ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The Boolean lattice on four elements: ranks, width, shadows, and a symmetric chain decomposition

Example

Let $A = {1, 2, 3, 4}$ . The Boolean lattice $B (A)$ has rank sizes

$1, 4, 6, 4, 1,$

so its width is $6$ . For $F = {{1, 2}, {1, 3}}$ , both its lower and upper shadows have three members:

$\partial F = {{1}, {2}, {3}}, \nabla F = {{1, 2, 3}, {1, 2, 4}, {1, 3, 4}} .$

Facts & Assumptions

Given: The set $A = {1, 2, 3, 4}$ and the family $F$ in the Example.

[F1]

The rank- $k$ level of $B (A)$ is the family of $k$ -subsets (The Boolean lattice of subsets of a finite set and its rank levels).

[F2]

Lower and upper shadows consist of the immediate subsets and supersets one rank away (The lower and upper shadows of a uniform set family).

[L1]

Sperner's theorem says the width of $B (A)$ is its middle binomial coefficient (Sperner's theorem and its equality cases: a largest antichain is a complete middle level).

[L2]

Every finite Boolean lattice has a symmetric chain decomposition (Every finite Boolean lattice has a symmetric chain decomposition).

Verification

technique · direct

1.1

Listing subsets by cardinality gives rank sizes $1, 4, 6, 4, 1$ , and [L1] gives width $6$ .

givenF1L1

1.2

Deleting one element from a member of $F$ gives exactly ${1}, {2}, {3}$ , while adjoining one element gives exactly $123, 124, 134$ . Thus the displayed shadows are correct.

givenF2

1.3

The following symmetric chains partition all sixteen subsets: $\emptyset \subset 1 \subset 12 \subset 123 \subset 1234$ ; $4 \subset 14 \subset 124$ ; $2 \subset 23 \subset 234$ ; $24$ ; $3 \subset 13 \subset 134$ ; and $34$ . Their endpoint ranks sum to $4$ , in agreement with [L2].

F1L2

2.1

Steps 1.1, 1.2, and 1.3 verify the ranks, width, shadows, and an explicit symmetric chain decomposition.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Keller and W. T. Trotter, Applied Combinatorics, §6.2 (standard reference, not scraped)