Barwise.

        One of these books names an operator after my major advisor. It is called
        the ``Kaiser Hull''. I think it is in the Barwise book.


        The class operators are interesting, but not as nicely behaved, as you can
        imagine, since relational logic is not equational.


        Matt Insall



        At 03:35 PM 9/14/00 +0200, you wrote:
        >Hello all,
        >
        >class operators for algebras (e.g. H...all homomorphic images or
        >S...all subalgebras) and how they are related (e.g HS>=SH) are well known.
        >
        >My question: Are there such considerations for relational systems
        >(or even first-order structures) including operators like e.g.
        >formation of retracts, products, reduced (wrt. a filter over the index set)
        >products, etc. ?
        >
        >Jens.
        >
        >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        >Jens-Uwe Grabowski email: grabo@...-dresden.de
        > WWW: www.math.tu-dresden.de/~grabo
        >
        >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        >
        >
      • mhebert
        There are many papers related to this subject, often in the context of what was called preservation theorems in model theory (because what one was looking
        Message 3 of 4 , Sep 14, 2000
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          There are many papers related to this subject, often in the context of what
          was called "preservation theorems" in model theory (because what one was
          looking for was a "syntactic characterization"(a "Birkhoff's HSP type"
          theorem). Also, there is a Category Theory version, often in the context of
          the so-called locally presentable categories.
          For the first trend, to Chang-Keisler book you may addthe recent
          Hodges, W., Model Theory,
          and consult the MathSciNet under the classification 03C40.
          For the Category Theory trend,
          Adamek, J., Rosicky,J., Accessible and locally presentable
          categories,Cambridge Un. Press 1994.
          and its bibliography is a good reference. Classifications 18C10, 18A30, 18A32,
          18A35 can be good places to search.

          Michel Hebert


          >===== Original Message From univalg@egroups.com =====
          >Hello all,
          >
          >class operators for algebras (e.g. H...all homomorphic images or
          >S...all subalgebras) and how they are related (e.g HS>=SH) are well known.
          >
          >My question: Are there such considerations for relational systems
          >(or even first-order structures) including operators like e.g.
          >formation of retracts, products, reduced (wrt. a filter over the index set)
          >products, etc. ?
          >
          >Jens.
          >
          >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
          >Jens-Uwe Grabowski email: grabo@...-dresden.de
          > WWW: www.math.tu-dresden.de/~grabo
          >
          >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        • Rob.Goldblatt@MCS.VUW.AC.NZ
          There is a duality between arbitrary relational structures and certain Boolean algebras with operators (BAO s). It uses special constructions on relational
          Message 4 of 4 , Sep 17, 2000
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            There is a duality between arbitrary relational structures and
            certain Boolean algebras with operators (BAO's). It uses special
            constructions on relational structures called

            inner substructures, bounded morphisms, disjoint unions

            which are dual to H,S, and P for BAO's. There are also "bounded
            unions" of structures which are dual to subdirect products, and
            "canonical extensions" of structures which are a kind of Stone-space
            construction.

            I have developed a calculus of such class operations and used it to
            formulate various interesting properties, such as the fact that
            bounded unions commute with ultraproducts of structures, and
            canonical extension are bounded morphic images of ultrapowers etc.

            Discussion of these class operations for relational structures can be
            found in the papers of mine listed below.

            cheers,
            Rob Goldblatt


            @Article{
            gold:elem95,
            author = "Robert Goldblatt",
            title = "{Elementary Generation and Canonicity for Varieties of
            Boolean Algebras with Operators}",
            journal = "Algebra Universalis",
            year = "1995",
            volume = "34",
            pages = "551--607"
            }


            @Article{
            gold:alge00,
            author = "Robert Goldblatt",
            title = "{Algebraic Polymodal Logic}",
            journal = "Logic Journal of the IGPL,
            \mbox{\rm Special Issue on
            Algebraic Logic edited by Istv{\'a}n N{\'e}meti and
            Ildik{\'o} Sain}",
            volume="8",
            number="4",
            publisher="Oxford University Press",
            pages="391--448",
            year="2000",
            month="July",
            note="Electronically available at: {\tt
            http://www3.oup.co.uk/igpl/Volume_08/Issue_04/ }"
            }

            @Article{
            gold:vari89,
            author = "Robert Goldblatt",
            title = "{Varieties of Complex Algebras}",
            journal = "Annals of Pure and Applied Logic",
            year = "1989",
            volume = "44",
            pages = "173--242"
            }


            >Hello all,
            >
            >class operators for algebras (e.g. H...all homomorphic images or
            >S...all subalgebras) and how they are related (e.g HS>=SH) are well known.
            >
            >My question: Are there such considerations for relational systems
            >(or even first-order structures) including operators like e.g.
            >formation of retracts, products, reduced (wrt. a filter over the index set)
            >products, etc. ?
            >
            >Jens.
            >
            >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            >Jens-Uwe Grabowski email: grabo@...-dresden.de
            > WWW: www.math.tu-dresden.de/~grabo
            >
            >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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