# Re: More on lists and sets

Date: 28 Mar 2006 11:04:53 -0800

Message-ID: <1143572693.169876.60560_at_j33g2000cwa.googlegroups.com>

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 - 21:04:53 CEST