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Mikito Harakiri wrote:
> 3. Homomorphisms into boolean algebras:
> X -> X /\ 00
> X -> X \/ 00
> X -> X /\ 11
> X -> X \/ 11
> This informal condition can be rewritten into a set of formal axioms,
> e.g. restrictive cases of distributivity
> A/\11 /\ ( (B/\11) \/ (C/\11) ) == (A /\ B /\ 11) \/ (A /\ C /\ 11)
>
> 4. More restrictive cases of distributivty, that doesn't seem to follow
> from the above
> 00 /\ (A \/ B) = (00 /\ A) \/ (00 /\ B)
> 11 \/ (A /\ B) = (11 \/ A) /\ (11 \/ B)
What am I writing? #4 follows from homomorphism definition! However
00 \/ (A /\ B) != (00 \/ A) /\ (00 \/ B)
11 /\ (A \/ B) != (11 /\ A) \/ (11 /\ B)
Therefore,
X -> X /\ 11
and
X -> X \/ 00
are *not* homomorphisms. Sublattices X /\ 11 and X \/ 00 are still
boolean algebras, though.
Received on Tue Mar 28 2006 - 13:04:53 CST