<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="//commons.wikimedia.org/wiki/File:Hypercubestar_binary.svg" class="image"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Hypercubestar_binary.svg/220px-Hypercubestar_binary.svg.png" decoding="async" width="220" height="206" class="thumbimage" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Hypercubestar_binary.svg/330px-Hypercubestar_binary.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Hypercubestar_binary.svg/440px-Hypercubestar_binary.svg.png 2x" data-file-width="373" data-file-height="350"></a>  <div class="thumbcaption"><div class="magnify"><a href="//commons.wikimedia.org/wiki/File:Hypercubestar_binary.svg" class="internal" title="Enlarge"></a></div>Tesseract</div></div></div>
<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="//commons.wikimedia.org/wiki/File:Weak_orderings_4.svg" class="image"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Weak_orderings_4.svg/220px-Weak_orderings_4.svg.png" decoding="async" width="220" height="220" class="thumbimage" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Weak_orderings_4.svg/330px-Weak_orderings_4.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Weak_orderings_4.svg/440px-Weak_orderings_4.svg.png 2x" data-file-width="695" data-file-height="694"></a>  <div class="thumbcaption"><div class="magnify"><a href="//commons.wikimedia.org/wiki/File:Weak_orderings_4.svg" class="internal" title="Enlarge"></a></div>The 75 ordered set partitions</div></div></div>
<p>Implications between the <span style="white-space:nowrap"><a href="https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="Sloane's"><img alt="Sloane's" src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/11px-OEISicon_light.svg.png" decoding="async" width="11" height="15" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/17px-OEISicon_light.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/22px-OEISicon_light.svg.png 2x" data-file-width="409" data-file-height="556"></a> <a href="//oeis.org/A000629" class="extiw" title="oeis:A000629">A000629</a></span>(4) = 150 statements with 4 variables in <a href="https://en.wikipedia.org/wiki/first-order_logic" class="extiw" title="w:first-order logic">first-order logic</a>
</p><p>The formulas are represented by <a href="https://en.wikipedia.org/wiki/Weak_ordering" class="extiw" title="w:Weak ordering">ordered set partitions</a>. Black stands for the universal <a href="https://en.wikipedia.org/wiki/Quantification" class="extiw" title="w:Quantification">quantifier</a>, red for the existential one.
</p><p>The basic structure of this Graph is like the <a href="https://en.wikipedia.org/wiki/Tesseract" class="extiw" title="w:Tesseract">tesseract</a> shown on the right.<br>
The eight little cubes correspond to tesseract vertices with rank 1 and 3, the six <a href="https://en.wikipedia.org/wiki/Rhombic_dodecahedron" class="extiw" title="w:Rhombic dodecahedron">rhombic dodecahedra</a> to those with rank 2.
</p><p><br><b>Examples:</b>
</p>
<table class="wikitable"><tbody><tr><th style="width:60px;">matrix</th>
        <th style="width:120px;">abbreviation</th>
        <th style="width:230px;">formula</th>
    </tr><tr style="text-align:center; height:40px;"><td><a href="//commons.wikimedia.org/wiki/File:Predicate_logic;_4_variables;_ordered_partition_a3e2a4e1.svg" class="image"><img alt="Predicate logic; 4 variables; ordered partition a3e2a4e1.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Predicate_logic%3B_4_variables%3B_ordered_partition_a3e2a4e1.svg/40px-Predicate_logic%3B_4_variables%3B_ordered_partition_a3e2a4e1.svg.png" decoding="async" width="40" height="40" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Predicate_logic%3B_4_variables%3B_ordered_partition_a3e2a4e1.svg/60px-Predicate_logic%3B_4_variables%3B_ordered_partition_a3e2a4e1.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Predicate_logic%3B_4_variables%3B_ordered_partition_a3e2a4e1.svg/80px-Predicate_logic%3B_4_variables%3B_ordered_partition_a3e2a4e1.svg.png 2x" data-file-width="243" data-file-height="243"></a></td>
        <td>a3 e2 a4 e1</td>
        <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall y~\exists x~\forall z~\exists w~Pwxyz}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi mathvariant="normal">∀<!-- ∀ --></mi><mi>y</mi><mtext> </mtext><mi mathvariant="normal">∃<!-- ∃ --></mi><mi>x</mi><mtext> </mtext><mi mathvariant="normal">∀<!-- ∀ --></mi><mi>z</mi><mtext> </mtext><mi mathvariant="normal">∃<!-- ∃ --></mi><mi>w</mi><mtext> </mtext><mi>P</mi><mi>w</mi><mi>x</mi><mi>y</mi><mi>z</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \forall y~\exists x~\forall z~\exists w~Pwxyz}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ce542c6192a5af3ec9d5fc1fc6184256ecf691" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:19.713ex; height:2.509ex;" alt="\forall y~\exists x~\forall z~\exists w~Pwxyz"></span></td>
    </tr><tr style="text-align:center; height:40px;"><td><a href="//commons.wikimedia.org/wiki/File:Predicate_logic;_4_variables;_ordered_partition_e1a3e24.svg" class="image"><img alt="Predicate logic; 4 variables; ordered partition e1a3e24.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Predicate_logic%3B_4_variables%3B_ordered_partition_e1a3e24.svg/40px-Predicate_logic%3B_4_variables%3B_ordered_partition_e1a3e24.svg.png" decoding="async" width="40" height="31" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Predicate_logic%3B_4_variables%3B_ordered_partition_e1a3e24.svg/60px-Predicate_logic%3B_4_variables%3B_ordered_partition_e1a3e24.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Predicate_logic%3B_4_variables%3B_ordered_partition_e1a3e24.svg/80px-Predicate_logic%3B_4_variables%3B_ordered_partition_e1a3e24.svg.png 2x" data-file-width="243" data-file-height="190"></a></td>
        <td>e1 a3 e24</td>
        <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists w~\forall y~\exists x~\exists z~Pwxyz}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi mathvariant="normal">∃<!-- ∃ --></mi><mi>w</mi><mtext> </mtext><mi mathvariant="normal">∀<!-- ∀ --></mi><mi>y</mi><mtext> </mtext><mi mathvariant="normal">∃<!-- ∃ --></mi><mi>x</mi><mtext> </mtext><mi mathvariant="normal">∃<!-- ∃ --></mi><mi>z</mi><mtext> </mtext><mi>P</mi><mi>w</mi><mi>x</mi><mi>y</mi><mi>z</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \exists w~\forall y~\exists x~\exists z~Pwxyz}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9bc331929518fcef65ef5810f7f1644a2430d6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:19.713ex; height:2.509ex;" alt="\exists w~\forall y~\exists x~\exists z~Pwxyz"></span></td>
    </tr><tr style="text-align:center; height:40px;"><td><a href="//commons.wikimedia.org/wiki/File:Predicate_logic;_4_variables;_ordered_partition_a1e3a24.svg" class="image"><img alt="Predicate logic; 4 variables; ordered partition a1e3a24.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/54/Predicate_logic%3B_4_variables%3B_ordered_partition_a1e3a24.svg/40px-Predicate_logic%3B_4_variables%3B_ordered_partition_a1e3a24.svg.png" decoding="async" width="40" height="31" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/5/54/Predicate_logic%3B_4_variables%3B_ordered_partition_a1e3a24.svg/60px-Predicate_logic%3B_4_variables%3B_ordered_partition_a1e3a24.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/5/54/Predicate_logic%3B_4_variables%3B_ordered_partition_a1e3a24.svg/80px-Predicate_logic%3B_4_variables%3B_ordered_partition_a1e3a24.svg.png 2x" data-file-width="243" data-file-height="190"></a></td>
        <td>a1 e3 a24</td>
        <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall w~\exists y~\forall x~\forall z~Pwxyz}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi mathvariant="normal">∀<!-- ∀ --></mi><mi>w</mi><mtext> </mtext><mi mathvariant="normal">∃<!-- ∃ --></mi><mi>y</mi><mtext> </mtext><mi mathvariant="normal">∀<!-- ∀ --></mi><mi>x</mi><mtext> </mtext><mi mathvariant="normal">∀<!-- ∀ --></mi><mi>z</mi><mtext> </mtext><mi>P</mi><mi>w</mi><mi>x</mi><mi>y</mi><mi>z</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \forall w~\exists y~\forall x~\forall z~Pwxyz}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6678a42f09de8b0d691a5d04e1cc2f01ac7d7d44" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:19.713ex; height:2.509ex;" alt="\forall w~\exists y~\forall x~\forall z~Pwxyz"></span></td>
    </tr><tr style="text-align:center; height:40px;"><td><a href="//commons.wikimedia.org/wiki/File:Predicate_logic;_4_variables;_ordered_partition_e3a124.svg" class="image"><img alt="Predicate logic; 4 variables; ordered partition e3a124.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/24/Predicate_logic%3B_4_variables%3B_ordered_partition_e3a124.svg/40px-Predicate_logic%3B_4_variables%3B_ordered_partition_e3a124.svg.png" decoding="async" width="40" height="23" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/2/24/Predicate_logic%3B_4_variables%3B_ordered_partition_e3a124.svg/60px-Predicate_logic%3B_4_variables%3B_ordered_partition_e3a124.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/2/24/Predicate_logic%3B_4_variables%3B_ordered_partition_e3a124.svg/80px-Predicate_logic%3B_4_variables%3B_ordered_partition_e3a124.svg.png 2x" data-file-width="243" data-file-height="138"></a></td>
        <td>e3 a124</td>
        <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists y~\forall w~\forall x~\forall z~Pwxyz}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi mathvariant="normal">∃<!-- ∃ --></mi><mi>y</mi><mtext> </mtext><mi mathvariant="normal">∀<!-- ∀ --></mi><mi>w</mi><mtext> </mtext><mi mathvariant="normal">∀<!-- ∀ --></mi><mi>x</mi><mtext> </mtext><mi mathvariant="normal">∀<!-- ∀ --></mi><mi>z</mi><mtext> </mtext><mi>P</mi><mi>w</mi><mi>x</mi><mi>y</mi><mi>z</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \exists y~\forall w~\forall x~\forall z~Pwxyz}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0506f94d10c6e4769270c0166573327fe5e37db8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:19.713ex; height:2.509ex;" alt="\exists y~\forall w~\forall x~\forall z~Pwxyz"></span></td>
    </tr></tbody></table><p><br></p>
<div class="thumb tleft"><div class="thumbinner" style="width:222px;"><a href="//commons.wikimedia.org/wiki/File:Concertina_tesseract;_vertices_1100.svg" class="image"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Concertina_tesseract%3B_vertices_1100.svg/220px-Concertina_tesseract%3B_vertices_1100.svg.png" decoding="async" width="220" height="260" class="thumbimage" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Concertina_tesseract%3B_vertices_1100.svg/330px-Concertina_tesseract%3B_vertices_1100.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Concertina_tesseract%3B_vertices_1100.svg/440px-Concertina_tesseract%3B_vertices_1100.svg.png 2x" data-file-width="246" data-file-height="291"></a>  <div class="thumbcaption"><div class="magnify"><a href="//commons.wikimedia.org/wiki/File:Concertina_tesseract;_vertices_1100.svg" class="internal" title="Enlarge"></a></div>The concertina tesseract is not a <a href="https://en.wikipedia.org/wiki/lattice_(order)" class="extiw" title="w:lattice (order)">lattice</a>. An example of two vertices with no unique <a href="https://en.wikipedia.org/wiki/Join_and_meet" class="extiw" title="w:Join and meet">join</a> above them are the two right above the lowest vertex in this rhombic dodecahedron.</div></div></div>
<table><tbody><tr><td style="padding: 15px;"><a href="//commons.wikimedia.org/wiki/File:Watchduck.svg" class="image"><img alt="Watchduck.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Watchduck.svg/40px-Watchduck.svg.png" decoding="async" width="40" height="46" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Watchduck.svg/60px-Watchduck.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Watchduck.svg/80px-Watchduck.svg.png 2x" data-file-width="703" data-file-height="806"></a>
</td>
<td><a href="//commons.wikimedia.org/wiki/User:Watchduck" title="User:Watchduck">Watchduck</a><br><span style="color: gray; font-size: 80%;">You can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck".</span>
</td></tr></tbody></table>
CC BY 3.0
<span class="int-own-work" lang="en">Own work</span>