update docs

README.md

@@ -518,9 +518,11 @@ are of the form `v < e` where `v` is a vertex that is an end point of the edge `
 
 * `zeta_matrix(p)` returns the zeta matrix of the poset. This is a `0,1`-matrix whose
   `i,j`-entry is `1` exactly when `p[i] ≤ p[j]`. 
+* `strict_zeta_matrix(p)` returns a  `0,1`-matrix whose `i,j` entry is `1` 
+  exactly when `p[i] < p[j]`.
 * `mobius_matrix(p)` returns the inverse of `zeta(p)`. 
 
-In both cases, the output is a dense, integer matrix. 
+In all cases, the output is a dense, integer matrix. 
 
 
 ## Operations

docs/build/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-11T10:27:38","documenter_version":"1.5.0"}}
+{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-16T15:43:17","documenter_version":"1.5.0"}}

docs/build/index.html

@@ -197,7 +197,7 @@
 {9, 8} directed simple Int64 graph
 
 julia> d == path_digraph(9)
-true</code></pre><p>Given a graph <code>g</code>, calling <code>vertex_edge_incidence_poset(p)</code> creates a poset whose elements correspond to the vertices and edges of <code>g</code>. In this poset the only relations are of the form <code>v &lt; e</code> where <code>v</code> is a vertex that is an end point of the edge <code>e</code>.</p><h2 id="Matrices"><a class="docs-heading-anchor" href="#Matrices">Matrices</a><a id="Matrices-1"></a><a class="docs-heading-anchor-permalink" href="#Matrices" title="Permalink"></a></h2><ul><li><code>zeta_matrix(p)</code> returns the zeta matrix of the poset. This is a <code>0,1</code>-matrix whose <code>i,j</code>-entry is <code>1</code> exactly when <code>p[i] ≤ p[j]</code>. </li><li><code>mobius_matrix(p)</code> returns the inverse of <code>zeta(p)</code>. </li></ul><p>In both cases, the output is a dense, integer matrix. </p><h2 id="Operations"><a class="docs-heading-anchor" href="#Operations">Operations</a><a id="Operations-1"></a><a class="docs-heading-anchor-permalink" href="#Operations" title="Permalink"></a></h2><h3 id="Dual"><a class="docs-heading-anchor" href="#Dual">Dual</a><a id="Dual-1"></a><a class="docs-heading-anchor-permalink" href="#Dual" title="Permalink"></a></h3><p>The dual of poset <code>p</code> is created using <code>reverse(p)</code>. This returns a new poset with the same elements as <code>p</code> in which all relations are reversed (i.e., <code>v &lt; w</code> in <code>p</code> if and  only if <code>w &lt; v</code> in <code>reverse(p)</code>). The dual (reverse) of <code>p</code> can also be created with <code>p&#39;</code>. </p><h3 id="Disjoint-union"><a class="docs-heading-anchor" href="#Disjoint-union">Disjoint union</a><a id="Disjoint-union-1"></a><a class="docs-heading-anchor-permalink" href="#Disjoint-union" title="Permalink"></a></h3><p>Given two posets <code>p</code> and <code>q</code>, the result of <code>p+q</code> is a new poset formed from the  disjoint union of <code>p</code> and <code>q</code>. Note that <code>p+q</code> and <code>q+p</code> are isomorphic, but  may be unequal because of the vertex numbering convention. </p><p>Alternatively <code>hcat(p,q)</code>.</p><h3 id="Stack"><a class="docs-heading-anchor" href="#Stack">Stack</a><a id="Stack-1"></a><a class="docs-heading-anchor-permalink" href="#Stack" title="Permalink"></a></h3><p>Given two posets <code>p</code> and <code>q</code>, the result of <code>p/q</code> is a new poset from a copy of <code>p</code>  and a copy of <code>q</code> with all elements of <code>p</code> above all elements of <code>q</code>. </p><p>Alternatively, <code>vcat(p,q)</code> or  <code>q\p</code>.</p><h3 id="Induced-subposet"><a class="docs-heading-anchor" href="#Induced-subposet">Induced subposet</a><a id="Induced-subposet-1"></a><a class="docs-heading-anchor-permalink" href="#Induced-subposet" title="Permalink"></a></h3><p>Given a poset <code>p</code> and a list of vertices <code>vlist</code>, use <code>induced_subposet(p)</code> to return a  pair <code>(q,vmap)</code>. The poset <code>q</code> is the induced subposet and the vector <code>vmap</code> maps the new vertices to the old ones  (the vertex <code>i</code> in the subposet corresponds to the vertex <code>vmap[i]</code> in <code>p</code>).</p><p>This is exactly analogous to <code>Graphs.induced_subgraph</code>. </p><h3 id="Intersection"><a class="docs-heading-anchor" href="#Intersection">Intersection</a><a id="Intersection-1"></a><a class="docs-heading-anchor-permalink" href="#Intersection" title="Permalink"></a></h3><p>Given two posets <code>p</code> and <code>q</code>, <code>intersect(p,q)</code> is a new poset in which <code>v &lt; w</code> if and only  if <code>v &lt; w</code> in both <code>p</code> and <code>q</code>. The number of elements is the smaller of <code>nv(p)</code> and <code>nv(q)</code>. This may also be invoked as <code>p ∩ q</code>. </p><p>For example, the intersection of a chain with its reversal has no relations:</p><pre><code class="nohighlight hljs">julia&gt; p = chain(5)
+true</code></pre><p>Given a graph <code>g</code>, calling <code>vertex_edge_incidence_poset(p)</code> creates a poset whose elements correspond to the vertices and edges of <code>g</code>. In this poset the only relations are of the form <code>v &lt; e</code> where <code>v</code> is a vertex that is an end point of the edge <code>e</code>.</p><h2 id="Matrices"><a class="docs-heading-anchor" href="#Matrices">Matrices</a><a id="Matrices-1"></a><a class="docs-heading-anchor-permalink" href="#Matrices" title="Permalink"></a></h2><ul><li><code>zeta_matrix(p)</code> returns the zeta matrix of the poset. This is a <code>0,1</code>-matrix whose <code>i,j</code>-entry is <code>1</code> exactly when <code>p[i] ≤ p[j]</code>. </li><li><code>strict_zeta_matrix(p)</code> returns a  <code>0,1</code>-matrix whose <code>i,j</code> entry is <code>1</code>  exactly when <code>p[i] &lt; p[j]</code>.</li><li><code>mobius_matrix(p)</code> returns the inverse of <code>zeta(p)</code>. </li></ul><p>In all cases, the output is a dense, integer matrix. </p><h2 id="Operations"><a class="docs-heading-anchor" href="#Operations">Operations</a><a id="Operations-1"></a><a class="docs-heading-anchor-permalink" href="#Operations" title="Permalink"></a></h2><h3 id="Dual"><a class="docs-heading-anchor" href="#Dual">Dual</a><a id="Dual-1"></a><a class="docs-heading-anchor-permalink" href="#Dual" title="Permalink"></a></h3><p>The dual of poset <code>p</code> is created using <code>reverse(p)</code>. This returns a new poset with the same elements as <code>p</code> in which all relations are reversed (i.e., <code>v &lt; w</code> in <code>p</code> if and  only if <code>w &lt; v</code> in <code>reverse(p)</code>). The dual (reverse) of <code>p</code> can also be created with <code>p&#39;</code>. </p><h3 id="Disjoint-union"><a class="docs-heading-anchor" href="#Disjoint-union">Disjoint union</a><a id="Disjoint-union-1"></a><a class="docs-heading-anchor-permalink" href="#Disjoint-union" title="Permalink"></a></h3><p>Given two posets <code>p</code> and <code>q</code>, the result of <code>p+q</code> is a new poset formed from the  disjoint union of <code>p</code> and <code>q</code>. Note that <code>p+q</code> and <code>q+p</code> are isomorphic, but  may be unequal because of the vertex numbering convention. </p><p>Alternatively <code>hcat(p,q)</code>.</p><h3 id="Stack"><a class="docs-heading-anchor" href="#Stack">Stack</a><a id="Stack-1"></a><a class="docs-heading-anchor-permalink" href="#Stack" title="Permalink"></a></h3><p>Given two posets <code>p</code> and <code>q</code>, the result of <code>p/q</code> is a new poset from a copy of <code>p</code>  and a copy of <code>q</code> with all elements of <code>p</code> above all elements of <code>q</code>. </p><p>Alternatively, <code>vcat(p,q)</code> or  <code>q\p</code>.</p><h3 id="Induced-subposet"><a class="docs-heading-anchor" href="#Induced-subposet">Induced subposet</a><a id="Induced-subposet-1"></a><a class="docs-heading-anchor-permalink" href="#Induced-subposet" title="Permalink"></a></h3><p>Given a poset <code>p</code> and a list of vertices <code>vlist</code>, use <code>induced_subposet(p)</code> to return a  pair <code>(q,vmap)</code>. The poset <code>q</code> is the induced subposet and the vector <code>vmap</code> maps the new vertices to the old ones  (the vertex <code>i</code> in the subposet corresponds to the vertex <code>vmap[i]</code> in <code>p</code>).</p><p>This is exactly analogous to <code>Graphs.induced_subgraph</code>. </p><h3 id="Intersection"><a class="docs-heading-anchor" href="#Intersection">Intersection</a><a id="Intersection-1"></a><a class="docs-heading-anchor-permalink" href="#Intersection" title="Permalink"></a></h3><p>Given two posets <code>p</code> and <code>q</code>, <code>intersect(p,q)</code> is a new poset in which <code>v &lt; w</code> if and only  if <code>v &lt; w</code> in both <code>p</code> and <code>q</code>. The number of elements is the smaller of <code>nv(p)</code> and <code>nv(q)</code>. This may also be invoked as <code>p ∩ q</code>. </p><p>For example, the intersection of a chain with its reversal has no relations:</p><pre><code class="nohighlight hljs">julia&gt; p = chain(5)
 {5, 10} Int64 poset
 
 julia&gt; p ∩ reverse(p)
@@ -207,4 +207,4 @@
 julia&gt; trim!(p)
 
 julia&gt; p
-{4, 6} Int64 poset</code></pre><h2 id="Implementation"><a class="docs-heading-anchor" href="#Implementation">Implementation</a><a id="Implementation-1"></a><a class="docs-heading-anchor-permalink" href="#Implementation" title="Permalink"></a></h2><p>A <code>Poset</code> is a structure that contains a single data element: a <code>DiGraph</code>.  Users should not be accessing this directly, but it may be useful to understand how posets are implemented. The directed graph is acyclic (including loopless) and transitively closed. This means if <span>$a \to b$</span> is an edge and <span>$b\to c$</span> is an edge, then <span>$a \to c$</span> is also an edge. The advantage to this structure is that checking if <span>$a \prec b$</span> in a poset is quick. There are two disadvantages.</p><p>First, the graph may be larger than needed. If we only kept cover edges  (the transitive reduction of the digraph) we might have many fewer edges.  For example, a linear order with <span>$n$</span> elements has <span>$\binom{n}{2} \sim n^2/2$</span>  edges in the digraph that represents it, whereas there are only <span>$n-1$</span> edges in  the cover digraph. However, this savings is an extreme example. A poset with <span>$n$</span> elements split into two antichains, with every element of the first antichain below every element of the second, has <span>$n^2/4$</span> edges in either representation.  So in either case, the representing digraph may have up to order <span>$n^2$</span> edges. </p><p>Second, the computational cost of adding (or deleting) a relation is nontrivial.  The <code>add_relation!</code> function first checks if the added relation would violate  transitivity; this is speedy because we can add the relation <span>$a \prec b$</span> so  long as we don&#39;t have <span>$b\prec a$</span> already in the poset. However, after the edge <span>$(a,b)$</span>  is inserted into the digraph, we execute <code>transitiveclosure!</code> and that takes some  work. Adding several relations to the poset, one at a time, can be slow. </p><p>This can be greatly accelerated by using <code>Posets.add_relations!</code> but (as discussed above) this function can cause severe problems if not used carefully.</p><h2 id="See-Also"><a class="docs-heading-anchor" href="#See-Also">See Also</a><a id="See-Also-1"></a><a class="docs-heading-anchor-permalink" href="#See-Also" title="Permalink"></a></h2><p>The <code>extras</code> folder includes additional code that may be useful in  working with <code>Posets</code>. See the <code>README</code> in the <code>extras</code> directory. </p><p>Of note is <code>extras/converter.jl</code> that defines the function <code>poset_converter</code> that can  be used to transform a <code>Poset</code> (defined in this module) to a <code>SimplePoset</code>  (defined in the <a href="https://github.com/scheinerman/SimplePosets.jl">SimplePosets</a> module). </p></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 11 August 2024 10:27">Sunday 11 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+{4, 6} Int64 poset</code></pre><h2 id="Implementation"><a class="docs-heading-anchor" href="#Implementation">Implementation</a><a id="Implementation-1"></a><a class="docs-heading-anchor-permalink" href="#Implementation" title="Permalink"></a></h2><p>A <code>Poset</code> is a structure that contains a single data element: a <code>DiGraph</code>.  Users should not be accessing this directly, but it may be useful to understand how posets are implemented. The directed graph is acyclic (including loopless) and transitively closed. This means if <span>$a \to b$</span> is an edge and <span>$b\to c$</span> is an edge, then <span>$a \to c$</span> is also an edge. The advantage to this structure is that checking if <span>$a \prec b$</span> in a poset is quick. There are two disadvantages.</p><p>First, the graph may be larger than needed. If we only kept cover edges  (the transitive reduction of the digraph) we might have many fewer edges.  For example, a linear order with <span>$n$</span> elements has <span>$\binom{n}{2} \sim n^2/2$</span>  edges in the digraph that represents it, whereas there are only <span>$n-1$</span> edges in  the cover digraph. However, this savings is an extreme example. A poset with <span>$n$</span> elements split into two antichains, with every element of the first antichain below every element of the second, has <span>$n^2/4$</span> edges in either representation.  So in either case, the representing digraph may have up to order <span>$n^2$</span> edges. </p><p>Second, the computational cost of adding (or deleting) a relation is nontrivial.  The <code>add_relation!</code> function first checks if the added relation would violate  transitivity; this is speedy because we can add the relation <span>$a \prec b$</span> so  long as we don&#39;t have <span>$b\prec a$</span> already in the poset. However, after the edge <span>$(a,b)$</span>  is inserted into the digraph, we execute <code>transitiveclosure!</code> and that takes some  work. Adding several relations to the poset, one at a time, can be slow. </p><p>This can be greatly accelerated by using <code>Posets.add_relations!</code> but (as discussed above) this function can cause severe problems if not used carefully.</p><h2 id="See-Also"><a class="docs-heading-anchor" href="#See-Also">See Also</a><a id="See-Also-1"></a><a class="docs-heading-anchor-permalink" href="#See-Also" title="Permalink"></a></h2><p>The <code>extras</code> folder includes additional code that may be useful in  working with <code>Posets</code>. See the <code>README</code> in the <code>extras</code> directory. </p><p>Of note is <code>extras/converter.jl</code> that defines the function <code>poset_converter</code> that can  be used to transform a <code>Poset</code> (defined in this module) to a <code>SimplePoset</code>  (defined in the <a href="https://github.com/scheinerman/SimplePosets.jl">SimplePosets</a> module). </p></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Friday 16 August 2024 15:43">Friday 16 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

docs/src/index.md

@@ -518,9 +518,11 @@ are of the form `v < e` where `v` is a vertex that is an end point of the edge `
 
 * `zeta_matrix(p)` returns the zeta matrix of the poset. This is a `0,1`-matrix whose
   `i,j`-entry is `1` exactly when `p[i] ≤ p[j]`. 
+* `strict_zeta_matrix(p)` returns a  `0,1`-matrix whose `i,j` entry is `1` 
+  exactly when `p[i] < p[j]`.
 * `mobius_matrix(p)` returns the inverse of `zeta(p)`. 
 
-In both cases, the output is a dense, integer matrix. 
+In all cases, the output is a dense, integer matrix. 
 
 
 ## Operations