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

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The diamond $M_{3}$ and pentagon $N_{5}$ violate distributivity by explicit joins and meets

Statement refuted

Every finite lattice is distributive.

Facts & Assumptions

Given: The diamond $M_{3} = {0, 1, a, b, c}$ , where $a, b, c$ are incomparable atoms, and the pentagon $N_{5} = {0, a, b, c, 1}$ , where $0 < a < b < 1$ , $0 < c < 1$ , and $c$ is incomparable with $a, b$ .

[F1]

Distributivity requires $x \land (y \lor z) = (x \land y) \lor (x \land z)$ for all elements (Lattices, distributive lattices, and order ideals).

Counterexample

technique · direct

1.1

In $M_{3}$ , one has $b \lor c = 1$ , $a \land b = 0$ , and $a \land c = 0$ . Therefore $a \land (b \lor c) = a$ , while $(a \land b) \lor (a \land c) = 0$ .

givenF1

1.2

In $N_{5}$ , one has $a \lor c = 1$ , $b \land a = a$ , and $b \land c = 0$ . Therefore $b \land (a \lor c) = b$ , while $(b \land a) \lor (b \land c) = a$ .

givenF1

2.1

Since $a \neq 0$ in $M_{3}$ and $a \neq b$ in $N_{5}$ , each lattice violates the distributive identity in [F1]. Both are finite, so either one refutes the Statement.

step 1.1step 1.2F1∎

Remarks


\node[font=\normalsize] at (2.8,3.35) {$N_5$}; \node[elt] (n0) at (2.2,0) {$0$}; \node[elt] (na) at (1.5,.95) {$a$}; \node[elt] (nb) at (1.5,2.05) {$b$}; \node[elt] (nc) at (4.1,1.45) {$c$}; \node[elt] (n1) at (2.2,2.9) {$1$}; \draw[gray!75] (n0)--(na)--(nb)--(n1) (n0)--(nc)--(n1); \end{tikzpicture} ```

Depends on

Lattices, distributive lattices, and order ideals

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

MIT OpenCourseWare 18.212, Lecture 16: Distributive lattices (standard reference, not scraped)

The diamond M3M3​ and pentagon N5N5​ violate distributivity by explicit joins and meets