## Algebras with a Cantor-Bernstein type theorem

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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
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@...
• 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,

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