ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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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 $\mathcal F=\{\{1,2\},\{1,3\}\}$ , both its lower and upper shadows have three members:

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

Facts & Assumptions

Given: The set $A=\{1,2,3,4\}$ and the family $\mathcal F$ 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 $\mathcal F$ 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: $\varnothing\subset1\subset12\subset123\subset1234$ ; $4\subset14\subset124$ ; $2\subset23\subset234$ ; $24$ ; $3\subset13\subset134$ ; 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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