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<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap2_mj.html">[Previous Chapter]</a>    <a href="chap4_mj.html">[Next Chapter]</a>   </div>

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<p><a id="X78FC7902804CED8E" name="X78FC7902804CED8E"></a></p>
<div class="ChapSects"><a href="chap3_mj.html#X78FC7902804CED8E">3 <span class="Heading">Mappings of many-object structures</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X82F856A086B93832">3.1 <span class="Heading">Homomorphisms of magmas with objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86E00FEA7FF38FEA">3.1-1 MagmaWithObjectsHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7C4D1AEE80D41A35">3.2 <span class="Heading">Homomorphisms of semigroups and monoids with objects</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X795C8DE37AED7B44">3.3 <span class="Heading">Homomorphisms to more than one piece</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AE44D9485EB50F1">3.3-1 HomomorphismByUnion</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C053B0379DDCE13">3.3-2 IsInjectiveOnObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7B9D71BB7BAEFAAA">3.4 <span class="Heading">Mappings defined by a function</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X81B64F9A855D265F">3.4-1 MappingWithObjectsByFunction</a></span>
</div></div>
</div>

<h3>3 <span class="Heading">Mappings of many-object structures</span></h3>

<p>A <em>homomorphism</em> <span class="SimpleMath">\(f\)</span> from a magma with objects <span class="SimpleMath">\(M\)</span> to a magma with objects <span class="SimpleMath">\(N\)</span> consists of</p>


<ul>
<li><p>a map <span class="SimpleMath">\(f_O\)</span> from the objects of <span class="SimpleMath">\(M\)</span> to those of <span class="SimpleMath">\(N\)</span>,</p>

</li>
<li><p>a map <span class="SimpleMath">\(f_A\)</span> from the arrows of <span class="SimpleMath">\(M\)</span> to those of <span class="SimpleMath">\(N\)</span>.</p>

</li>
</ul>
<p>The map <span class="SimpleMath">\(f_A\)</span> is required to be compatible with the tail and head maps and to preserve multiplication:</p>

<p class="center">\[
f_A(a : u \to v) * f_A(b : v \to w) ~=~ f_A(a*b : u \to w) 
\]</p>

<p>with tail <span class="SimpleMath">\(f_O(u)\)</span> and head <span class="SimpleMath">\(f_O(w)\)</span>.</p>

<p>When the underlying magma of <span class="SimpleMath">\(M\)</span> is a monoid or group, the map <span class="SimpleMath">\(f_A\)</span> is required to preserve identities and inverses.</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">‣ HomomorphismFromSinglePiece</code>( <var class="Arg">src</var>, <var class="Arg">rng</var>, <var class="Arg">hom</var>, <var class="Arg">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</var>, <var class="Arg">rng</var>, <var class="Arg">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">‣ MappingToSinglePieceData</code>( <var class="Arg">mwohom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PiecesOfMapping</code>( <var class="Arg">mwohom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismNewObjects</code>( <var class="Arg">src</var>, <var class="Arg">objlist</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>There are a variety of homomorphism constructors.</p>

<p>The simplest construction gives a homomorphism <span class="SimpleMath">\(M \to N\)</span> with both <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(N\)</span> connected. It is implemented as <code class="code">IsMappingToSinglePieceRep</code> with attributes <code class="code">Source</code>, <code class="code">Range</code> and <code class="code">MappingToSinglePieceData</code>. The operation requires the following information:</p>


<ul>
<li><p>a magma homomorphism <code class="code">hom</code> from the underlying magma of <span class="SimpleMath">\(M\)</span> to the underlying magma of <span class="SimpleMath">\(N\)</span>,</p>

</li>
<li><p>a list <code class="code">imobs</code> of the images of the objects of <span class="SimpleMath">\(M\)</span>.</p>

</li>
</ul>
<p>In the first example we construct endomappings of <code class="code">m</code> and <code class="code">M78</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">tup1 := [ DirectProductElement([m1,m2]), DirectProductElement([m2,m1]), </span>
<span class="GAPprompt">></span> <span class="GAPinput">             DirectProductElement([m3,m4]), DirectProductElement([m4,m3]) ];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f1 := GeneralMappingByElements( m, m, tup1 ); </span>
<general mapping: m -> m >
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMagmaHomomorphism( f1 ); </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">hom1 := MagmaWithObjectsHomomorphism( M78, M78, f1, [-7,-8] ); </span>
magma with objects homomorphism : M78 -> M78
[ [ <mapping: m -> m >, [ -7, -8 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">[ Source( hom1 ), Range( hom1 ) ]; </span>
[ M78, M78 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">b87;</span>
[m4 : -8 -> -7]
<span class="GAPprompt">gap></span> <span class="GAPinput">im1 := ImageElm( hom1, b87 );</span>
[m3 : -7 -> -8]
<span class="GAPprompt">gap></span> <span class="GAPinput">i65 := IsomorphismNewObjects( M78, [-6,-5] ); </span>
magma with objects homomorphism : [ [ IdentityMapping( m ), [ -6, -5 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ib87 := ImageElm( i65, b87 );</span>
[m4 : -6 -> -5]
<span class="GAPprompt">gap></span> <span class="GAPinput">M65 := Range( i65);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( M65, "M65" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">j65 := InverseGeneralMapping( i65 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImagesOfObjects( j65 ); </span>
[ -8, -7 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">comp := j65 * hom1;</span>
magma with objects homomorphism : M65 -> M78
[ [ <mapping: m -> m >, [ -7, -8 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( comp, ib87 );</span>
[m3 : -7 -> -8]

</pre></div>

<p>A homomorphism <em>to</em> a connected magma with objects may have a source with several pieces, and so is a union of homomorphisms <em>from</em> single pieces.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">M4 := UnionOfPieces( [ M78, M65 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">images := [ MappingToSinglePieceData( hom1 )[1], </span>
<span class="GAPprompt">></span> <span class="GAPinput">MappingToSinglePieceData( j65 )[1] ]; </span>
[ [ <mapping: m -> m >, [ -7, -8 ] ], [ IdentityMapping( m ), [ -8, -7 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">map4 := HomomorphismToSinglePiece( M4, M78, images ); </span>
magma with objects homomorphism : 
[ [ <mapping: m -> m >, [ -7, -8 ] ], [ IdentityMapping( m ), [ -8, -7 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( map4, b87 ); </span>
[m3 : -7 -> -8]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( map4, ib87 );</span>
[m4 : -8 -> -7]

</pre></div>

<p><a id="X7C4D1AEE80D41A35" name="X7C4D1AEE80D41A35"></a></p>

<h4>3.2 <span class="Heading">Homomorphisms of semigroups and monoids with objects</span></h4>

<p>The next example exhibits a homomorphism between transformation semigroups with objects.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">t2 := Transformation( [2,2,4,1] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s2 := Transformation( [1,1,4,4] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">r2 := Transformation( [4,1,3,3] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sgp2 := Semigroup( [ t2, s2, r2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( sgp2, "sgp<t2,s2,r2>" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">##  apparently no method for transformation semigroups available for: </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">##  nat := NaturalHomomorphismByGenerators( sgp, sgp2 );  so we use: </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">##  in the function flip below t is a transformation on [1..n] </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">flip := function( t ) </span>
<span class="GAPprompt">></span> <span class="GAPinput">    local i, j, k, L, L2, n; </span>
<span class="GAPprompt">></span> <span class="GAPinput">    n := DegreeOfTransformation( t );  </span>
<span class="GAPprompt">></span> <span class="GAPinput">    L := ImageListOfTransformation( t ); </span>
<span class="GAPprompt">></span> <span class="GAPinput">    if IsOddInt(n) then n:=n+1; L1:=Concatenation(L,[n]); </span>
<span class="GAPprompt">></span> <span class="GAPinput">                   else L1:=L; fi; </span>
<span class="GAPprompt">></span> <span class="GAPinput">    L2 := ShallowCopy( L1 );</span>
<span class="GAPprompt">></span> <span class="GAPinput">    for i in [1..n] do </span>
<span class="GAPprompt">></span> <span class="GAPinput">        if IsOddInt(i) then j:=i+1; else j:=i-1; fi; </span>
<span class="GAPprompt">></span> <span class="GAPinput">        k := L1[j]; </span>
<span class="GAPprompt">></span> <span class="GAPinput">        if IsOddInt(k) then L2[i]:=k+1; else L2[i]:=k-1; fi; </span>
<span class="GAPprompt">></span> <span class="GAPinput">    od; </span>
<span class="GAPprompt">></span> <span class="GAPinput">    return( Transformation( L2 ) ); </span>
<span class="GAPprompt">></span> <span class="GAPinput">end;; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">smap := MappingByFunction( sgp, sgp2, flip );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ok := RespectsMultiplication( smap ); </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">[ t, ImageElm( smap, t ) ]; </span>
[ Transformation( [ 1, 1, 2, 3 ] ), Transformation( [ 2, 2, 4, 1 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">[ s, ImageElm( smap, s ) ]; </span>
[ Transformation( [ 2, 2, 3, 3 ] ), Transformation( [ 1, 1, 4, 4 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">[ r, ImageElm( smap, r ) ]; </span>
[ Transformation( [ 2, 3, 4, 4 ] ), Transformation( [ 4, 1, 3, 3 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( smap, "smap" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">T123 := SemigroupWithObjects( sgp2, [-13,-12,-11] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">shom := MagmaWithObjectsHomomorphism( S123, T123, smap, [-11,-12,-13] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">it12 := ImageElm( shom, t12 );;  [ t12, it12 ]; </span>
[ [Transformation( [ 1, 1, 2, 3 ] ) : -1 -> -2], 
  [Transformation( [ 2, 2, 4, 1 ] ) : -13 -> -12] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">is23 := ImageElm( shom, s23 );;  [ s23, is23 ]; </span>
[ [Transformation( [ 2, 2, 3, 3 ] ) : -2 -> -3], 
  [Transformation( [ 1, 1, 4, 4 ] ) : -12 -> -11] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ir31 := ImageElm( shom, r31 );;  [ r31, ir31 ]; </span>
[ [Transformation( [ 2, 3, 4, 4 ] ) : -3 -> -1], 
  [Transformation( [ 4, 1, 3, 3 ] ) : -11 -> -13] ]

</pre></div>

<p><a id="X795C8DE37AED7B44" name="X795C8DE37AED7B44"></a></p>

<h4>3.3 <span class="Heading">Homomorphisms to more than one piece</span></h4>

<p><a id="X7AE44D9485EB50F1" name="X7AE44D9485EB50F1"></a></p>

<h5>3.3-1 HomomorphismByUnion</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomomorphismByUnion</code>( <var class="Arg">src</var>, <var class="Arg">rng</var>, <var class="Arg">homs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>When <span class="SimpleMath">\(f : M \to N\)</span> and <span class="SimpleMath">\(N\)</span> has more than one connected component, then <span class="SimpleMath">\(M\)</span> also has more than one component and <span class="SimpleMath">\(f\)</span> is a union of homomorphisms, one for each piece in the range.</p>

<p>See section <a href="chap5_mj.html#X795C8DE37AED7B44"><span class="RefLink">5.5</span></a> for the equivalent operation with groupoids.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">N4 := UnionOfPieces( [ M78, T123 ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h14 := HomomorphismByUnionNC( N1, N4, [ hom1, shom ] ); </span>
magma with objects homomorphism : 
[ magma with objects homomorphism : M78 -> M78
    [ [ <mapping: m -> m >, [ -7, -8 ] ] ], magma with objects homomorphism : 
    [ [ smap, [ -11, -12, -13 ] ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( h14, a78 );</span>
[m1 : -8 -> -7]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( h14, r31 );</span>
[Transformation( [ 4, 1, 3, 3 ] ) : -11 -> -13]

</pre></div>

<p><a id="X7C053B0379DDCE13" name="X7C053B0379DDCE13"></a></p>

<h5>3.3-2 IsInjectiveOnObjects</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInjectiveOnObjects</code>( <var class="Arg">mwohom</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSurjectiveOnObjects</code>( <var class="Arg">mwohom</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBijectiveOnObjects</code>( <var class="Arg">mwohom</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEndomorphismWithObjects</code>( <var class="Arg">mwohom</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAutomorphismWithObjects</code>( <var class="Arg">mwohom</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>The meaning of these five properties is obvious.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">IsInjectiveOnObjects( h14 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSurjectiveOnObjects( h14 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBijectiveOnObjects( h14 ); </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEndomorphismWithObjects( h14 ); </span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAutomorphismWithObjects( h14 ); </span>
false

</pre></div>

<p><a id="X7B9D71BB7BAEFAAA" name="X7B9D71BB7BAEFAAA"></a></p>

<h4>3.4 <span class="Heading">Mappings defined by a function</span></h4>

<p><a id="X81B64F9A855D265F" name="X81B64F9A855D265F"></a></p>

<h5>3.4-1 MappingWithObjectsByFunction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MappingWithObjectsByFunction</code>( <var class="Arg">src</var>, <var class="Arg">rng</var>, <var class="Arg">fun</var>, <var class="Arg">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">‣ IsMappingWithObjectsByFunction</code>( <var class="Arg">map</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingFunction</code>( <var class="Arg">map</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>More general mappings, which need not preserve multiplication, are available using this operation. See chapter <a href="chap6_mj.html#X803E01577A2B37D2"><span class="RefLink">6</span></a> for an application.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">swap := function(a) return Arrow(M78,a![1],a![3],a![2]); end;      </span>
function( a ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">swapmap := MappingWithObjectsByFunction( M78, M78, swap, [-7,-8] );</span>
magma with objects mapping by function : M78 -> M78
function: function ( a )
    return Arrow( M78, a![1], a![3], a![2] );
end

<span class="GAPprompt">gap></span> <span class="GAPinput">a78; ImageElm( swapmap, a78 );              </span>
[m2 : -7 -> -8]
[m2 : -8 -> -7]

</pre></div>


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