Fig. 2: FREE LATTICE EXTENSION OF COMPLEMENTED DISTRIBUTIVE LATTICE ON 30
(PL p. 146, Lattice Theory, Garett Birkhoff, on "Free distributive lattice". To Birkhoff, a
complemented distributive lattice (CDL) always has what is labeled (in this dictionary
) as a MAX and a MIN: elements not generated from "lattice atoms" (PL),
rather assigned. So the number of elements in such a CDL -- or the free lattice
drrived from it -- is always two more than our formulations. Thus, on this page,
Birkhoff writes f(3) = 20 for the free CDL on 3 atoms. Then, ignoring the MAX, MIN,
find 20 - 2 = 18, agreeing with what is found below.)
30 (RANK 7)
/\
/ |\
/ | \
/ | \
/ | \
/ | \
6 10 15 (RANK 6)
/ /|\ \
/ / | \ \
/ / | \ \
| / | \ \
| / | \ \
| / | \ |
| / | \ |
6
10 615 1015 (RANK 5)
/ \ |\ \ \__________
/ \ | \________\_________ |
/ \______|___________\_______ | |
/ | \ \| |
2 3 5 61015 (RANK 4)
/ \ /|\ /\
/ \ / | \ / \
/ \ / | \ / \
/ \ / | \ / \
/ \ / | \ / \
/ \ / | \ / \
/ \/ | \/ \
(23)
(25) (23) (35) (25)(35) (RANK 3)
| \ \ / | \ / / |
| \ \ / | \ / / |
| \ \ / | \ / / |
| \ \ / | \ / / |
| \ \/ | \/ / |
| \ /\ | /\ / |
| \/ \ | / \/ |
| /\ \ | / /\ |
| / \ \ | / / \ |
| / \ \ | / / \ |
| ___/______\___\_______|_______/_ _/ \ |
|| / \ \ | / \ |
|| / \___\_____|_____/______________\__ |
||/ \ | / \ | |
2 3 2 5 3 5 (RANK 2)
| | |
| | |
| | |
| | |
| | |
| | |
-------------------2 3 5 (RANK 1)----------------------
|
|
1 (RANK 0)
Please note the Rank-Ballot Distribution: 1, 3, 3, 4, 3, 3, 1 -- definitely not Binomial. (PL
free database.)