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Algebras with a Cantor-Bernstein type theorem

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  • Gejza Jenca
    It is well-known, that in the class of sigma-complete Boolean algebras we have (1) A x B x C = A implies that A x B = A, where x stands for the direct product
    Message 1 of 2 , May 6, 2002
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      It is well-known, that in the class of sigma-complete
      Boolean algebras we have

      (1) A x B x C = A implies that A x B = A,

      where x stands for the direct product and = stands
      for the isomorphism of Boolean algebras. It is
      easy to check that (1) implies that,

      (2) D x F = E and E x G = D => D = E.

      Recently, I proved that (1) [and thus (2)]
      is valid for sigma-complete effect algebras
      (to appear in Alg. Univ.).

      My questions : are there some other classes of algebras
      satisfying (1) ? In particular, is (1) valid for
      sigma-complete lattices or some class of
      groups/rings/modules ?
      Is there anybody interested in such questions ?

      --
      Gejza Jenca, PhD.
      Department of Mathematics
      Faculty of Electrical Engineering and Information Technology
      Slovak University of Technology
      Ilkovicova 3
      SK-812 19 Bratislava
      SLOVAKIA

      E-mail : jenca@...
    • Friedrich Wehrung
      Dear Gejza, about your question on Cantor-Bernstein type theorems, ... %%%%%%%%%%%%%%%%% Antisymmetry of order is valid in a very general class of algebras,
      Message 2 of 2 , May 7, 2002
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        Dear Gejza,

        about your question on Cantor-Bernstein type theorems,

        %%%%%%%%%%%%%%%%%
        >It is well-known, that in the class of sigma-complete
        >Boolean algebras we have
        >
        >(1) A x B x C = A implies that A x B = A,
        >
        >where x stands for the direct product and = stands
        >for the isomorphism of Boolean algebras. It is
        >easy to check that (1) implies that,
        >
        >(2) D x F = E and E x G = D => D = E.
        >
        >Recently, I proved that (1) [and thus (2)]
        >is valid for sigma-complete effect algebras
        >(to appear in Alg. Univ.).
        >
        >My questions : are there some other classes of algebras
        >satisfying (1) ? In particular, is (1) valid for
        >sigma-complete lattices or some class of
        >groups/rings/modules ?
        >Is there anybody interested in such questions ?
        %%%%%%%%%%%%%%%%%

        Antisymmetry of order is valid in a very general class of algebras, called
        "cardinal algebras". The reference is Tarski's book, "Cardinal Algebras",
        Oxford University Press, 1949. It is of course out of print.

        A cardinal algebra is a commutative monoid, endowed with an infinitary
        addition (defined on sequences of elements), subjected to very natural
        axioms.

        Isomorphism types of $\sigma$-complete BA's form a cardinal algebra.

        About $\sigma$-complete lattices I am not sure, some continuity conditions
        may be necessary, more or less related to completeness of the center (see
        my paper on decompositions of non-algebraic complete lattices, can be
        downloaded from my URL below), but it is probably easy to check.

        About modules, yes, there are results, for example, right self-injective
        modules over regular rings, see K.R. Goodearl's book "von Neumann Regular
        Rings".

        I am not sure whether isomorphism types of $\sigma$-complete effect
        algebras form a cardinal algebra, but this may be worthwhile checking.

        Best regards,

        Friedrich WEHRUNG (office S3 114)
        CNRS, UMR 6139
        Universit\'e de Caen, Campus 2
        D\'epartement de Math\'ematiques, B.P. 5186
        14032 Caen cedex
        FRANCE

        tel-fax H (+33) (0) 2 31 06 02 23
        tel O (+33) (0) 2 31 56 74 29
        tel. secr. math. (+33) (0) 2 31 56 73 22
        fax secr. math. (+33) (0) 2 31 56 73 20

        E-mail wehrung@...
        URL http://www.math.unicaen.fr/~wehrung
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