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  25 <div class="section boost_icl_semantics_collectors__maps_of_sets" lang="en">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="boost_icl.semantics.collectors__maps_of_sets"></a><a class="link" href="collectors__maps_of_sets.html" title="Collectors: Maps of Sets">Collectors:
  28       Maps of Sets</a>
  29 </h3></div></div></div>
  30 <p>
  31         Icl <code class="computeroutput"><span class="identifier">Collectors</span></code>, behave like
  32         <code class="computeroutput"><span class="identifier">Sets</span></code>. This can be understood
  33         easily, if we consider, that every map of sets can be transformed to an equivalent
  34         set of pairs. For instance in the pseudocode below map <code class="computeroutput"><span class="identifier">m</span></code>
  35
  36 </p>
  37 <pre class="programlisting"><span class="identifier">icl</span><span class="special">::</span><span class="identifier">map</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">,</span><span class="identifier">set</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">&gt;</span> <span class="special">&gt;</span>  <span class="identifier">m</span> <span class="special">=</span> <span class="special">{(</span><span class="number">1</span><span class="special">-&gt;{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">}),</span> <span class="special">(</span><span class="number">2</span><span class="special">-&gt;{</span><span class="number">1</span><span class="special">})};</span>
  38 </pre>
  39 <p>
  40         is equivalent to set <code class="computeroutput"><span class="identifier">s</span></code>
  41 </p>
  42 <pre class="programlisting"><span class="identifier">icl</span><span class="special">::</span><span class="identifier">set</span><span class="special">&lt;</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">,</span><span class="keyword">int</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">s</span> <span class="special">=</span> <span class="special">{(</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">),</span>   <span class="comment">//representing 1-&gt;{1,2}
  43 </span>                              <span class="special">(</span><span class="number">2</span><span class="special">,</span><span class="number">1</span><span class="special">)</span>       <span class="special">};</span> <span class="comment">//representing 2-&gt;{1}
  44 </span></pre>
  45 <p>
  46       </p>
  47 <p>
  48         Also the results of add, subtract and other operations on <code class="computeroutput"><span class="identifier">map</span>
  49         <span class="identifier">m</span></code> and <code class="computeroutput"><span class="identifier">set</span>
  50         <span class="identifier">s</span></code> preserves the equivalence of
  51         the containers <span class="emphasis"><em><span class="bold"><strong>almost</strong></span></em></span>
  52         perfectly:
  53 </p>
  54 <pre class="programlisting"><span class="identifier">m</span> <span class="special">+=</span> <span class="special">(</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">);</span>
  55 <span class="identifier">m</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">-&gt;{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">}),</span> <span class="special">(</span><span class="number">2</span><span class="special">-&gt;{</span><span class="number">1</span><span class="special">})};</span> <span class="comment">//aggregated on collision of key value 1
  56 </span><span class="identifier">s</span> <span class="special">+=</span> <span class="special">(</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">);</span>
  57 <span class="identifier">s</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">),</span>   <span class="comment">//representing 1-&gt;{1,2,3}
  58 </span>      <span class="special">(</span><span class="number">2</span><span class="special">,</span><span class="number">1</span><span class="special">)</span>             <span class="special">};</span> <span class="comment">//representing 2-&gt;{1}
  59 </span></pre>
  60 <p>
  61       </p>
  62 <p>
  63         The equivalence of <code class="computeroutput"><span class="identifier">m</span></code> and
  64         <code class="computeroutput"><span class="identifier">s</span></code> is only violated if an
  65         empty set occurres in <code class="computeroutput"><span class="identifier">m</span></code> by
  66         subtraction of a value pair:
  67 </p>
  68 <pre class="programlisting"><span class="identifier">m</span> <span class="special">-=</span> <span class="special">(</span><span class="number">2</span><span class="special">,</span><span class="number">1</span><span class="special">);</span>
  69 <span class="identifier">m</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">-&gt;{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">}),</span> <span class="special">(</span><span class="number">2</span><span class="special">-&gt;{})};</span> <span class="comment">//aggregated on collision of key value 2
  70 </span><span class="identifier">s</span> <span class="special">-=</span> <span class="special">(</span><span class="number">2</span><span class="special">,</span><span class="number">1</span><span class="special">);</span>
  71 <span class="identifier">s</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">)</span>   <span class="comment">//representing 1-&gt;{1,2,3}
  72 </span>                       <span class="special">};</span> <span class="comment">//2-&gt;{} is not represented in s
  73 </span></pre>
  74 <p>
  75       </p>
  76 <p>
  77         This problem can be dealt with in two ways.
  78       </p>
  79 <div class="orderedlist"><ol type="1">
  80 <li>
  81           Deleting value pairs form the Collector, if it's associated value becomes
  82           a neutral value or <code class="computeroutput"><span class="identifier">identity_element</span></code>.
  83         </li>
  84 <li>
  85           Using a different equality, called distinct equality in the laws to validate.
  86           Distinct equality only accounts for value pairs that that carry values
  87           unequal to the <code class="computeroutput"><span class="identifier">identity_element</span></code>.
  88         </li>
  89 </ol></div>
  90 <p>
  91         Solution (1) led to the introduction of map traits, particularly trait <span class="emphasis"><em><span class="bold"><strong>partial_absorber</strong></span></em></span>, which is the default
  92         setting in all icl's map templates.
  93       </p>
  94 <p>
  95         Solution (2), is applied to check the semantics of icl::Maps for the partial_enricher
  96         trait that does not delete value pairs that carry identity elements. Distinct
  97         equality is implemented by a non member function called <code class="computeroutput"><span class="identifier">is_distinct_equal</span></code>.
  98         Throughout this chapter distinct equality in pseudocode and law denotations
  99         is denoted as <code class="computeroutput"><span class="special">=</span><span class="identifier">d</span><span class="special">=</span></code> operator.
 100       </p>
 101 <p>
 102         The validity of the sets of laws that make up <code class="computeroutput"><span class="identifier">Set</span></code>
 103         semantics should now be quite evident. So the following text shows the laws
 104         that are validated for all <code class="computeroutput"><span class="identifier">Collector</span></code>
 105         types <code class="computeroutput"><span class="identifier">C</span></code>. Which are <code class="computeroutput"><a class="link" href="../../boost/icl/map.html" title="Class template map">icl::map</a></code><code class="computeroutput"><span class="special">&lt;</span><span class="identifier">D</span><span class="special">,</span><span class="identifier">S</span><span class="special">,</span><span class="identifier">T</span><span class="special">&gt;</span></code>,
 106         <code class="computeroutput"><a class="link" href="../../boost/icl/interval_map.html" title="Class template interval_map">interval_map</a></code><code class="computeroutput"><span class="special">&lt;</span><span class="identifier">D</span><span class="special">,</span><span class="identifier">S</span><span class="special">,</span><span class="identifier">T</span><span class="special">&gt;</span></code> and <code class="computeroutput"><a class="link" href="../../boost/icl/split_interval_map.html" title="Class template split_interval_map">split_interval_map</a></code><code class="computeroutput"><span class="special">&lt;</span><span class="identifier">D</span><span class="special">,</span><span class="identifier">S</span><span class="special">,</span><span class="identifier">T</span><span class="special">&gt;</span></code> where <code class="computeroutput"><span class="identifier">CodomainT</span></code>
 107         type <code class="computeroutput"><span class="identifier">S</span></code> is a model of <code class="computeroutput"><span class="identifier">Set</span></code> and <code class="computeroutput"><span class="identifier">Trait</span></code>
 108         type <code class="computeroutput"><span class="identifier">T</span></code> is either <code class="computeroutput"><a class="link" href="../../boost/icl/partial_absorber.html" title="Struct partial_absorber">partial_absorber</a></code> or <code class="computeroutput"><a class="link" href="../../boost/icl/partial_enricher.html" title="Struct partial_enricher">partial_enricher</a></code>.
 109       </p>
 110 <a name="boost_icl.semantics.collectors__maps_of_sets.laws_on_set_union__set_intersection_and_set_difference"></a><h6>
 111 <a name="id1090371"></a>
 112         <a class="link" href="collectors__maps_of_sets.html#boost_icl.semantics.collectors__maps_of_sets.laws_on_set_union__set_intersection_and_set_difference">Laws
 113         on set union, set intersection and set difference</a>
 114       </h6>
 115 <p>
 116
 117 </p>
 118 <pre class="programlisting"><span class="identifier">Associativity</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,+,==</span> <span class="special">&gt;:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+(</span><span class="identifier">b</span><span class="special">+</span><span class="identifier">c</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)+</span><span class="identifier">c</span>
 119 <span class="identifier">Neutrality</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,+,==</span> <span class="special">&gt;</span>   <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">;</span>       <span class="identifier">a</span><span class="special">+</span><span class="identifier">C</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">a</span>
 120 <span class="identifier">Commutativity</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,+,==</span> <span class="special">&gt;:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span>       <span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">+</span><span class="identifier">a</span>
 121
 122 <span class="identifier">Associativity</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,&amp;,==</span> <span class="special">&gt;:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span><span class="special">&amp;(</span><span class="identifier">b</span><span class="special">&amp;</span><span class="identifier">c</span><span class="special">)</span> <span class="special">==(</span><span class="identifier">a</span><span class="special">&amp;</span><span class="identifier">b</span><span class="special">)&amp;</span><span class="identifier">c</span>
 123 <span class="identifier">Commutativity</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,&amp;,==</span> <span class="special">&gt;:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span>       <span class="identifier">a</span><span class="special">&amp;</span><span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">&amp;</span><span class="identifier">a</span>
 124
 125 <span class="identifier">RightNeutrality</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,-,==</span> <span class="special">&gt;:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">;</span>   <span class="identifier">a</span><span class="special">-</span><span class="identifier">C</span><span class="special">()</span> <span class="special">==</span>  <span class="identifier">a</span>
 126 <span class="identifier">Inversion</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</span>     <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">;</span>   <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">a</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="identifier">C</span><span class="special">()</span>
 127 </pre>
 128 <p>
 129       </p>
 130 <p>
 131         All the fundamental laws could be validated for all icl Maps in their instantiation
 132         as Maps of Sets or Collectors. As expected, Inversion only holds for distinct
 133         equality, if the map is not a <code class="computeroutput"><span class="identifier">partial_absorber</span></code>.
 134       </p>
 135 <p>
 136
 137 </p>
 138 <pre class="programlisting">                             <span class="special">+</span>    <span class="special">&amp;</span>    <span class="special">-</span>
 139 <span class="identifier">Associativity</span>                <span class="special">==</span>   <span class="special">==</span>
 140 <span class="identifier">Neutrality</span>                   <span class="special">==</span>        <span class="special">==</span>
 141 <span class="identifier">Commutativity</span>                <span class="special">==</span>   <span class="special">==</span>
 142 <span class="identifier">Inversion</span> <span class="identifier">partial_absorber</span>             <span class="special">==</span>
 143           <span class="identifier">partial_enricher</span>             <span class="special">=</span><span class="identifier">d</span><span class="special">=</span>
 144 </pre>
 145 <p>
 146       </p>
 147 <a name="boost_icl.semantics.collectors__maps_of_sets.distributivity_laws"></a><h6>
 148 <a name="id1091034"></a>
 149         <a class="link" href="collectors__maps_of_sets.html#boost_icl.semantics.collectors__maps_of_sets.distributivity_laws">Distributivity
 150         Laws</a>
 151       </h6>
 152 <p>
 153
 154 </p>
 155 <pre class="programlisting">     <span class="identifier">Distributivity</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,+,&amp;,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">&amp;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span>
 156      <span class="identifier">Distributivity</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,&amp;,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">&amp;</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">)</span>
 157 <span class="identifier">RightDistributivity</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,+,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
 158 <span class="identifier">RightDistributivity</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,&amp;,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">&amp;</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
 159 </pre>
 160 <p>
 161       </p>
 162 <p>
 163         Results for the distributivity laws are almost identical to the validation
 164         of sets except that for a <code class="computeroutput"><span class="identifier">partial_enricher</span>
 165         <span class="identifier">map</span></code> the law <code class="computeroutput"><span class="special">(</span><span class="identifier">a</span> <span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span>
 166         <span class="identifier">c</span> <span class="special">==</span>
 167         <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
 168         <span class="special">&amp;</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span></code> holds for lexicographical equality.
 169       </p>
 170 <p>
 171
 172 </p>
 173 <pre class="programlisting">                                                   <span class="special">+,&amp;</span>    <span class="special">&amp;,+</span>
 174      <span class="identifier">Distributivity</span>  <span class="identifier">joining</span>                       <span class="special">==</span>     <span class="special">==</span>
 175                      <span class="identifier">splitting</span>   <span class="identifier">partial_absorber</span>  <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>    <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>
 176                                  <span class="identifier">partial_enricher</span>  <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>    <span class="special">==</span>
 177
 178                                                    <span class="special">+,-</span>    <span class="special">&amp;,-</span>
 179 <span class="identifier">RightDistributivity</span>  <span class="identifier">joining</span>                       <span class="special">==</span>     <span class="special">==</span>
 180                      <span class="identifier">splitting</span>                     <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>    <span class="special">==</span>
 181 </pre>
 182 <p>
 183       </p>
 184 <a name="boost_icl.semantics.collectors__maps_of_sets.demorgan_s_law_and_symmetric_difference"></a><h6>
 185 <a name="id1091906"></a>
 186         <a class="link" href="collectors__maps_of_sets.html#boost_icl.semantics.collectors__maps_of_sets.demorgan_s_law_and_symmetric_difference">DeMorgan's
 187         Law and Symmetric Difference</a>
 188       </h6>
 189 <p>
 190
 191 </p>
 192 <pre class="programlisting"><span class="identifier">DeMorgan</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,+,&amp;,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">&amp;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
 193 <span class="identifier">DeMorgan</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,&amp;,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
 194 </pre>
 195 <p>
 196       </p>
 197 <p>
 198
 199 </p>
 200 <pre class="programlisting">                         <span class="special">+,&amp;</span>     <span class="special">&amp;,+</span>
 201 <span class="identifier">DeMorgan</span>  <span class="identifier">joining</span>        <span class="special">==</span>      <span class="special">==</span>
 202           <span class="identifier">splitting</span>      <span class="special">==</span>      <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>
 203 </pre>
 204 <p>
 205       </p>
 206 <p>
 207
 208 </p>
 209 <pre class="programlisting"><span class="identifier">SymmetricDifference</span><span class="special">&lt;</span><span class="identifier">C</span><span class="special">,==</span> <span class="special">&gt;</span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">*</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">)</span>
 210 </pre>
 211 <p>
 212       </p>
 213 <p>
 214         Reviewing the validity tables above shows, that the sets of valid laws for
 215         <code class="computeroutput"><span class="identifier">icl</span> <span class="identifier">Sets</span></code>
 216         and <code class="computeroutput"><span class="identifier">icl</span> <span class="identifier">Maps</span>
 217         <span class="identifier">of</span> <span class="identifier">Sets</span></code>
 218         that are <span class="emphasis"><em>identity absorbing</em></span> are exactly the same. As
 219         expected, only for Maps of Sets that represent empty sets as associated values,
 220         called <span class="emphasis"><em>identity enrichers</em></span>, there are marginal semantic
 221         differences.
 222       </p>
 223 </div>
 224 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 225 <td align="left"></td>
 226 <td align="right"><div class="copyright-footer">Copyright &#169; 2007 -2010 Joachim Faulhaber<br>Copyright &#169; 1999 -2006 Cortex Software GmbH<p>
 227         Distributed under the Boost Software License, Version 1.0. (See accompanying
 228         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
 229       </p>
 230 </div></td>
 231 </tr></table>
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