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<title>Distributiver Verband</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Distributiver Verband</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <b>distributiver Verband</b> ist eine spezielle <a href="Struktur_(erste_Stufe)" title="Struktur (erste Stufe)">Struktur</a> der <a href="Mathematik" title="Mathematik">Mathematik</a>. Gegenüber allgemeinen <a href="Verband_(Mathematik)" title="Verband (Mathematik)">Verbänden</a>, in denen für die beiden (zweistelligen) Operationen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }">

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<annotation encoding="application/x-tex">{\displaystyle \vee }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b76220c6805c9b465d6efbc7686c624f49f3023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }">

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<annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1caa4004cb216ef2930bb12fe805a76870caed94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }" loading="lazy"></span> nur die <a href="Assoziativgesetz" title="Assoziativgesetz">Assoziativgesetze</a>, die <a href="Kommutativgesetz" title="Kommutativgesetz">Kommutativgesetze</a> und die <a href="Absorptionsgesetz_(Verbandstheorie)" class="mw-redirect" title="Absorptionsgesetz (Verbandstheorie)">Absorptionsgesetze</a> gefordert werden, gelten in einem distributiven Verband noch zusätzlich <a href="Distributivgesetz" title="Distributivgesetz">Distributivgesetze</a> <i>für beide Richtungen</i>.

</p><p>Die Gültigkeit der Distributivgesetze macht Verbände interessanter. Sie lassen sich einfacher untersuchen, da auftretende Terme sich leichter umformen lassen und es in gewissem Sinne einfache Darstellungen gibt. Dabei treten distributive Verbände sehr häufig auf, auch in Bereichen außerhalb der Mathematik. <a href="Boolesche_Algebra" title="Boolesche Algebra">Boolesche Algebren</a> sind spezielle distributive Verbände.

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<div class="mw-heading mw-heading2"><h2 id="Präzisierung"><span id="Pr.C3.A4zisierung"></span>Präzisierung</h2></div>

<p>Im Folgenden meinen wir mit dem Verband <i>V</i> stets den Verband <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(V,\vee ,\wedge \right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(V,\vee ,\wedge \right)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1647c443a9c210ecd8dac7a9c0db0171e27dd84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.765ex; height:2.843ex;" alt="{\displaystyle \left(V,\vee ,\wedge \right)}" loading="lazy"></span>.

</p><p>Ein Verband <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> heißt <b>distributiver Verband</b>, wenn für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in V}">

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<annotation encoding="application/x-tex">{\displaystyle a,b,c\in V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a66290044f65b599c4850dfbb9d0d1f1b671099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.93ex; height:2.509ex;" alt="{\displaystyle a,b,c\in V}" loading="lazy"></span> gilt:

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<annotation encoding="application/x-tex">{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9e48067e5f9e4745cc98498fc56e1819ef2eec7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.138ex; height:2.843ex;" alt="{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c)}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77c0c416c820fd1d6cdb6f0d7cee7f540bde5947.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.138ex; height:2.843ex;" alt="{\displaystyle a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)}" loading="lazy"></span>.</li></ul>

<p>Man kann jede der beiden Aussagen aus der anderen mit Hilfe der Verbandsaxiome ableiten.<sup id="cite_ref-AequivalenzD1D2_1-0" class="reference"><a href="#cite_note-AequivalenzD1D2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Daher genügt es, die Gültigkeit eines dieser beiden Distributivgesetze zu fordern.

</p><p>Jeder distributive Verband ist <a href="Modularer_Verband" title="Modularer Verband">modular</a>, aber nicht umgekehrt.

</p><p>Ein modularer Verband, der nicht distributiv ist, enthält immer den Verband <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle M_{3}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eba9967ecd86d57a9d85001e5fe15f93686861d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{3}}" loading="lazy"></span>, den Verband der Untergruppen der <a href="Kleinsche_Vierergruppe" title="Kleinsche Vierergruppe">Kleinschen Vierergruppe</a>, als Unterverband.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

Dies ergibt das Kriterium:

</p>

<ul><li>Hat ein Verband weder einen Unterverband der Form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{5}}">

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<annotation encoding="application/x-tex">{\displaystyle N_{5}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc559e5307902950d3652ca3bacde6108b65aadb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{5}}" loading="lazy"></span> noch einen der Form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle M_{3}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eba9967ecd86d57a9d85001e5fe15f93686861d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{3}}" loading="lazy"></span>, dann ist er distributiv.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<p>Distributive Verbände kann man in vielen Gebieten innerhalb und außerhalb der Mathematik finden. Distributive Verbände sind:

</p>

<ul><li>jede <a href="Ordnungsrelation" title="Ordnungsrelation">total geordnete</a> Menge</li>

<li>für jede natürliche Zahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> die Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy"></span> ihrer <a href="Teilbarkeit" title="Teilbarkeit">Teiler</a> mit der Teilbarkeit als Ordnungsrelation (also <a href="GgT" class="mw-redirect" title="GgT">ggT</a> und <a href="KgV" class="mw-redirect" title="KgV">kgV</a> als Verknüpfungen)</li>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"></span> (mit oder ohne 0) mit der Teilbarkeit als Ordnungsrelation (also <a href="GgT" class="mw-redirect" title="GgT">ggT</a> und <a href="KgV" class="mw-redirect" title="KgV">kgV</a> als Verknüpfungen)</li>

<li>jeder <a href="Mengenverband" title="Mengenverband">Mengenverband</a> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cap }">

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<annotation encoding="application/x-tex">{\displaystyle \cap }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d4e886e6f5a28a33e073fb108440c152ecfe2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cup }">

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<annotation encoding="application/x-tex">{\displaystyle \cup }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8ff7d0293ad19b43524a133ae5129f3d71f2040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }" loading="lazy"></span></li>

<li>jede <a href="Heyting-Algebra" title="Heyting-Algebra">Heyting-Algebra</a>, daher auch

<ul><li>jede <a href="Boolesche_Algebra" title="Boolesche Algebra">Boolesche Algebra</a></li>

<li>die <a href="Offene_Menge" title="Offene Menge">offenen Mengen</a> eines <a href="Topologischer_Raum" title="Topologischer Raum">topologischen Raumes</a> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subseteq }">

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<annotation encoding="application/x-tex">{\displaystyle \subseteq }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a924f8dcb2847bb8871edfdbf4c6b5cca0669228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \subseteq }" loading="lazy"></span> als Ordnung</li></ul></li></ul>

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</style><div class="thumb tleft vl-mehrere-bilder" style="width:611px;"><div class="thumbinner"><div class="vl-mehrere-bilder-kopf" style="text-align:center;">Beispiele für distributive Verbände</div><div class="vl-mehrere-bilder-horizontal" style="width:202px;"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Verband der Teilmengen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x,y,z\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x,y,z\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87c4fef0d3ac75471707c57a9e2e624d80666de6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.966ex; height:2.843ex;" alt="{\displaystyle \{x,y,z\}}" loading="lazy"></span> durch Teilmengenrelation geordnet</div></div><div class="vl-mehrere-bilder-horizontal" style="width:192px;"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Verband der Teiler von 60, mit <a href="GgT" class="mw-redirect" title="GgT">ggT</a> und <a href="KgV" class="mw-redirect" title="KgV">kgV</a></div></div><div class="vl-mehrere-bilder-horizontal" style="width:199px;"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} _{0}\times \mathbb {N} _{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{0}\times \mathbb {N} _{0}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d176ce63bbee939f044d97ab09bbcba56be3ab96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.305ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} _{0}\times \mathbb {N} _{0}}" loading="lazy"></span> mit der Produkt-Ordnung</div></div><div style="clear:both;"></div>

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<div class="thumb tright vl-mehrere-bilder" style="width:322px;"><div class="thumbinner"><div class="vl-mehrere-bilder-kopf" style="text-align:center;">nicht-distributive Verbände</div><div class="vl-mehrere-bilder-horizontal" style="width:154px;"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{5}}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N_{5}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc559e5307902950d3652ca3bacde6108b65aadb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{5}}" loading="lazy"></span>, der minimale nicht-modulare Verband</div></div><div class="vl-mehrere-bilder-horizontal" style="width:154px;"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eba9967ecd86d57a9d85001e5fe15f93686861d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{3}}" loading="lazy"></span>, der minimale modulare, nicht-distributive Verband: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge (b\vee c)=a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\wedge (b\vee c)=a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8309b780148259474ef51d214cfbc863aa3a65e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.537ex; height:2.843ex;" alt="{\displaystyle a\wedge (b\vee c)=a}" loading="lazy"></span>, aber <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\wedge b)\vee (a\wedge c)=0}">

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<annotation encoding="application/x-tex">{\displaystyle (a\wedge b)\vee (a\wedge c)=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc0a4c138d37a707229f84dce78add69cf83c1c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.091ex; height:2.843ex;" alt="{\displaystyle (a\wedge b)\vee (a\wedge c)=0}" loading="lazy"></span></div></div><div style="clear:both;"></div>

</div></div>

<div class="mw-heading mw-heading2"><h2 id="Kürzungsregel"><span id="K.C3.BCrzungsregel"></span>Kürzungsregel</h2></div>

<p>In einem distributiven Verband gilt die Kürzungsregel:

Gelten für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b,c\in V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a66290044f65b599c4850dfbb9d0d1f1b671099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.93ex; height:2.509ex;" alt="{\displaystyle a,b,c\in V}" loading="lazy"></span> die <i>beiden</i> Gleichungen

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\wedge b=a\wedge c}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/364530425ae68dd6893812b8a110de1e836a8430.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.728ex; height:2.176ex;" alt="{\displaystyle a\wedge b=a\wedge c}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee b=a\vee c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\vee b=a\vee c}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/725987e395a5d918d5d2291165b8204073f52e10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.728ex; height:2.176ex;" alt="{\displaystyle a\vee b=a\vee c}" loading="lazy"></span> folgt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=c}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b133a00dc90e54130a96482c99750f845feb955e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.103ex; height:2.176ex;" alt="{\displaystyle b=c}" loading="lazy"></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul>

<p>Das Beispiel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eba9967ecd86d57a9d85001e5fe15f93686861d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{3}}" loading="lazy"></span> zeigt, dass diese Regel in beliebigen Verbänden nicht gilt.

Sie ist in dem folgenden Sinn typisch für distributive Verbände:

</p>

<ul><li>Ist die Kürzungsregel für beliebige Wahl von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"></span> in einem Verband V gültig, dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> distributiv.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Komplemente_in_distributiven_Verbänden"><span id="Komplemente_in_distributiven_Verb.C3.A4nden"></span>Komplemente in distributiven Verbänden</h2></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Komplement_(Verbandstheorie)" title="Komplement (Verbandstheorie)">Komplement (Verbandstheorie)</a></i></div>

<p>Für ein gegebenes Element a eines beschränkten Verbandes nennt man ein Element b mit der Eigenschaft

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\wedge b=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f60aa2ad9de8a4268a5398e6bb5e3fc458172df8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.071ex; height:2.176ex;" alt="{\displaystyle a\wedge b=0}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee b=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\vee b=1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a34a0cb9f1ad05364290d2b467f7a4a0cbd108c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.071ex; height:2.176ex;" alt="{\displaystyle a\vee b=1}" loading="lazy"></span></li></ul>

<p>ein Komplement von a.

</p><p>Während es im Allgemeinen zu einem Element <i>mehrere</i> komplementäre Elemente geben kann, gilt:

</p>

<ul><li>wenn in einem distributiven Verband ein Komplement von <i>a</i> existiert, dann ist es eindeutig bestimmt.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul>

<p>Man bezeichnet ein <i>eindeutig bestimmtes</i> Komplement von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{c}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{c}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/798ebc74b8765a306c1e5c26ca4edc35d281dfdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.174ex; height:2.343ex;" alt="{\displaystyle a^{c}}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e72f6b2a9120b875c42a17235dbf8d417e9abbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:1.676ex;" alt="{\displaystyle \neg a}" loading="lazy"></span> (vor allem bei Anwendungen in der Logik) oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {a}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {a}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/032e261791bd07a59cf1419352fc66f7901d4b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.345ex; height:2.343ex;" alt="{\displaystyle {\overline {a}}}" loading="lazy"></span>.

</p><p>Ein distributiver Verband, in dem jedes Element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> ein (eindeutig bestimmtes) Komplement <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e72f6b2a9120b875c42a17235dbf8d417e9abbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:1.676ex;" alt="{\displaystyle \neg a}" loading="lazy"></span> hat, heißt <b>Boolesche Algebra</b>.

</p>

<div class="sieheauch" role="navigation" style="font-style:italic;"><span class="sieheauch-text">Siehe auch</span>: <a href="Boolesche_Algebra" title="Boolesche Algebra">Boolesche Algebra</a></div>

<p>Auch in einem nicht-distributiven Verband kann jedes Element genau ein Komplement haben. Damit man die Distributivität folgern kann, muss man mehr fordern:

</p>

<ul><li>Ein Verband <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> ist distributiv, wenn <i>jedes</i> Element in <i>jedem</i> Intervall <i>höchstens ein</i> <a href="Komplement_(Verbandstheorie)#Relative_Komplemente" title="Komplement (Verbandstheorie)">relatives Komplement</a> besitzt.</li></ul>

<p>Ist <i>V</i> ein distributiver Verband und haben <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b\in V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/163bc2cb570fa92aff90855138320a7c23d4d295.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.889ex; height:2.509ex;" alt="{\displaystyle a,b\in V}" loading="lazy"></span> Komplemente, dann haben auch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\wedge b}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1dc496a5b5da3e9b94eb72f04a54167dfe022e45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\displaystyle a\wedge b}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\vee b}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3393da6721f85fa89a1d3a8c28e82c679abe4032.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\displaystyle a\vee b}" loading="lazy"></span> Komplemente und es gilt

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a\wedge b)^{c}=a^{c}\vee b^{c}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/304fa4216a9a2fa8e1a69add33f32f3645956952.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.36ex; height:2.843ex;" alt="{\displaystyle (a\wedge b)^{c}=a^{c}\vee b^{c}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\vee b)^{c}=a^{c}\wedge b^{c}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a\vee b)^{c}=a^{c}\wedge b^{c}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/461da9cbe30d3cfc01baa939d4f60a58db32239b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.36ex; height:2.843ex;" alt="{\displaystyle (a\vee b)^{c}=a^{c}\wedge b^{c}}" loading="lazy"></span></li></ul>

<p>Dies ist eine andere Formulierung der <a href="De_Morgansche_Gesetze" class="mw-redirect" title="De Morgansche Gesetze">de Morganschen Regeln</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Repräsentationssatz_für_distributive_Verbände"><span id="Repr.C3.A4sentationssatz_f.C3.BCr_distributive_Verb.C3.A4nde"></span>Repräsentationssatz für distributive Verbände</h2></div>

<p>Distributive Verbände sind auch anders zu charakterisieren, denn <a href="Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff</a> (1933) und <a href="Marshall_Harvey_Stone" title="Marshall Harvey Stone">Stone</a> (1936) haben gezeigt:

</p>

<ul><li>Ein Verband ist genau dann distributiv, wenn er isomorph zu einem Mengen-Ring ist.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul>

<p>Hieraus folgt natürlich, dass sich jeder distributive Verband in eine Boolesche Algebra einbetten lässt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Weitere_Eigenschaften">Weitere Eigenschaften</h2></div>

<p>Jeder Unterverband eines distributiven Verbandes ist distributiv, dagegen sind <i>Teilverbände</i> nicht immer distributiv.

</p><p>Das <a href="Verband_(Mathematik)#Homomorphismen" title="Verband (Mathematik)">homomorphe Bild</a> eines distributiven Verbandes ist distributiv.

</p><p>Das <a href="Direktes_Produkt" title="Direktes Produkt">direkte Produkt</a> beliebig vieler distributiver Verbände ist distributiv.

</p>

<div class="mw-heading mw-heading2"><h2 id="Vollständige_Distributivität"><span id="Vollst.C3.A4ndige_Distributivit.C3.A4t"></span>Vollständige Distributivität</h2></div>

<p>Ein Verband heißt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1caa4004cb216ef2930bb12fe805a76870caed94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }" loading="lazy"></span>-volldistributiv, wenn für jede Wahl von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ce41f6c8dc3510b3168437d70388ab1cb227d66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.858ex; height:2.176ex;" alt="{\displaystyle a\in V}" loading="lazy"></span> und jede Teilmenge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\subseteq V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\subseteq V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55f39aa11d96d82180ac69e1235b17dcd79b6115.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.328ex; height:2.343ex;" alt="{\displaystyle M\subseteq V}" loading="lazy"></span> gilt

</p>

</mrow>

</munder>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\wedge \bigvee _{x\in M}x=\bigvee _{x\in M}(a\wedge x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c052f3593ce97e60e3203575ab094fecfd4b77a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.106ex; height:5.676ex;" alt="{\displaystyle a\wedge \bigvee _{x\in M}x=\bigvee _{x\in M}(a\wedge x)}" loading="lazy"></span>.</dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vee }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b76220c6805c9b465d6efbc7686c624f49f3023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }" loading="lazy"></span>-Volldistributivität wird dual definiert.

</p><p>Der Begriff <b>Volldistributivität</b> ohne Zusatz wird unterschiedlich verwendet:

</p>

<ul><li>Es kann bedeuten, dass eine von diesen beiden Bedingungen erfüllt ist und im anderen Fall spricht man von dual-volldistributiv oder verwendet explizit die obige Bezeichnung.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>Es kann bedeuten, dass beide Bedingungen erfüllt sind.</li>

<li>Es kann bedeuten, dass das folgende unendliche Distributivgesetz und die dazu duale Form gilt</li></ul>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \emptyset \neq I,J\subseteq V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7983cfbe82b08a8488966e629d02a508d492264.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.824ex; height:2.843ex;" alt="{\displaystyle \emptyset \neq I,J\subseteq V}" loading="lazy"></span> gilt: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge \left\{\bigvee \left\{a_{ij}|j\in J\right\}|i\in I\right\}=\bigvee \left\{\bigwedge \left\{a_{i\varphi (i)}|i\in I\right\}|\varphi \colon I\to J\right\}}">

<mrow>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge \left\{\bigvee \left\{a_{ij}|j\in J\right\}|i\in I\right\}=\bigvee \left\{\bigwedge \left\{a_{i\varphi (i)}|i\in I\right\}|\varphi \colon I\to J\right\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0d84835214053c1c2bea0f276ffe36b0e1e5185.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.941ex; height:4.843ex;" alt="{\displaystyle \bigwedge \left\{\bigvee \left\{a_{ij}|j\in J\right\}|i\in I\right\}=\bigvee \left\{\bigwedge \left\{a_{i\varphi (i)}|i\in I\right\}|\varphi \colon I\to J\right\}}" loading="lazy"></span> <sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl>

<p>Für alle drei Begriffe gilt:

</p><p>Jeder volldistributive Verband ist distributiv und jeder endliche distributive Verband ist volldistributiv.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p><p>Ein vollständiger distributiver Verband braucht nicht volldistributiv sein, wie das Beispiel zeigt.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise_und_Anmerkungen">Einzelnachweise und Anmerkungen</h2></div>

Der Beweis ist eine Gleichungsumformung. Wir nehmen an, dass D2 gilt, und wollen D1 zeigen:<br>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a\vee b)\wedge (a\vee c)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a794eb2cc95a788cca7c892c46be7bbafecd5f7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.83ex; height:2.843ex;" alt="{\displaystyle (a\vee b)\wedge (a\vee c)}" loading="lazy"></span>; Anwendung des zweiten Axioms:<br>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =((a\vee b)\wedge a)\vee ((a\vee b)\wedge c)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15125131371998ac7bf47f14f498a5d514fcab7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.295ex; height:2.843ex;" alt="{\displaystyle =((a\vee b)\wedge a)\vee ((a\vee b)\wedge c)}" loading="lazy"></span>; nach Absoptionsgesetz:<br>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =a\vee ((a\vee b)\wedge c)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68551fcf41f3768fd0b2973196d9e46c48d6b760.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.284ex; height:2.843ex;" alt="{\displaystyle =a\vee ((a\vee b)\wedge c)}" loading="lazy"></span>; Anwendung des zweiten Axioms in Klammer:<br>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =a\vee ((a\wedge c)\vee (b\wedge c))}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/483ffa3edb12df8d4dd1562b90d9db6c5c3937ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.682ex; height:2.843ex;" alt="{\displaystyle =a\vee ((a\wedge c)\vee (b\wedge c))}" loading="lazy"></span>; nach Assoziativgesetz:<br>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =(a\vee (a\wedge c))\vee (b\wedge c)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1319b275fc86b9b2e41bc743a3b23bb33a42acc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.682ex; height:2.843ex;" alt="{\displaystyle =(a\vee (a\wedge c))\vee (b\wedge c)}" loading="lazy"></span>; die linke Seite entspricht nach dem Absorptionsgesetz <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>:<br>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =a\vee (b\wedge c)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/795728202d31f060809d02f484d4d53e085c51df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.662ex; height:2.843ex;" alt="{\displaystyle =a\vee (b\wedge c)}" loading="lazy"></span>. <br>

Die Gegenrichtung folgt dual.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Der Beweis (mit mehreren Zwischenschritten) findet sich z. B. in: H. Gericke, Theorie der Verbände, Mannheim, ²1967, S. 111</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Auch dies wird mit einer einfachen Folge von Gleichungen bewiesen, in der das Absorptionsgesetz, das Distributivgesetz und die Voraussetzungen verwendet werden:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=b\vee (a\wedge b)=b\vee (a\wedge c)=(b\vee a)\wedge (b\vee c)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fc60f775dbe15603660f0f058f935c8879a4788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.299ex; height:2.843ex;" alt="{\displaystyle b=b\vee (a\wedge b)=b\vee (a\wedge c)=(b\vee a)\wedge (b\vee c)}" loading="lazy"></span><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =(a\vee c)\wedge (b\vee c)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =(a\vee c)\wedge (b\vee c)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa771d8d3c12385f90641743e3613fab1288a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.061ex; height:2.843ex;" alt="{\displaystyle =(a\vee c)\wedge (b\vee c)}" loading="lazy"></span><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =(a\wedge b)\vee c=(a\wedge c)\vee a=c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =(a\wedge b)\vee c=(a\wedge c)\vee a=c}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0932d62828901e25505f16ef9cef1642eca48f6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.307ex; height:2.843ex;" alt="{\displaystyle =(a\wedge b)\vee c=(a\wedge c)\vee a=c}" loading="lazy"></span>; nach H. Gericke, Theorie der Verbände, ²1967, S. 114</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Die Beweisidee ist, dass in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{5}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N_{5}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc559e5307902950d3652ca3bacde6108b65aadb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{5}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eba9967ecd86d57a9d85001e5fe15f93686861d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{3}}" loading="lazy"></span> jeweils die Kürzungsregel nicht gilt. Vgl. H. Gericke, Theorie der Verbände, ²1967, S. 113f</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Dies folgt unmittelbar aus der Kürzungsregel</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">G.Grätzer, <i>Lattice Theory</i>, 1971, S. 75</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">So z. B. H. Gericke, Theorie der Verbände, ²1967, S. 114</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Diese Form wurde aus G. Grätzer, Lattice Theory, p 118, Exercise 7 übernommen.</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text">H. Gericke, Theorie der Verbände, Mannheim, ²1967, S. 114 f.</span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text">Der Verband ohne die 1 ist als Produkt von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \times \{0,1\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \times \{0,1\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19592aa1108b7e2a9a96badf3f7225067f390ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.202ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} \times \{0,1\}}" loading="lazy"></span> distributiv. Dass der ganze Verband vollständig und distributiv ist, sieht man leicht. Das Beispiel findet sich (mit etwas anderem Hasse-Diagramm) in H. Gericke, Theorie der Verbände, Mannheim, ²1967, S. 115</span>

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Garrett_Birkhoff" title="Garrett Birkhoff">Garrett Birkhoff</a>: <cite style="font-style:italic">Lattice Theory</cite>. 3. Auflage. AMS, Providence, RI 1973, ISBN 0-8218-1025-1.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Distributiver+Verband&rft.au=Garrett+Birkhoff&rft.btitle=Lattice+Theory&rft.date=1973&rft.edition=3.&rft.genre=book&rft.isbn=0821810251&rft.place=Providence%2C+RI&rft.pub=AMS" style="display:none"> </span></li>

<li>Marcel Erné: <cite style="font-style:italic">Einführung in die Ordnungstheorie</cite>. Bibliographisches Institut, Mannheim 1982, ISBN 3-411-01638-8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Distributiver+Verband&rft.au=Marcel+Ern%C3%A9&rft.btitle=Einf%C3%BChrung+in+die+Ordnungstheorie&rft.date=1982&rft.genre=book&rft.isbn=3411016388&rft.place=Mannheim&rft.pub=Bibliographisches+Institut" style="display:none"> </span></li>

<li>George Grätzer: <cite style="font-style:italic">General Lattice Theory</cite>. 2. Auflage. Birkhäuser, 1998, ISBN 978-0-8176-5239-5.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Distributiver+Verband&rft.au=George+Gr%C3%A4tzer&rft.btitle=General+Lattice+Theory&rft.date=1998&rft.edition=2.&rft.genre=book&rft.isbn=9780817652395&rft.pub=Birkh%C3%A4user" style="display:none"> </span></li>

<li><a href="Hans_Hermes" title="Hans Hermes">Hans Hermes</a>: <cite style="font-style:italic">Einführung in die Verbandstheorie</cite>. 2. Auflage. Springer, Berlin/Heidelberg 1967.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Distributiver+Verband&rft.au=Hans+Hermes&rft.btitle=Einf%C3%BChrung+in+die+Verbandstheorie&rft.date=1967&rft.edition=2.&rft.genre=book&rft.place=Berlin%2FHeidelberg&rft.pub=Springer" style="display:none"> </span></li></ul></div><div><div class="zim-footer">

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