<html><head><title>[ref] 55 Modules</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href="../index.htm">Top</a>] [<a href = "chapters.htm">Up</a>] [<a href ="CHAP054.htm">Previous</a>] [<a href ="CHAP056.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>55 Modules</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP055.htm#SECT001">Generating modules</a> <li> <A HREF="CHAP055.htm#SECT002">Submodules</a> <li> <A HREF="CHAP055.htm#SECT003">Free Modules</a> </ol><p> <p> <p> <h2><a name="SECT001">55.1 Generating modules</a></h2> <p><p> <a name = "SSEC001.1"></a> <li><code>IsLeftOperatorAdditiveGroup( </code><var>D</var><code> ) C</code> <p> A domain <var>D</var> lies in <code>IsLeftOperatorAdditiveGroup</code> if it is an additive group that is closed under scalar multplication from the left, and such that λ*(<i>x</i>+<i>y</i>)=λ*<i>x</i>+λ*<i>y</i> for all scalars λ and elements <i>x</i>,<i>y</i> ∈ <i>D</i>. <p> <a name = "SSEC001.2"></a> <li><code>IsLeftModule( </code><var>M</var><code> ) C</code> <p> A domain <var>M</var> lies in <code>IsLeftModule</code> if it lies in <code>IsLeftOperatorAdditiveGroup</code>, <em>and</em>the set of scalars forms a ring, <em>and</em>(λ+μ)*<i>x</i>=λ*<i>x</i>+μ*<i>x</i> for scalars λ,μ and <i>x</i> ∈ <i>M</i>, <em>and</em>scalar multiplication satisfies λ*(μ*<i>x</i>) = (λ*μ)*<i>x</i> for scalars λ,μ and <i>x</i> ∈ <i>M</i>. <p> <pre> gap> V:= FullRowSpace( Rationals, 3 ); ( Rationals^3 ) gap> IsLeftModule( V ); true </pre> <p> <a name = "SSEC001.3"></a> <li><code>GeneratorsOfLeftOperatorAdditiveGroup( </code><var>D</var><code> ) A</code> <p> returns a list of elements of <var>D</var> that generates <var>D</var> as a left operator additive group. <p> <a name = "SSEC001.4"></a> <li><code>GeneratorsOfLeftModule( </code><var>M</var><code> ) A</code> <p> returns a list of elements of <var>M</var> that generate <var>M</var> as a left module. <p> <pre> gap> V:= FullRowSpace( Rationals, 3 );; gap> GeneratorsOfLeftModule( V ); [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] </pre> <p> <a name = "SSEC001.5"></a> <li><code>AsLeftModule( </code><var>R</var><code>, </code><var>D</var><code> ) O</code> <p> if the domain <var>D</var> forms an additive group and is closed under left multiplication by the elements of <var>R</var>, then <code>AsLeftModule( </code><var>R</var><code>, </code><var>D</var><code> )</code> returns the domain <var>D</var> viewed as a left module. <p> <pre> gap> coll:= [ [0*Z(2),0*Z(2)], [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ]; [ [ 0*Z(2), 0*Z(2) ], [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0 ] ] gap> AsLeftModule( GF(2), coll ); <vector space of dimension 2 over GF(2)> </pre> <p> <a name = "SSEC001.6"></a> <li><code>IsRightOperatorAdditiveGroup( </code><var>D</var><code> ) C</code> <p> A domain <var>D</var> lies in <code>IsRightOperatorAdditiveGroup</code> if it is an additive group that is closed under scalar multplication from the right, and such that (<i>x</i>+<i>y</i>)*λ = <i>x</i>*λ+<i>y</i>*λ for all scalars λ and elements <i>x</i>,<i>y</i> ∈ <i>D</i>. <p> <a name = "SSEC001.7"></a> <li><code>IsRightModule( </code><var>M</var><code> ) C</code> <p> A domain <var>M</var> lies in <code>IsRightModule</code> if it lies in <code>IsRightOperatorAdditiveGroup</code>, <em>and</em>the set of scalars forms a ring, <em>and</em><i>x</i>*(λ+μ) = <i>x</i>*λ+<i>x</i>*μ for scalars λ,μ and <i>x</i> ∈ <i>M</i>, <em>and</em>scalar multiplication satisfies (<i>x</i>*μ)*λ = <i>x</i>*(μ*λ) for scalars λ,μ and <i>x</i> ∈ <i>M</i>. <p> <a name = "SSEC001.8"></a> <li><code>GeneratorsOfRightOperatorAdditiveGroup( </code><var>D</var><code> ) A</code> <p> returns a list of elements of <var>D</var> that generates <var>D</var> as a right operator additive group. <p> <a name = "SSEC001.9"></a> <li><code>GeneratorsOfRightModule( </code><var>M</var><code> ) A</code> <p> returns a list of elements of <var>M</var> that generate <var>M</var> as a left module. <p> <a name = "SSEC001.10"></a> <li><code>LeftModuleByGenerators( </code><var>R</var><code>, </code><var>gens</var><code> ) O</code> <li><code>LeftModuleByGenerators( </code><var>R</var><code>, </code><var>gens</var><code>, </code><var>zero</var><code> ) O</code> <p> returns the left module over <var>R</var> generated by <var>gens</var>. <p> <pre> gap> coll:= [ [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ];; gap> V:= LeftModuleByGenerators( GF(16), coll ); <vector space over GF(2^4), with 3 generators> </pre> <p> <a name = "SSEC001.11"></a> <li><code>LeftActingDomain( </code><var>D</var><code> ) A</code> <p> Let <var>D</var> be an external left set, that is, <var>D</var> is closed under the action of a domain <i>L</i> by multiplication from the left. Then <i>L</i> can be accessed as value of <code>LeftActingDomain</code> for <var>D</var>. <p> <p> <h2><a name="SECT002">55.2 Submodules</a></h2> <p><p> <a name = "SSEC002.1"></a> <li><code>Submodule( </code><var>M</var><code>, </code><var>gens</var><code> ) F</code> <li><code>Submodule( </code><var>M</var><code>, </code><var>gens</var><code>, "basis" ) F</code> <p> is the left module generated by the collection <var>gens</var>, with parent module <var>M</var>. The second form generates the submodule of <var>M</var> for that the list <var>gens</var> is known to be a list of basis vectors; in this case, it is <strong>not</strong> checked whether <var>gens</var> really are linearly independent and whether all in <var>gens</var> lie in <var>M</var>. <p> <pre> gap> coll:= [ [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ];; gap> V:= LeftModuleByGenerators( GF(16), coll );; gap> W:= Submodule( V, [ coll[1], coll[2] ] ); <vector space over GF(2^4), with 2 generators> gap> Parent( W ) = V; true </pre> <p> <a name = "SSEC002.2"></a> <li><code>SubmoduleNC( </code><var>M</var><code>, </code><var>gens</var><code> ) F</code> <li><code>SubmoduleNC( </code><var>M</var><code>, </code><var>gens</var><code>, "basis" ) F</code> <p> <code>SubmoduleNC</code> does the same as <code>Submodule</code>, except that it does not check whether all in <var>gens</var> lie in <var>M</var>. <p> <a name = "SSEC002.3"></a> <li><code>ClosureLeftModule( </code><var>M</var><code>, </code><var>m</var><code> ) O</code> <p> is the left module generated by the left module generators of <var>M</var> and the element <var>m</var>. <p> <pre> gap> V:= LeftModuleByGenerators( Rationals, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] ); <vector space over Rationals, with 2 generators> gap> ClosureLeftModule( V, [ 1, 1, 1 ] ); <vector space over Rationals, with 3 generators> </pre> <p> <a name = "SSEC002.4"></a> <li><code>TrivialSubmodule( </code><var>M</var><code> ) A</code> <p> returns the zero submodule of <var>M</var>. <p> <pre> gap> V:= LeftModuleByGenerators( Rationals, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] );; gap> TrivialSubmodule( V ); <vector space over Rationals, with 0 generators> </pre> <p> <p> <h2><a name="SECT003">55.3 Free Modules</a></h2> <p><p> <a name = "SSEC003.1"></a> <li><code>IsFreeLeftModule( </code><var>M</var><code> ) C</code> <p> A left module is free as module if it is isomorphic to a direct sum of copies of its left acting domain. <p> Free left modules can have bases. <p> The characteristic (see <a href="CHAP030.htm#SSEC010.1">Characteristic</a>) of a free left module is defined as the characteristic of its left acting domain (see <a href="CHAP055.htm#SSEC001.11">LeftActingDomain</a>). <p> <a name = "SSEC003.2"></a> <li><code>FreeLeftModule( </code><var>R</var><code>, </code><var>gens</var><code> ) F</code> <li><code>FreeLeftModule( </code><var>R</var><code>, </code><var>gens</var><code>, </code><var>zero</var><code> ) F</code> <li><code>FreeLeftModule( </code><var>R</var><code>, </code><var>gens</var><code>, "basis" ) F</code> <li><code>FreeLeftModule( </code><var>R</var><code>, </code><var>gens</var><code>, </code><var>zero</var><code>, "basis" ) F</code> <p> <code>FreeLeftModule( </code><var>R</var><code>, </code><var>gens</var><code> )</code> is the free left module over the ring <var>R</var>, generated by the vectors in the collection <var>gens</var>. <p> If there are three arguments, a ring <var>R</var> and a collection <var>gens</var> and an element <var>zero</var>, then <code>FreeLeftModule( </code><var>R</var><code>, </code><var>gens</var><code>, </code><var>zero</var><code> )</code> is the <var>R</var>-free left module generated by <var>gens</var>, with zero element <var>zero</var>. <p> If the last argument is the string <code>"basis"</code> then the vectors in <var>gens</var> are known to form a basis of the free module. <p> It should be noted that the generators <var>gens</var> must be vectors, that is, they must support an addition and a scalar action of <var>R</var> via left multiplication. (See also Section <a href="CHAP030.htm#SECT003">Constructing Domains</a> for the general meaning of ``generators'' in <font face="Gill Sans,Helvetica,Arial">GAP</font>.) In particular, <code>FreeLeftModule</code> is <strong>not</strong> an equivalent of commands such as <code>FreeGroup</code> (see <a href="CHAP035.htm#SSEC002.1">FreeGroup</a>) in the sense of a constructor of a free group on abstract generators; Such a construction seems to be unnecessary for vector spaces, for that one can use for example row spaces (see <a href="CHAP059.htm#SSEC008.4">FullRowSpace</a>) in the finite dimensional case and polynomial rings (see <a href="CHAP064.htm#SSEC014.1">PolynomialRing</a>) in the infinite dimensional case. Moreover, the definition of a ``natural'' addition for elements of a given magma (for example a permutation group) is possible via the construction of magma rings (see Chapter <a href="../ref/CHAP063.htm">Magma Rings</a>). <p> <pre> gap> V:= FreeLeftModule( Rationals, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ], "basis" ); <vector space of dimension 2 over Rationals> </pre> <p> <a name = "SSEC003.3"></a> <li><code>AsFreeLeftModule( </code><var>F</var><code>, </code><var>D</var><code> ) O</code> <p> if the domain <var>D</var> is a free left module over <var>F</var>, then <code>AsFreeLeftModule( </code><var>F</var><code>, </code><var>D</var><code> )</code> returns the domain <var>D</var> viewed as free left module over <var>F</var>. <p> <a name = "SSEC003.4"></a> <li><code>Dimension( </code><var>M</var><code> ) A</code> <p> A free left module has dimension <i>n</i> if it is isomorphic to a direct sum of <i>n</i> copies of its left acting domain. <p> (We do <strong>not</strong> mark <code>Dimension</code> as invariant under isomorphisms since we want to call <code>UseIsomorphismRelation</code> also for free left modules over different left acting domains.) <p> <pre> gap> V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );; gap> Dimension( V ); 2 </pre> <p> <a name = "SSEC003.5"></a> <li><code>IsFiniteDimensional( </code><var>M</var><code> ) P</code> <p> is <code>true</code> if <var>M</var> is a free left module that is finite dimensional over its left acting domain, and <code>false</code> otherwise. <p> <pre> gap> V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );; gap> IsFiniteDimensional( V ); true </pre> <p> <a name = "SSEC003.6"></a> <li><code>UseBasis( </code><var>V</var><code>, </code><var>gens</var><code> ) O</code> <p> The vectors in the list <var>gens</var> are known to form a basis of the free left module <var>V</var>. <code>UseBasis</code> stores information in <var>V</var> that can be derived form this fact, namely <ul> <li> <var>gens</var> are stored as left module generators if no such generators were bound (this is useful especially if <var>V</var> is an algebra), <li> the dimension of <var>V</var> is stored. </ul> <p> <pre> gap> V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );; gap> UseBasis( V, [ [ 1, 0 ], [ 1, 1 ] ] ); gap> V; # now V knows its dimension <vector space of dimension 2 over Rationals> </pre> <p> <a name = "SSEC003.7"></a> <li><code>IsRowModule( </code><var>V</var><code> ) P</code> <p> A <strong>row module</strong> is a free left module whose elements are row vectors. <p> <a name = "SSEC003.8"></a> <li><code>IsMatrixModule( </code><var>V</var><code> ) P</code> <p> A <strong>matrix module</strong> is a free left module whose elements are matrices. <p> <a name = "SSEC003.9"></a> <li><code>IsFullRowModule( </code><var>M</var><code> ) P</code> <p> A <strong>full row module</strong> is a module <i>R</i><sup><i>n</i></sup>, for a ring <i>R</i> and a nonnegative integer <i>n</i>. <p> More precisely, a full row module is a free left module over a ring <i>R</i> such that the elements are row vectors with entries in <i>R</i> and such that the dimension is equal to the length of the row vectors. <p> Several functions delegate their tasks to full row modules, for example <code>Iterator</code> and <code>Enumerator</code>. <p> <a name = "SSEC003.10"></a> <li><code>FullRowModule( </code><var>R</var><code>, </code><var>n</var><code> ) F</code> <p> is the row module <code></code><var>R</var><code>^</code><var>n</var><code></code>, for a ring <var>R</var> and a nonnegative integer <var>n</var>. <p> <pre> gap> V:= FullRowModule( Integers, 5 ); ( Integers^5 ) </pre> <p> <a name = "SSEC003.11"></a> <li><code>IsFullMatrixModule( </code><var>M</var><code> ) P</code> <p> A <strong>full matrix module</strong> is a module <i>R</i><sup>[<i>m</i>,<i>n</i>]</sup>, for a ring <i>R</i> and two nonnegative integers <i>m</i>, <i>n</i>. <p> More precisely, a full matrix module is a free left module over a ring <i>R</i> such that the elements are matrices with entries in <i>R</i> and such that the dimension is equal to the number of entries in each matrix. <p> <a name = "SSEC003.12"></a> <li><code>FullMatrixModule( </code><var>R</var><code>, </code><var>m</var><code>, </code><var>n</var><code> ) F</code> <p> is the row module <code></code><var>R</var><code>^[</code><var>m</var><code>,</code><var>n</var><code>]</code>, for a ring <var>R</var> and nonnegative integers <var>m</var> and <var>n</var>. <p> <pre> gap> FullMatrixModule( GaussianIntegers, 3, 6 ); ( GaussianIntegers^[ 3, 6 ] ) </pre> <p> <a name = "SSEC003.13"></a> <li><code>IsHandledByNiceBasis( </code><var>M</var><code> ) C</code> <p> For a free left module <var>M</var> in this category, essentially all operations are performed using a ``nicer'' free left module, which is usually a row module. <p> <p> [<a href="../index.htm">Top</a>] [<a href = "chapters.htm">Up</a>] [<a href ="CHAP054.htm">Previous</a>] [<a href ="CHAP056.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <font face="Gill Sans,Helvetica,Arial">GAP 4 manual<br>December 2008 </font></body></html>
