TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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Every finite Boolean lattice has a symmetric chain decomposition

Statement

For every finite set $A$ , the Boolean lattice $B(A)$ can be partitioned into saturated chains whose least and greatest ranks sum to $|A|$ .

Facts & Assumptions

Given: A finite set $A$ .

[L1]

A symmetric chain decomposition of $B(X)$ lifts to one of $B(X\cup\{x\})$ whenever $x\notin X$ (A symmetric chain decomposition of one Boolean lattice lifts to the next Boolean lattice).

[L2]

The principle of induction on $\mathbb N$ (The principle of mathematical induction).

Proof

technique · induction

1.1

For $|A|=0$ , the Boolean lattice consists only of $\varnothing$ ; the one-term chain has endpoint ranks $0$ and $0$ , so it is symmetric.

base

1.2

Assume every Boolean lattice on an $n$ -element set has a symmetric chain decomposition, and let $A$ have $n+1$ elements. Choose $x\in A$ and put $X=A\setminus\{x\}$ , so $|X|=n$ .

ihL2choose

2.1

The induction hypothesis gives a symmetric chain decomposition of $B(X)$ , and [L1] lifts it to a symmetric chain decomposition of $B(X\cup\{x\})=B(A)$ .

step 1.2L1

3.1

The base case and induction step prove the assertion for every finite cardinality, hence for every finite set $A$ .

step 1.1step 2.1discharge-induction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 9 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

Symmetric chain decomposition background, Electronic Journal of Combinatorics (standard reference, not scraped)