- 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,

%%%%%%%%%%%%%%%%%>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

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E-mail wehrung@...

URL http://www.math.unicaen.fr/~wehrung