False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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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False: every maximal antichain in a finite poset has maximum cardinality

Statement

Every maximal antichain in a finite poset has cardinality equal to the width.

Facts & Assumptions

Given: The three-element poset $P = {a, b, c}$ with $a < b$ , $a < c$ , and $b, c$ incomparable.

[F1]

A maximal antichain is inclusion-maximal, while a maximum antichain has greatest cardinality; the width is the cardinality of a maximum antichain (Antichains, chain covers, and antichain covers of a poset, Height and width of a nonempty finite poset).

Refutation

technique · direct

1.1

The singleton ${a}$ is an antichain, and it is maximal because both remaining elements $b$ and $c$ are comparable with $a$ .

givenF1

1.2

The set ${b, c}$ is an antichain of cardinality $2$ , so the width of $P$ is at least $2$ and ${a}$ is not maximum.

givenF1

2.1

Thus the finite poset $P$ has a maximal antichain that is not maximum, refuting the Statement.

step 1.1step 1.2∎

Depends on

Used by

A maximal antichain of size one in a finite poset of width two Counterexample

Dependency tree · two levels

7 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, §6.4 (standard reference, not scraped)