Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/groupoids/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 11.8.2025 mit Größe 20 kB image not shown  

Quellcode-Bibliothek chap3.html   Sprache: HTML

 
 products/sources/formale Sprachen/GAP/pkg/groupoids/doc/chap3.html


<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (groupoids) - Chapter 3: Mappings of many-object structures</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap3"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap2.html">[Previous Chapter]</a>    <a href="chap4.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap3_mj.html">[MathJax on]</a></p>
<p><a id="X78FC7902804CED8E" name="X78FC7902804CED8E"></a></p>
<div class="ChapSects"><a href="chap3.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.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.html#X86E00FEA7FF38FEA">3.1-1 MagmaWithObjectsHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.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.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.html#X7AE44D9485EB50F1">3.3-1 HomomorphismByUnion</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C053B0379DDCE13">3.3-2 IsInjectiveOnObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.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.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="pcenter">
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 -> 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 -> 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.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.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>


<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap2.html">[Previous Chapter]</a>    <a href="chap4.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

100%


¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.0.50Bemerkung:  (vorverarbeitet)  ¤

*© Formatika GbR, Deutschland




Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.