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<h1>5 Transformation nearrings</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP005.htm#SECT001">Constructing transformation nearrings</a>
<li> <A HREF="CHAP005.htm#SECT002">Nearrings of transformations</a>
<li> <A HREF="CHAP005.htm#SECT003">The group a transformation nearring acts on</a>
<li> <A HREF="CHAP005.htm#SECT004">Transformation nearrings and other nearrings</a>
<li> <A HREF="CHAP005.htm#SECT005">Noetherian quotients for transformation nearrings</a>
<li> <A HREF="CHAP005.htm#SECT006">Zerosymmetric mappings</a>
</ol>

In the previous chapter we introduced mappings on groups, and we
called them endomappings. We also introduced the operation of
pointwise addition <code>+</code> for endomappings. Now we are able to use these
mappings together with pointwise addition <code>+</code> and composition <code>*</code> to
construct left nearrings. These nearrings satisfy the distributive
law x * (y + z) = x * y + x * z.

A transformation nearring is a set of mappings on a group G that
is closed under pointwise addition of mappings, under forming the
additive inverse and under functional composition. For more
information we suggest <a href="biblio.htm#Pilz:Nearrings"><[>Pilz:Nearrings</cite></a>], <a href="biblio.htm#meldrum85:NATLWG"><[>meldrum85:NATLWG</cite></a>],
and <a href="biblio.htm#Clay:Nearrings"><[>Clay:Nearrings</cite></a>],

The algorithms used can be found in
<a href="biblio.htm#aichingereckernoebauer00:TUOCINT"><[>aichingereckernoebauer00:TUOCINT</cite></a>] and <a href="biblio.htm#aichingerea00:CWN"><[>aichingerea00:CWN</cite></a>].

The elements of a transformation nearring are given as endomappings on
the group G (cf. Chapter ``Functions on groups that are not
necessarily homomorphisms: EndoMappings'').


<h2><a name="SECT001">5.1 Constructing transformation nearrings</a></h2>

<a name = "SSEC001.1"></a>
<li><code>TransformationNearRingByGenerators( </code><var>G</var><code>, </code><var>endomaplist</var><code> )</code>

For a (possibly empty) list <var>endomaplist</var> of endomappings on a group
<var>G</var>, the constructor function <code>TransformationNearRingByGenerators</code> returns the
nearring generated by these mappings. All of them must be
endomappings on the group <var>G</var>.

<pre>
 gap> g := AlternatingGroup ( 4 );
 Alt( [ 1 .. 4 ] )
 gap> AsSortedList ( g );
 [ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2),
 (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ]
 gap> t := EndoMappingByPositionList ( g, [1,3,4,5,2,1,1,1,1,1,1,1] );
 <mapping: AlternatingGroup( [ 1 .. 4 ] ) -> AlternatingGroup(
 [ 1 .. 4 ] ) >
 gap> m := TransformationNearRingByGenerators ( g, [t] );
 TransformationNearRingByGenerators(
 [ <mapping: AlternatingGroup( [ 1 .. 4 ] ) -> AlternatingGroup(
 [ 1 .. 4 ] ) > ])
 gap> Size (m); # may take a few moments
 20736
 gap> IsCommutative ( m );
 false
</pre>

<a name = "SSEC001.2"></a>
<li><code>TransformationNearRingByAdditiveGenerators( </code><var>G</var><code>, </code><var>endomaplist</var><code> )</code>

If a transformation nearring is known to be additively generated by a
set of endomappings on a group (as for example the distributively
generated nearrings E(G), A(G) and I(G)), the function
<code>TransformationNearRingByAdditiveGenerators</code> allows to
construct this nearring. The only difference between
<code>TransformationNearRingByGenerators</code> and
<code>TransformationNearRingByAdditiveGenerators</code> is that
<code>TransformationNearRingByAdditiveGenerators</code> is much faster.

<pre>
 gap> G := SymmetricGroup(3);;
 gap> endos := Endomorphisms ( G );
 [ [ (1,2,3), (1,2) ] -> [ (), () ], [ (1,2,3), (1,2) ] -> [ (), (2,3) ],
 [ (1,2,3), (1,2) ] -> [ (), (1,2) ], [ (1,2,3), (1,2) ] -> [ (), (1,3) ],
 [ (1,2,3), (1,2) ] -> [ (1,2,3), (2,3) ],
 [ (1,2,3), (1,2) ] -> [ (1,3,2), (2,3) ],
 [ (1,2,3), (1,2) ] -> [ (1,3,2), (1,2) ],
 [ (1,2,3), (1,2) ] -> [ (1,2,3), (1,2) ],
 [ (1,2,3), (1,2) ] -> [ (1,2,3), (1,3) ],
 [ (1,2,3), (1,2) ] -> [ (1,3,2), (1,3) ] ]
 gap> Endo := TransformationNearRingByAdditiveGenerators ( G, endos );
 < transformation nearring with 10 generators >
 gap> Size( Endo );
 54
</pre>


<h2><a name="SECT002">5.2 Nearrings of transformations</a></h2>

<a name = "SSEC002.1"></a>
<li><code>MapNearRing( </code><var>G</var><code> )</code>

<a name = "SSEC002.2"></a>
<li><code>TransformationNearRing( </code><var>G</var><code> )</code>

<code>MapNearRing</code> and <code>TransformationNearRing</code> both return the nearring of all
mappings on <var>G</var>.

<pre>
 gap> m := MapNearRing ( GTW32_12 );
 TransformationNearRing(32/12)
 gap> Size ( m );
 1461501637330902918203684832716283019655932542976
 gap> NearRingIdeals ( m );
 [ < nearring ideal >, < nearring ideal > ]
</pre>

<a name = "SSEC002.3"></a>
<li><code>IsFullTransformationNearRing( </code><var>tfmnr</var><code> )</code>

The function <code>IsFullTransformationNearRing</code> returns <code>true</code> if the
transformation nearring <var>tfmnr</var> is the nearring of all mappings over
the group.

<pre>
 gap> g := CyclicGroup ( 4 );
 <pc group of size 4 with 2 generators>
 gap> m := MapNearRing ( g );
 TransformationNearRing(<pc group of size 4 with 2 generators>)
 gap> gens := Filtered ( AsList ( m ),
 > f -> IsFullTransformationNearRing (
 > TransformationNearRingByGenerators ( g, [ f ] )));;
 gap> Length(gens);
 12
</pre>

<a name = "SSEC002.4"></a>
<li><code>PolynomialNearRing( </code><var>G</var><code> )</code>

<code>PolynomialNearRing</code> returns the nearring of all polynomial functions on <var>G</var>.

<pre>
 gap> P := PolynomialNearRing ( GTW16_6 );
 PolynomialNearRing( 16/6 )
 gap> Size ( P );
 256
</pre>

<a name = "SSEC002.5"></a>
<li><code>EndomorphismNearRing( </code><var>G</var><code> )</code>

<code>EndomorphismNearRing</code> returns the nearring generated by all endomorphisms
on <var>G</var>.

<pre>
 gap> ES4 := EndomorphismNearRing ( SymmetricGroup ( 4 ) );
 EndomorphismNearRing( Sym( [ 1 .. 4 ] ) )
 gap> Size ( ES4 );
 927712935936
</pre>

<a name = "SSEC002.6"></a>
<li><code>AutomorphismNearRing( </code><var>G</var><code> )</code>

<code>AutomorphismNearRing</code> returns the nearring generated by all automorphisms
on <var>G</var>.

<pre>
 gap> A := AutomorphismNearRing ( DihedralGroup ( 8 ) );
 AutomorphismNearRing( <pc group of size 8 with 3 generators> )
 gap> Length(NearRingRightIdeals ( A ));
 28
 gap> Size (A);
 32
</pre>

<a name = "SSEC002.7"></a>
<li><code>InnerAutomorphismNearRing( </code><var>G</var><code> )</code>

<code>InnerAutomorphismNearRing</code> returns the nearring generated by all inner
automorphisms on <var>G</var>.

<pre>
 gap> I := InnerAutomorphismNearRing ( AlternatingGroup ( 4 ) );
 InnerAutomorphismNearRing( Alt( [ 1 .. 4 ] ) )
 gap> Size ( I );
 3072
 gap> m := Enumerator( I )[1000];
 <mapping: AlternatingGroup( [ 1 .. 4 ] ) -> AlternatingGroup( [ 1 .. 4 ] ) >
 gap> graph := List ( AsList ( AlternatingGroup ( 4 ) ),
 > x -> [x, Image (m, x)] );
 [ [ (), () ], [ (2,3,4), (1,4)(2,3) ], [ (2,4,3), (1,4)(2,3) ],
 [ (1,2)(3,4), (1,2)(3,4) ], [ (1,2,3), (1,3)(2,4) ],
 [ (1,2,4), (1,4)(2,3) ], [ (1,3,2), (1,4)(2,3) ], [ (1,3,4), (1,2)(3,4) ],
 [ (1,3)(2,4), (1,3)(2,4) ], [ (1,4,2), () ], [ (1,4,3), (1,4)(2,3) ],
 [ (1,4)(2,3), (1,4)(2,3) ] ]
</pre>

<a name = "SSEC002.8"></a>
<li><code>CompatibleFunctionNearRing( </code><var>G</var><code> )</code>

<code>CompatibleFunctionNearRing</code> returns the nearring of all compatible functions
on the group <var>G</var>. A function m:G → G is compatible iff for every normal
subgroup N of G and all g,h ∈ G if g and h are in the same coset of N
then their images under m are in the same coset of G.

<a name = "SSEC002.9"></a>
<li><code>ZeroSymmetricCompatibleFunctionNearRing( </code><var>G</var><code> )</code>

<code>ZeroSymmetricCompatibleFunctionNearRing</code> returns the nearring of all zerosymmetric
compatible functions on the group <var>G</var>. This function is also called by
<code>CompatibleFunctionNearRing</code>.

<a name = "SSEC002.10"></a>
<li><code>IsCompatibleEndoMapping( </code><var>m</var><code> )</code>

<code>IsCompatibleEndoMapping</code> returns <code>true</code> iff <var>m</var> is a compatible function on its
source.

<a name = "SSEC002.11"></a>
<li><code>Is1AffineComplete( </code><var>G</var><code> )</code>

A group <var>G</var> is called 1-affine complete, iff every compatible function on <var>G</var> is
polynomial. <code>Is1AffineComplete</code> returns <code>true</code> iff <var>G</var> is 1-affine complete.

<a name = "SSEC002.12"></a>
<li><code>CentralizerNearRing( </code><var>G</var><code>, </code><var>endos</var><code> )</code>

<code>CentralizerNearRing</code> returns the nearring of all functions
m:G → G such that for all endomorphisms e in <var>endos</var> the
equality m °e = e °m holds.

<pre>
 gap> autos := Automorphisms ( GTW8_4 );
 [ IdentityMapping( 8/4 ), ^(2,4),
 [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,2)(3,4) ],
 [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,2)(3,4) ], ^(1,4)(2,3),
 ^(1,2,3,4), [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,4)(2,3) ],
 [ (1,4)(2,3), (1,4,3,2) ] -> [ (2,4), (1,2,3,4) ] ]
 gap> C := CentralizerNearRing ( GTW8_4, autos );
 CentralizerNearRing( 8/4, ... )
 gap> C0 := ZeroSymmetricPart ( C );
 < transformation nearring with 4 generators >
 gap> Size ( C0 );
 32
 gap> Is := NearRingIdeals ( C0 );
 [ < nearring ideal >, < nearring ideal >, < nearring ideal >,
 < nearring ideal >, < nearring ideal >, < nearring ideal >,
 < nearring ideal >, < nearring ideal >, < nearring ideal >,
 < nearring ideal >, < nearring ideal >, < nearring ideal >,
 < nearring ideal > ]
 gap> List (Is, Size);
 [ 1, 2, 4, 2, 4, 8, 8, 16, 4, 8, 16, 16, 32 ]
</pre>

<a name = "SSEC002.13"></a>
<li><code>RestrictedEndomorphismNearRing( </code><var>G</var><code>, </code><var>U</var><code> )</code>

<code>RestrictedEndomorphismNearRing</code> returns the nearring generated by all
endomorphisms e on G with e(G) ⊆ U.

<pre>
 gap> G := GTW16_8;
 16/8
 gap> U := First ( NormalSubgroups ( G ),
 > x -> Size (x) = 2 );
 Group([ ( 1, 5)( 2,10)( 3,11)( 4,12)( 6,15)( 7,16)( 8, 9)(13,14) ])
 gap> HGU := RestrictedEndomorphismNearRing (G, U);
 RestrictedEndomorphismNearRing( 16/8, Group(
 [ ( 1, 5)( 2,10)( 3,11)( 4,12)( 6,15)( 7,16)( 8, 9)(13,14) ]) )
 gap> Size (HGU);
 8
 gap> IsDistributiveNearRing ( HGU );
 true
 gap> Filtered ( AsList ( HGU),
 > x -> x = x * x );
 [ <mapping: 16/8 -> 16/8 > ]
</pre>

<a name = "SSEC002.14"></a>
<li><code>LocalInterpolationNearRing( </code><var>tfmnr</var><code>, </code><var>m</var><code> )</code>

<code>LocalInterpolationNearRing</code> returns the nearring of all mappings on
G that can be interpolated at any set of m places by a mapping in
<var>tfmnr</var>, where G is the domain and codomain of the elements in
<var>tfmnr</var>.

<pre>
 gap> P := PolynomialNearRing ( GTW8_5 );
 PolynomialNearRing( 8/5 )
 gap> L := LocalInterpolationNearRing ( P, 2 );
 LocalInterpolationNearRing( PolynomialNearRing( 8/5 ), 2 )
 gap> Size ( L ) / Size ( P );
 16
</pre>


<h2><a name="SECT003">5.3 The group a transformation nearring acts on</a></h2>

<a name = "SSEC003.1"></a>
<li><code>Gamma( </code><var>tfmnr</var><code> )</code>

The function <code>Gamma</code> returns the group on which the mappings of the
nearring <var>tfmnr</var> act.

<pre>
 gap> Gamma ( PolynomialNearRing ( CyclicGroup ( 25 ) ) );
 <pc group of size 25 with 2 generators>
 gap> IsCyclic (last);
 true
</pre>


<h2><a name="SECT004">5.4 Transformation nearrings and other nearrings</a></h2>

<a name = "SSEC004.1"></a>
<li><code>AsTransformationNearRing( </code><var>nr</var><code> )</code>

Provided that <var>nr</var> is not already a transformation nearring,
<code>AsTransformationNearRing</code> returns a transformation nearring that is isomorphic
to the nearring <var>nr</var>.

<pre>
 gap> L := LibraryNearRing (GTW8_3, 12);
 LibraryNearRing(8/3, 12)
 gap> Lt := AsTransformationNearRing ( L );
 < transformation nearring with 3 generators >
 gap> Gamma ( Lt );
 8/3 x C_2
</pre>

<a name = "SSEC004.2"></a>
<li><code>AsExplicitMultiplicationNearRing( </code><var>nr</var><code> )</code>

Provided that <var>nr</var> is not already an explicit multiplication nearring
(i. e. a transformation nearring), <code>AsExplicitMultiplicationNearRing</code> returns
an explicit multiplication nearring that is isomorphic to the nearring <var>nr</var>.

<pre>
 gap> P := PolynomialNearRing ( GTW4_2 );
 PolynomialNearRing( 4/2 )
 gap> n := AsExplicitMultiplicationNearRing ( P );
 ExplicitMultiplicationNearRing ( Group(
 [ ( 1, 2)( 5, 6)( 9,10)(13,14), ( 3, 4)( 7, 8)(11,12)(15,16),
 ( 7, 8)( 9,10)(13,14)(15,16) ]) , multiplication )
</pre>


<h2><a name="SECT005">5.5 Noetherian quotients for transformation nearrings</a></h2>

<a name = "SSEC005.1"></a>
<li><code>NoetherianQuotient( </code><var>tfmnr</var><code>, </code><var>target</var><code>, </code><var>source</var><code> )</code>

<code>NoetherianQuotient</code> returns the set of all mappings t in <var>tfmnr</var>
with t(<tt>source</tt>) ⊆ <tt>target</tt>.

<pre>
 gap> G := SymmetricGroup ( 4 );
 Sym( [ 1 .. 4 ] )
 gap> V := First ( NormalSubgroups ( G ), x -> Size ( x ) = 4 );
 Group([ (1,4)(2,3), (1,3)(2,4) ])
 gap> P := InnerAutomorphismNearRing ( G );
 InnerAutomorphismNearRing( Sym( [ 1 .. 4 ] ) )
 gap> N := NoetherianQuotient ( P, V, G );
 NoetherianQuotient( Group([ (1,4)(2,3), (1,3)(2,4) ]) ,Sym(
 [ 1 .. 4 ] ) )
 gap> Size ( P ) / Size ( N );
 54
</pre>

<a name = "SSEC005.2"></a>
<li><code>CongruenceNoetherianQuotient( </code><var>P</var><code>, </code><var>A</var><code>, </code><var>B</var><code>, </code><var>C</var><code> )</code>

<code>CongruenceNoetherianQuotient</code> returns the ideal of all those mappings in <var>P</var> that
map every element of the group Gamma(P) into <var>C</var>, and maps two elements that
are congruent modulo <var>B</var> into elements that are congruent modulo <var>A</var>.
Input conditions: (1) <var>P</var> is the nearring of polynomial functions on a group G,
 (2) <var>A</var> is a normal subgroup of G,
 (3) <var>B</var> is a normal subgroup of G,
 (4) <var>C</var> is a normal subgroup of G,
 (5) [C,B] is less or equal to A.
<pre>
 gap> G := GTW8_4;
 8/4
 gap> P := PolynomialNearRing (G);
 PolynomialNearRing( 8/4 )
 gap> A := TrivialSubgroup (G);
 Group(())
 gap> B := DerivedSubgroup (G);
 Group([ (1,3)(2,4) ])
 gap> C := G;
 8/4
 gap> I := CongruenceNoetherianQuotient (P, A, B, C);
 < nearring ideal >
 gap> Size (P/I);
 2
</pre>

<a name = "SSEC005.3"></a>
<li><code>CongruenceNoetherianQuotientForInnerAutomorphismNearRings (</code><var>I</var><code>, </code><var>A</var><code>, </code><var>B</var><code>, </code><var>C</var><code> )</code>

<code>CongruenceNoetherianQuotientForInnerAutomorphismNearRings</code> returns the ideal of all those mappings in <var>I</var> that
map every element of the group Gamma(I) into <var>C</var>, and maps two elements that
are congruent modulo <var>B</var> into elements that are congruent modulo <var>A</var>.
Input conditions: (1) <var>P</var> is the nearring of polynomial functions on a group G,
 (2) <var>A</var> is a normal subgroup of G,
 (3) <var>B</var> is a normal subgroup of G,
 (4) <var>C</var> is a normal subgroup of G,
 (5) [C,B] is less or equal to A.
<pre>
 gap> G := GTW8_4;
 8/4
 gap> I := InnerAutomorphismNearRing (G);
 InnerAutomorphismNearRing( 8/4 )
 gap> A := TrivialSubgroup (G);
 Group(())
 gap> B := DerivedSubgroup (G);
 Group([ (1,3)(2,4) ])
 gap> C := G;
 8/4
 gap> j := CongruenceNoetherianQuotientForInnerAutomorphismNearRings (I,A,B,C);
 < nearring ideal >
 gap> Size (I/j);
 2
</pre>


<h2><a name="SECT006">5.6 Zerosymmetric mappings</a></h2>

<a name = "SSEC006.1"></a>
<li><code>ZeroSymmetricPart( </code><var>tfmnr</var><code> )</code>

<code>ZeroSymmetricPart</code> returns the nearring of all mappings t in
 <var>tfmnr</var> with t(0) = 0.

<pre>
 gap> g := GTW8_4;
 8/4
 gap> P := PolynomialNearRing ( g );
 PolynomialNearRing( 8/4 )
 gap> Zp := ZeroSymmetricPart ( P );
 < transformation nearring with 4 generators >
 gap> InnerAutomorphismNearRing ( g ) = Zp;
 true
</pre>


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