A typo in a statement demanding properties for a lattice to be a logic?
- From: "Peter van Emburg" <petervanemburg@xxxxxxxxx>
- Date: Sat, 10 Sep 2005 12:18:53 +0200
Dear lattice theorist (e.g. Marc Olschok),
from p. 281 in "Geometry and Meaning", by D. Widdows, who quotes V. S.
Varadarajan's "Geometry of Quantum Theory", I learned that:
-----------------------------------------------------------
One of the properties that a lattice L in which each element x has a unique
complement x' must have to represent a _logic_ is that for a_1, a_2, b in L:
[i] (a_1 <= a_2) ==> there is a b such that (b <= a_1') & (b /\ a_1 = a_2)
-----------------------------------------------------------
But (b /\ a_1 = a_2) ==> (a_2 <= a_1 & a_2 <= b) and because the antecedent
is a_1 <= a_2 we see that only the case in which a_1 = a_2 could work,
excluding choice of any a_1 < a_2, which choice should be included.
On the other hand,
[ii] (a_1 <= a_2) ==> there is a b such that (b <= a_1') & (b \/ a_1 = a_2)
_does_ always hold because (b \/ a_1 = a_2) ==> (a_1 <= a_2 & b <= a_2) and
there is always a b in L such that a_1 <= b <= a_2.
Conclusion: I seem to have spotted a typo in Widdow's text ('/\' in [i]
should be '\/' as in [ii]). But I am still insufficiently confident
mathematically to suspect my reasoning to be flawed in some place or at
least unnecessarily inelegant. Can someone improve on it?
......................................................
Remark 1: I do not yet fully understand why the stated property is demanded
for L to be a logic.
Remark 2: that can be interpreted as defining a characteristic function.
Then the 1 of L is labeled 1111 and the 0 is labeled 0000. Next, select
elements a_1 and a_2 so that a_1 <= a_2, for example, a_1 = 0010, a_2 =
1011. Then the complement a_1' = 1101. If the b to be found must satisfy b
<= a_1' then b must be in the principal down-set of a_1', which is {0000,
0001, 0100, 1000, 0101, 1001, 1100, 1101}. You will find that there is
neither a b in there such that b /\ a_1 = a_2 nor one for which b /\ a_2 =
a_1. But there is a b satisfying b \/ a_1 = a_2, namely, b = 1001.
......................................................
Peter van Emburg,
Leiden.
.
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