Embedding Boolean Algebras

From: Noel Vaillant (vaillant_at_probability.net)
Date: 10/28/04 

Date: 28 Oct 2004 06:14:06 -0700

>if a Boolean algebra E is isomorphic to a product FxG, and
>if F has at least one atom (say a), then F can be embedded
>in E as follows: define f:F->E by f(x)=(x,1) if a<=x
>and f(x)=(x,0) if a<=~x.

In fact, there is no need to assume the existence of an
atom in F, nor the case of a finite product: if E is
isomorphic to prod_{i in I} Ei, then each Ei can be
imbedded in E. Indeed, since E is isomorphic to
Ei x prod_{i<>j}Ej, without loss of generality we are
back to the finite case, i.e. E=FxG. To find an embedding
F->E, we consider J maximal (proper) ideal of F, and we
define q:F->G by setting q(x)= 0 if x in J and q(x)=1
otherwise. Then f:F->E defined by f(x)=(x,q(x)) is a
monomorphism.

Noel.

Relevant Pages

  • Re: Embedding Boolean Algebras
    ... Subject: Embedding Boolean Algebras ... The Boolean algebra of clopen subsets of Q, ... The existence of an atom, gives a coatom which generates a maximal ideal. ... any surjective map f:B -> A gives an embedding of Pinto P. ...
    (sci.math)
  • Re: Embedding Boolean Algebras
    ... Subject: Embedding Boolean Algebras ... as I seem to have remembered, when A proper subset B, ... >if F has at least one atom, ...
    (sci.math)