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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Event structure</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Computer_science" title="Computer science">computer science</a>, an <b>event structure</b> describes sequences of events that can be triggered by combinations of other events, with certain forbidden combinations of events. Different sources provide more or less flexible mathematical formalizations of the way events can be triggered and which combinations are forbidden.
</p><p>The most general of these formalizations is given by Glynn Winskel. Winskel formalizes an event structure can be formalized as a triple <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 (E,{\mathcal {C}},\vdash )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
</mrow>
</mrow>
<mo>,</mo>
<mo>⊢<!-- ⊢ --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {C}},\vdash )}</annotation>
</semantics>
</math></span><img src="./00d0b17ecd01111a20ca608993589cce72b41126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.312ex; height:2.843ex;" alt="{\displaystyle (E,{\mathcal {C}},\vdash )}" loading="lazy"></span>, in which:
</p>
<ul><li><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 E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> is a set of events, not necessarily finite.</li>
<li><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 {\mathcal {C}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>
</semantics>
</math></span><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"></span> is a family of finite subsets of <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 E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>, the subsets that are deemed to be consistent (not forbidden). If <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 C\in {\mathcal {C}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C\in {\mathcal {C}}}</annotation>
</semantics>
</math></span><img src="./c472ed9c4e8e46f94dcfc8a0c25f1bfb73f561ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle C\in {\mathcal {C}}}" loading="lazy"></span> is one of these consistent sets, then every subset of <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 C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> must also be consistent. That is, <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 {\mathcal {C}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>
</semantics>
</math></span><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"></span> must be closed under the operation of taking subsets.</li>
<li><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 \vdash }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>⊢<!-- ⊢ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation>
</semantics>
</math></span><img src="./a0c0d30cf8cb7dba179e317fcde9583d842e80f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }" loading="lazy"></span> is a <a href="Binary_relation" title="Binary relation">binary relation</a> from consistent sets to elements of <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 E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>. The relation <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 C\vdash e}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>⊢<!-- ⊢ --></mo>
<mi>e</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C\vdash e}</annotation>
</semantics>
</math></span><img src="./68644b353d9e4a17bda3a5f26972e622cbcbb87a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.56ex; height:2.176ex;" alt="{\displaystyle C\vdash e}" loading="lazy"></span>, for <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 C\in {\mathcal {C}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C\in {\mathcal {C}}}</annotation>
</semantics>
</math></span><img src="./c472ed9c4e8e46f94dcfc8a0c25f1bfb73f561ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle C\in {\mathcal {C}}}" loading="lazy"></span> and <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 e\in E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>e</mi>
<mo>∈<!-- ∈ --></mo>
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e\in E}</annotation>
</semantics>
</math></span><img src="./34778736d9c6d607a4da3d25594b38dd3e8c82ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.7ex; height:2.176ex;" alt="{\displaystyle e\in E}" loading="lazy"></span> is interpreted as meaning that when the events so far form set <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 C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span>, this <i>enables</i> <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 e}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>e</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e}</annotation>
</semantics>
</math></span><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> to be the next event. When <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 C\vdash e}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>⊢<!-- ⊢ --></mo>
<mi>e</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C\vdash e}</annotation>
</semantics>
</math></span><img src="./68644b353d9e4a17bda3a5f26972e622cbcbb87a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.56ex; height:2.176ex;" alt="{\displaystyle C\vdash e}" loading="lazy"></span>, it is required that <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 C'\vdash e}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>C</mi>
<mo>′</mo>
</msup>
<mo>⊢<!-- ⊢ --></mo>
<mi>e</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C'\vdash e}</annotation>
</semantics>
</math></span><img src="./7dbe2e72db936a721d129a139bb4ea3eb012f930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.277ex; height:2.509ex;" alt="{\displaystyle C'\vdash e}" loading="lazy"></span> for every consistent superset <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 C'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>C</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C'}</annotation>
</semantics>
</math></span><img src="./b3f732043ebc78ca911b9d7801fd8787b1291b2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.483ex; height:2.509ex;" alt="{\displaystyle C'}" loading="lazy"></span> (with <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 C\subset C'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>⊂<!-- ⊂ --></mo>
<msup>
<mi>C</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C\subset C'}</annotation>
</semantics>
</math></span><img src="./ad917af6d0d1ca99e049f2ebceef027b6c26dd04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.347ex; height:2.509ex;" alt="{\displaystyle C\subset C'}" loading="lazy"></span> and <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 C'\in {\mathcal {C}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>C</mi>
<mo>′</mo>
</msup>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C'\in {\mathcal {C}}}</annotation>
</semantics>
</math></span><img src="./8ed81d85284ee4d11ef4eb547b54f8fe6c4d3102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.562ex; height:2.509ex;" alt="{\displaystyle C'\in {\mathcal {C}}}" loading="lazy"></span>).</li></ul>
<p>According to Winskel's definitions, a <i>configuration</i> of an event structure is a subset of <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 E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> all of whose finite subsets are consistent and whose events are all <i>secured</i>. Here, an event is secured when it belongs to a finite sequence of events from the configuration, each of which is enabled by the subset of earlier events from the same sequence.<sup id="cite_ref-winskel_1-0" class="reference"><a href="#cite_note-winskel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>The <a href="Nlab" class="mw-redirect" title="Nlab">nlab</a> simplifies these definitions in two ways:
</p>
<ul><li>It replaces the family of consistent events by an <a href="Irreflexive" class="mw-redirect" title="Irreflexive">irreflexive</a> <a href="Symmetric" class="mw-redirect" title="Symmetric">symmetric</a> relation <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 \#}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">#<!-- # --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \#}</annotation>
</semantics>
</math></span><img src="./f72e8254fd59fa4060c66c9310acbaf6df2ce894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.936ex; height:2.509ex;" alt="{\displaystyle \#}" loading="lazy"></span> called <i>incompatibility</i> (or <i>conflict</i>), such that a finite set of events is consistent if and only if it contains no incompatible pair.</li>
<li>And (either separately or with both simplifications together) it replaces the enabling relation by a <a href="Partial_order" class="mw-redirect" title="Partial order">partial order</a> relation on <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 E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> called <i>causal dependency</i>, such that each event has finitely many predecessors, all of which must have happened earlier to enable the event.</li></ul>
<p>For the event structures with both simplifications, which nlab calls <i>prime event structures</i>, the configurations are the downward-closed subsets of the partial order that include no incompatible pairs.<sup id="cite_ref-nlab_2-0" class="reference"><a href="#cite_note-nlab-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Antimatroid" title="Antimatroid">Antimatroid</a>, a system of events ordered by enabling subsets but without a consistency requirement</li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-winskel-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-winskel_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFWinskel1986" class="citation cs2">Winskel, Glynn (1986), <a rel="nofollow" class="external text" href="https://www.cl.cam.ac.uk/~gw104/EvStr.pdf">"Event structures"</a> <span class="cs1-format">(PDF)</span>, in Brauer, Wilfried; Reisig, Wolfgang; Rozenberg, Grzegorz (eds.), <i>Petri Nets: Central Models and Their Properties, Advances in Petri Nets 1986, Part II, Proceedings of an Advanced Course, Bad Honnef, Germany, 8-19 September 1986</i>, Lecture Notes in Computer Science, vol. 255, Springer, pp. <span class="nowrap">325–</span>392, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-17906-2_31">10.1007/3-540-17906-2_31</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-17906-1</bdi></cite></span>
</li>
<li id="cite_note-nlab-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-nlab_2-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/event+structure">Event structure</a> at the <a href="NLab" title="NLab"><i>n</i>Lab</a></span>
</li>
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