# complemented lattice

## Primary tabs

Defines:
related elements in lattice, complement, complemented element
Synonym:
perspective elements, complemented
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

### Lattices and power sets

Hi!
A small question:

Which additional properties are necessary/sufficient to make a lattice $(L, <)$ isomorphic to a power set (\Pot(M), \subsetneq)?

Ok, the following properties are necessary:
- complete (every subset has an infimum and a supremum)
- complemented
- distributive

But what set of properties is sufficient?

### Re: Lattices and power sets

A complemented distributive lattice is a Boolean algebra. A complete Boolean algebra A is lattice isomorphic to the power set of some set S if A is atomic (every element of A has an atom below it). You can show that S is in fact the set of all atoms of A. Check out the book Intro to Lattices and Order by Davey and Priestley.

### Re: Lattices and power sets

Ok, so this would mean, a lattice that is
- complete and
- complemented and
- distributive and
- atomic
is order-isomorphic to a power set. Any such isomorphism will identify the atoms of the lattice with the singletons of the power set.

Are all the four requirements are necessary? Don't have the book at hand...

### Re: Lattices and power sets

I think so. I saw a counter-example in the book if, say, the atomic condition is dropped. I'll find out.

### Re: Lattices and power sets

I found the counter-example:

Consider the power set of the reals R. Consider the Boolean subalgebra A of R generated by the following intervals

(-oo, a)
[b,c}
[d,oo)
empty set

where a,b,c,d are reals.

Then A is complete and atomless, and A is not lattice isomorphic to any power set of a set.

### Re: Lattices and power sets

Hi, thanks, but..
Sorry I don't get it!

Do you mean a < b < c < d ?
Do you mean A \subseteq R or A \subseteq \Pot(R) ?
With (-oo, a), [b, c), [d, oo) \in A ?
or (-oo, a), [b, c), [d, oo) \subseteq A ?
Maybe a sup-generated subset of the power set \Pot(R) ??

Regards, Schneemann.

### Re: Lattices and power sets

Sorry for the sloppiness. First, a, b, c, d are arbitrary real numbers, there are no specific orderings on these numbers. Second, the Boolean subalgebra A I am refering to is the set of all finite unions of those intervals that I mentioned. You can show that A is indeed a Boolean algebra under union and intersection and complementation, etc... Furthermore, it is complete. You can derive that A is atomless by using a proof of contradiction.

### Re: Lattices and power sets

At the beginning I thought you mean a, b, c, d to be fixed numbers. I'm still not sure, but now I think you mean this set:

S := { (oo,b) | b in RR }
cup { [a, b) | a, b in RR }
cup { [a,oo) | a in RR }
cup { emptyset }

Now A is generated from S by building finite unions.. right? Ok, I'll think about that. Thanks!

Regards. Schneemann