4
$\begingroup$

Let $(L,\wedge,\vee)$ be a finite distributive lattice, and let $1$ its greatest element.
An element $a \in L$ is called maximal if $a \le a' < 1$ implies $a = a'$.
Let $b$ be the meet of all the maximal elements, then $[b,1]$ is called the top interval of $L$.
A boolean lattice ($B_n$) is the subsets lattice of a set (of $n$ elements); for example $B_3$ is the following:
enter image description here

Question: Is the top interval of a finite distributive lattice, a boolean lattice?

$\endgroup$

1 Answer 1

10
$\begingroup$

Any finite distributive lattice whose smallest element is the meet of its maximal elements is a boolean algebra (= hypercube lattice), so the answer to your question is yes. The proof follows easily, for instance, from the characterization (up to isomorphism) of finite distributive lattices as order ideals of finite posets. See Enumerative Combinatorics, vol. 1, second ed., items a-i on pages 254-255.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.