Atomic vs. atomistic
Victor Porton
http://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined
is written:
"Such an algebra can be defined equivalently as a complete Boolean
algebra that is atomic, meaning that every element is a sup of some
set of atoms."
However in this Wikipedia page:
http://en.wikipedia.org/wiki/Lattice_(order)
is written:
* Atomic if for every nonzero element x of L, there exists an atom a
of L such that a ≤ x ;
* Atomistic if every element of L is a supremum of atoms. That is, for
all a, b in L such that a\nleq b, there exists an atom x of L such
that x\leq a and x\nleq b.
Thus, accordingly the latter, the Boolean_algebras_canonically_defined
should speak about "atomistic" rather than about "atomic" lattices.
Where is the error? and what is the correct usage of terms?
Virgil
<efede2b0-f7da-44da...@g26g2000yqe.googlegroups.com>,
Victor Porton <por...@narod.ru> wrote:
Shouldn't the issue of "Atomic versus atomistic" be brought up by
Victor PROTON rather than Victor Porton?
William Elliot
> "Such an algebra can be defined equivalently as a complete Boolean
> algebra that is atomic, meaning that every element is a sup of some
> set of atoms."
>
> * Atomic if for every nonzero element x of L, there exists an atom a
> of L such that a G�~n x ;
What, in simple text, is G^I~n x?
> * Atomistic if every element of L is a supremum of atoms. That is, for
> all a, b in L such that a\nleq b, there exists an atom x of L such
> that x\leq a and x\nleq b.
>
> Thus, accordingly the latter, the Boolean_algebras_canonically_defined
> should speak about "atomistic" rather than about "atomic" lattices.
>
> Where is the error? and what is the correct usage of terms?
>
The error is in using Tex and other special characters.
Butch Malahide
Why do you think there is an error? Your first quotation speaks of
complete Boolean algebras. In the special case of complete Boolean
algebras, "atomic" is equivalent to "atomistic". So what?
Victor Porton
It is a valuable note that <<In the special case of complete Boolean
algebras, "atomic" is equivalent to "atomistic">>. I was not knowing
that.
However it is said without a proof. This is a problem and should be
corrected in Wikipedia.
Where I can read about complete atomic boolean algebras _with proofs_?
Butch Malahide
>
> It is a valuable note that <<In the special case of complete Boolean
> algebras, "atomic" is equivalent to "atomistic">>.
Did I say that? I'm sorry, I misspoke. Actually, completeness has
nothing to do with it: every atomic Boolean algebra is "atomistic".
*Complete* atomic Boolean algebras are much more special: every such
algebra is isomorphic to the algebra of all subsets of a set, namely
its set of atoms.
> I was not knowing that.
But now you do. If you have not already verified it, I recommend it to
you as an easy exercise.
> However it is said without a proof. This is a problem and should be
> corrected in Wikipedia.
It seems to me that encyclopedias usually contain few or no proofs.
> Where I can read about complete atomic boolean algebras _with proofs_?
In a good book on Boolean algebras or lattice theory?
porky_...@my-deja.com
> In article
> <efede2b0-f7da-44da-9b2f-58f2d02f7...@g26g2000yqe.googlegroups.com>,
> Victor Porton <por...@narod.ru> wrote:
>
> Shouldn't the issue of "Atomic versus atomistic" be brought up by
> Victor PROTON rather than Victor Porton?
what if he collides with Archimedes PLUTONIUM of atomic (or should I
say "atomistic") totality fame?