How to check distributivity of a finite lattice
Victor Porton
check whether it is distributive?
Should one (or a program) to check all combinations of elements?
Should I write a program which does this?
Particularly I'm interested to know distributivity the following
lattice which William Elliot proposed as the solution of my problem
1
|
a
/ \
x y
\ /
0
See http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice
for the updated version of the problem which requires the involved
lattices to be distributive.
If the above William Elliot's lattice is distributive then it solves
the renewed problem. If it is not distributive we should search for an
other example for my conjecture. Is it distributive?
Mariano Suárez-Alvarez
> When we have a finite lattice expressed with a Hasse diagram, how to
> check whether it is distributive?
>
> Should one (or a program) to check all combinations of elements?
>
> Should I write a program which does this?
>
> Particularly I'm interested to know distributivity the following
> lattice which William Elliot proposed as the solution of my problem
> 1
> |
> a
> / \
> x y
> \ /
> 0
>
> for the updated version of the problem which requires the involved
> lattices to be distributive.
>
> If the above William Elliot's lattice is distributive then it solves
> the renewed problem. If it is not distributive we should search for an
> other example for my conjecture. Is it distributive?
A lattice is distributive iff it does not contain
as sublattices the two lattices you'll see pictured
at <http://en.wikipedia.org/wiki/
Distributive_lattice#Characteristic_properties>
In particular, every 5-element lattice which is not one of
those two is distributive.
-- m
victor_me...@yahoo.co.uk
> When we have a finite lattice expressed with a Hasse diagram, how to
> check whether it is distributive?
Use the finite form of Stone's theorem. It is distributive
iff it's isomorphic to the lattice of down-sets in the lattice
of its join-irreducible elements.
> Particularly I'm interested to know distributivity the following
> lattice which William Elliot proposed as the solution of my problem
> 1
> |
> a
> / \
> x y
> \ /
> 0
> If the above William Elliot's lattice is distributive then it solves
> the renewed problem. If it is not distributive we should search for an
> other example for my conjecture. Is it distributive?
Yes. It's isomorphic to a lattice of sets via
0 -> {}
x -> {1}
y -> {2}
a -> {1,2}
1 -> {1,2,3}.
Do you still want your Abel prize?
Victor Meldrew
"I don't believe it!"
Victor Porton
> On 12 Sep, 14:37, Victor Porton <por...@narod.ru> wrote:
> Do you still want your Abel prize?
Yes, and 15 September has not yet passed, so that you can nominate my
works for Abel Prize:
http://www.mathematics21.org/abel-prize.html
My works (see http://www.mathematics21.org/algebraic-general-topology.html)
are great even despite of some my ignorance in lattice theory such as
not remembering the criterion for a lattice to be distributive:
http://en.wikipedia.org/wiki/Distributive_lattice#Characteristic_properties