<p>In this short chapter, we describe how to compute with residue-class-wise affine monoids. <em>Residue-class-wise affine</em> monoids, or <em>rcwa</em> monoids for short, are monoids whose elements are residue-class-wise affine mappings.</p>
<p>As any other monoids in <strong class="pkg">GAP</strong>, residue-class-wise affine monoids can be constructed by <code class="code">Monoid</code> or <code class="code">MonoidByGenerators</code>.</p>
<p>There are methods for the operations <code class="code">View</code>, <code class="code">Display</code>, <code class="code">Print</code> and <code class="code">String</code> which are applicable to rcwa monoids. All rcwa monoids over a ring <span class="SimpleMath">\(R\)</span> are submonoids of Rcwa(<span class="SimpleMath">\(R\)</span>). The monoid Rcwa(<span class="SimpleMath">\(R\)</span>) itself is not finitely generated, thus cannot be constructed as described above. It is handled as a special case:</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Rcwa</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the monoid Rcwa(<var class="Arg">R</var>) of all residue-class-wise affine mappings of the ring <var class="Arg">R</var>.</p>
<p>In our methods to construct rcwa groups, two kinds of mappings played a crucial role, namely the restriction monomorphisms (cf. <code class="func">Restriction</code> (<a href="chap3_mj.html#X852EF2C079E4D7FF"><span class="RefLink">3.1-6</span></a>)) and the induction epimorphisms (cf. <code class="func">Induction</code> (<a href="chap3_mj.html#X82171D7287CBED95"><span class="RefLink">3.1-7</span></a>)). The restriction monomorphisms extend in a natural way to the monoids Rcwa(<span class="SimpleMath">\(R\)</span>), and the induction epimorphisms have corresponding generalizations, also. Therefore the operations <code class="code">Restriction</code> and <code class="code">Induction</code> can be applied to rcwa monoids as well:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M2 := Restriction(M,2*One(Rcwa(Integers)));</span>
<rcwa monoid over Z with 2 generators, of size 11>
<span class="GAPprompt">gap></span> <span class="GAPinput">Support(M2);</span>
0(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Action(M2,ResidueClass(1,2));</span>
Trivial rcwa monoid over Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Induction(M2,2*One(Rcwa(Integers))) = M;</span>
true
<h4>4.2 <span class="Heading">Computing with residue-class-wise affine monoids</span></h4>
<p>There is a method for <code class="code">Size</code> which computes the order of an rcwa monoid. Further there is a method for <code class="code">in</code> which checks whether a given rcwa mapping lies in a given rcwa monoid (membership test), and there is a method for <code class="code">IsSubset</code> which checks for a submonoid relation.</p>
<p>There are also methods for <code class="code">Support</code>, <code class="code">Modulus</code>, <code class="code">IsTame</code>, <code class="code">PrimeSet</code>, <code class="code">IsIntegral</code>, <code class="code">IsClassWiseOrderPreserving</code> and <code class="code">IsSignPreserving</code> available for rcwa monoids.</p>
<p>The <em>support</em> of an rcwa monoid is the union of the supports of its elements. The <em>modulus</em> of an rcwa monoid is the lcm of the moduli of its elements in case such an lcm exists and 0 otherwise. An rcwa monoid is called <em>tame</em> if its modulus is nonzero, and <em>wild</em> otherwise. The <em>prime set</em> of an rcwa monoid is the union of the prime sets of its elements. An rcwa monoid is called <em>integral</em>, <em>class-wise order-preserving</em> or <em>sign-preserving</em> if all of its elements are so.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ShortOrbits</code>( <var class="Arg">M</var>, <var class="Arg">S</var>, <var class="Arg">maxlng</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of finite forward orbits of the rcwa monoid <var class="Arg">M</var> of length at most <var class="Arg">maxlng</var> which start at points in the set <var class="Arg">S</var>.</p>
<p>Balls of given radius around an element of an rcwa monoid can be computed by the operation <code class="code">Ball</code>. This operation can also be used for computing forward orbits or subsets of such under the action of an rcwa monoid:</p>
<h5>4.2-2 <span class="Heading">
Ball (for monoid, element and radius or monoid, point, radius and action)
</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ball</code>( <var class="Arg">M</var>, <var class="Arg">f</var>, <var class="Arg">r</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ball</code>( <var class="Arg">M</var>, <var class="Arg">p</var>, <var class="Arg">r</var>, <var class="Arg">action</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: the ball of radius <var class="Arg">r</var> around the element <var class="Arg">f</var> in the monoid <var class="Arg">M</var>, respectively the ball of radius <var class="Arg">r</var> around the point <var class="Arg">p</var> under the action <var class="Arg">action</var> of the monoid <var class="Arg">M</var>.</p>
<p>All balls are understood with respect to <code class="code">GeneratorsOfMonoid(<var class="Arg">M</var>)</code>. As membership tests can be expensive, the first-mentioned method does not check whether <var class="Arg">f</var> is indeed an element of <var class="Arg">M</var>. The methods require that point- / element comparisons are cheap. They are not only applicable to rcwa monoids. If the option <var class="Arg">Spheres</var> is set, the ball is split up and returned as a list of spheres.</p>
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