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<div class="ChapSects"><a href="chap3.html#X85297B407ACAED81">3 <span class="Heading">Homomorphisms of many-object structures</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X82F856A086B93832">3.1 <span class="Heading">Homomorphisms of magmas with objects</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86E00FEA7FF38FEA">3.1-1 MagmaWithObjectsHomomorphism</a></span>
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<h3>3 <span class="Heading">Homomorphisms of many-object structures</span></h3>
<p>A <em>homomorphism</em> f from a magma with objects M to a magma with objects N consists of a map f_O from the objects of M to those of N together with a map f_A from the arrows of M to those of N which is compatible with tail and head and which preserves multiplication:</p>
<p class="pcenter">
f_A((a : u \to v)*f(b : v \to w)) ~=~ f_A(a*b : u \to w)
</p>
<p>with tail f_O(u) and head f_O(v).</p>
<p><a id="X82F856A086B93832" name="X82F856A086B93832"></a></p>
<h4>3.1 <span class="Heading">Homomorphisms of magmas with objects</span></h4>
<p><a id="X86E00FEA7FF38FEA" name="X86E00FEA7FF38FEA"></a></p>
<h5>3.1-1 MagmaWithObjectsHomomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MagmaWithObjectsHomomorphism</code>( <var class="Arg">args</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MagmaHomomorphismFromSinglePiece</code>( <var class="Arg">src, rng, hom, imobs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> HomomorphismToSinglePiece</code>( <var class="Arg">src, rng, images</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> HomomorphismByUnion</code>( <var class="Arg">src, rng, homs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>As usual, there are a variety of homomorphism constructors. The basic construction is a homomorphism M -> N with both M and N connected, which is implemented as <code class="code">IsHomomorphismToSinglePieceRep</code> with attributes <code class="code">Source</code>, <code class="code">Range</code> and <code class="code">PieceImages</code>. We require the following information:</p>
<ul>
<li><p>a magma homomorphism <code class="code">f</code> from the underlying of M to the underlying magma of N.</p>
</li>
<li><p>a list <code class="code">imobs</code> of the images of the objects of M;</p>
</li>
</ul>
<p>In the example we construct endomappings of m and M78.</p>
<table class="example">
<tr><td><pre>
gap> tup1 := [ Tuple([m1,m2]), Tuple([m2,m1]), Tuple([m3,m4]), Tuple([m4,m3]) ];
gap> f1 := GeneralMappingByElements( m, m, tup1 );
f1 = <general mapping: m -> m >
gap> IsMagmaHomomorphism( f1 );
true
gap> tup2 := [ Tuple([m1,m1]), Tuple([m2,m1]), Tuple([m3,m1]), Tuple([m4,m1]) ];;
gap> f2 := GeneralMappingByElements( m, m, tup2 );;
gap> IsMagmaHomomorphism( f2 );
true
gap> map1 := HomomorphismFromSinglePiece( M78, M78, [-8,-7], f1 );
magma with objects homomorphism : M78 -> M78
gap> Display( map1 );
Mapping to single piece magma:
[ M78 ] -> [ M78 ]
magma mapping: <mapping: m -> m >
object map: [ -8, -7 ] -> [ -8, -7 ]
Homomorphism to connected magma:
[ M78 ] -> [ M78 ]
object map = [ [ -8, -7 ], [ -8, -7 ] ]
homomorphism = <homomorphism: m -> m >
gap> idm := f1*f1;;
gap> idmap := HomomorphismFromSinglePiece( M78, M78, idm, [-7,-8] );
gap> map2 := HomomorphismFromSinglePiece( M78, M78, f2, [-7,-8] );
</pre></td></tr></table>
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