<html><head><title>[SONATA] 3 The nearring library</title></head>
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<h1>3 The nearring library</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP003.htm#SECT001">Extracting nearrings from the library</a>
<li> <A HREF="CHAP003.htm#SECT002">Identifying nearrings</a>
<li> <A HREF="CHAP003.htm#SECT003">IsLibraryNearRing</a>
<li> <A HREF="CHAP003.htm#SECT004">Accessing the information about a nearring stored in the library</a>
</ol><p>
<p>
The nearring library contains all nearrings up to order 15 and all
nearrings with identity up to order 31. All nearrings in the library are
nearrings constructed via <code>ExplicitMultiplicationNearRingNC</code>, so all functions
for these nearrings are applicable to <code>LibraryNearRing</code>s.
<p>
<p>
<h2><a name="SECT001">3.1 Extracting nearrings from the library</a></h2>
<p><p>
<a name = "SECT001"></a>
<li><code>LibraryNearRing( </code><var>G</var><code>, </code><var>num</var><code> )</code>
<p>
<code>LibraryNearRing</code> retrieves a nearring from the nearrings library files.
<var>G</var> must be a group of order <var>le15</var>. <var>num</var> must be an integer which
indicates the number of the class of nearrings on the specified group.
<p>
(<var>Remark:</var> due to the large number of nearrings on <var>D<sub>12</sub></var>, make sure
that you have enough main memory - say at least 32 MB - available if you want
to get a library nearring on <var>D<sub>12</sub></var>).
<p>
If <var>G</var> is given as a <code>TWGroup</code>, then a nearring is returned whose group reduct
is <strong>equal to</strong> <var>G</var>. Otherwise the result is a nearring whose group reduct is
<strong>isomorphic to</strong> <var>G</var>, and a warning is issued.
<p>
The number of nearrings definable on a certain group <var>G</var> can be accessed via
<p>
<a name = "SECT001"></a>
<li><code>NumberLibraryNearRings( </code><var>G</var><code> )</code>
<p>
<a name = "SECT001"></a>
<li><code>AllLibraryNearRings( </code><var>G</var><code> )</code>
<p>
returns a list of all nearrings (in the library) that have the group <var>G</var> as
group reduct.
<p>
<pre>
gap> l := AllLibraryNearRings( GTW3_1 );
[ LibraryNearRing(3/1, 1), LibraryNearRing(3/1, 2),
LibraryNearRing(3/1, 3), LibraryNearRing(3/1, 4),
LibraryNearRing(3/1, 5) ]
gap> Filtered( l, IsNearField );
[ LibraryNearRing(3/1, 3) ]
gap> NumberLibraryNearRings( GTW14_2 );
1821
gap> LN14_2_1234 := LibraryNearRing( GTW14_2, 1234 );
LibraryNearRing(14/2, 1234)
</pre>
<p>
<a name = "SECT001"></a>
<li><code>LibraryNearRingWithOne( </code><var>G</var><code>, </code><var>num</var><code> )</code>
<p>
<code>LibraryNearRingWithOne</code> retrieves a nearring from the nearrings library
files.
<var>G</var> must be one of the predefined groups of order <var>le31</var>.
<var>num</var> must be an integer which indicates the number of the class of
nearrings with identity on the specified group.
<p>
The number of nearrings with identity definable on a certain group <var>G</var>
can be accessed via
<p>
<a name = "SECT001"></a>
<li><code>NumberLibraryNearRingsWithOne( </code><var>G</var><code> )</code>
<p>
<a name = "SECT001"></a>
<li><code>AllLibraryNearRingsWithOne( </code><var>G</var><code> )</code>
<p>
returns a list of all nearrings with identity (in the library) that have
the group <var>G</var> as group reduct.
<p>
<pre>
gap> NumberLibraryNearRingsWithOne( GTW24_6 );
0
gap> NumberLibraryNearRingsWithOne( GTW24_4 );
10
gap> LNwI24_4_8 := LibraryNearRingWithOne( GTW24_4, 8 );
LibraryNearRingWithOne(24/4, 8)
gap> AllLibraryNearRingsWithOne( GTW24_6 );
[ ]
</pre>
<p>
<p>
<h2><a name="SECT002">3.2 Identifying nearrings</a></h2>
<p><p>
<a name = "SECT002"></a>
<li><code>IdLibraryNearRing( </code><var>nr</var><code> )</code>
<p>
The function <code>IdLibraryNearRing</code> returns a pair [<var>G</var>, <var>n</var>] such that the
nearring <var>nr</var> is isomorphic to the <var>n</var>th library nearring on the group <var>G</var>.
<p>
<pre>
gap> p := PolynomialNearRing( GTW4_2 );
PolynomialNearRing( 4/2 )
gap> IdLibraryNearRing( p );
[ 8/3, 833 ]
gap> n := LibraryNearRing( GTW3_1, 4 );
LibraryNearRing(3/1, 4)
gap> d := DirectProductNearRing( n, n );
DirectProductNearRing( LibraryNearRing(3/1, 4), LibraryNearRing(3/1, 4)\
)
gap> IdLibraryNearRing( d );
[ 9/2, 220 ]
</pre>
<p>
<a name = "SECT002"></a>
<li><code>IdLibraryNearRingWithOne( </code><var>nr</var><code> )</code>
<p>
The function <code>IdLibraryNearRingWithOne</code> returns a pair [<var>G</var>, <var>n</var>] such
that the nearring <var>nr</var> is isomorphic to the <var>n</var>th library nearring with
identity on the group <var>G</var>. This function can only be applied to nearrings
which have an identity.
<p>
<pre>
gap> l := LibraryNearRingWithOne( GTW12_3, 1 );
LibraryNearRingWithOne(12/3, 1)
gap> IdLibraryNearRing( l ); #this command requires time and memory!!!
[ 12/3, 37984 ]
gap> IdLibraryNearRingWithOne( l );
[ 12/3, 1 ]
</pre>
<p>
<p>
<h2><a name="SECT003">3.3 IsLibraryNearRing</a></h2>
<p><p>
<a name = "SECT003"></a>
<li><code>IsLibraryNearRing( </code><var>nr</var><code> )</code>
<p>
The function <code>IsLibraryNearRing</code> returns <code>true</code> if the nearring <var>nr</var> has been
read from the nearring library and <code>false</code> otherwise.
<p>
<pre>
gap> IsLibraryNearRing( LNwI24_4_8 );
true
</pre>
<p>
<p>
<h2><a name="SECT004">3.4 Accessing the information about a nearring stored in the library</a></h2>
<p><p>
<a name = "SECT004"></a>
<li><code>LibraryNearRingInfo( </code><var>group</var><code>, </code><var>list</var><code>, </code><var>string</var><code> )</code>
<p>
This function provides information about the specified library nearrings
in a way similar to how nearrings are presented in the appendix of
[Pil??].
The parameter <var>group</var> specifies a predefined group; valid
names are exactly those which are also valid for the function
<code>LibraryNearrings</code> (cf. Section <a href="CHAP003.htm#SECT001">LibraryNearRing</a>).
<p>
The parameter <var>list</var> must be a non-empty list of integers defining the
classes of nearrings to be printed. Naturally, these integers must all fit
into the ranges described in Section <a href="CHAP003.htm#SECT001">LibraryNearRing</a> for the according
groups.
<p>
The third parameter <var>string</var> is optional. <var>string</var> must be a string
consisting of one or more (or all) of the following characters:
<code>l</code>, <code>m</code>, <code>i</code>, <code>v</code>, <code>s</code>, <code>e</code>, <code>a</code>.
Per default, (i.e. if this parameter is not specified), the output is
minimal. According to each specified character, the following is added:
<p>
<dl compact>
<dt> <code>a</code> <dd> list the nearring automorphisms.
<p>
<dt> <code>c</code> <dd> print the meaning of the letters used in the output.
<p>
<dt> <code>e</code> <dd> list the nearring endomorphisms.
<p>
<dt> <code>g</code> <dd> list the endomorphisms of the group reduct.
<p>
<dt> <code>i</code> <dd> list the ideals.
<p>
<dt> <code>l</code> <dd> list the left ideals.
<p>
<dt> <code>m</code> <dd> print the multiplication tables.
<p>
<dt> <code>r</code> <dd> list the right ideals.
<p>
<dt> <code>s</code> <dd> list the subnearrings.
<p>
<dt> <code>v</code> <dd> list the invariant subnearrings.
<p>
</dl>
<p>
<strong>Examples:</strong>
<p>
<code>LibraryNearRingInfo( GTW3_1, [ 3 ], "lmivsea" )</code> will print all
available information about the third class of nearrings on the group
<var>Z<sub>3</sub></var>.
<p>
<code>LibraryNearRingInfo( GTW4_1, [ 1..12 ] )</code> will provide a minimal output
for all classes of nearrings on <var>Z<sub>4</sub></var>.
<p>
<code>LibraryNearRingInfo( GTW6_2, [ 5, 18, 21 ], "mi" )</code> will print
the minimal information plus the multiplication tables plus the ideals for
the classes 5, 18, and 21 of nearrings on the group <var>S<sub>3</sub></var>.
<p>
<p>
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<P>
<address>SONATA manual<br>November 2008
</address></body></html>
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