<html><head><title>[ref] 53 Additive Magmas</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href="../index.htm">Top</a>] [<a href = "chapters.htm">Up</a>] [<a href ="CHAP052.htm">Previous</a>] [<a href ="CHAP054.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>53 Additive Magmas</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP053.htm#SECT001">(Near-)Additive Magma Categories</a> <li> <A HREF="CHAP053.htm#SECT002">(Near-)Additive Magma Generation</a> <li> <A HREF="CHAP053.htm#SECT003">Attributes and Properties for (Near-)Additive Magmas</a> <li> <A HREF="CHAP053.htm#SECT004">Operations for (Near-)Additive Magmas</a> </ol><p> <p> This chapter deals with domains that are closed under addition <code>+</code>, which are called <strong>near-additive magmas</strong> in <font face="Gill Sans,Helvetica,Arial">GAP</font>. Together with the domains closed under multiplication <code>*</code>, (see <a href="CHAP033.htm">Magmas</a>), they are the basic algebraic structures. In many cases, the addition is commutative (see <a href="CHAP053.htm#SSEC003.1">IsAdditivelyCommutative</a>), the domain is called an <strong>additive magma</strong> then; every module (see <a href="CHAP055.htm">Modules</a>), vector space (see <a href="CHAP059.htm">Vector Spaces</a>), ring (see <a href="CHAP054.htm">Rings</a>), or field (see <a href="CHAP056.htm">Fields and Division Rings</a>) is an additive magma. In the cases of all <strong>(near-)additive magma-with-zero</strong> or <strong>(near-)additive magma-with-inverses</strong>, additional additive structure is present (see <a href="CHAP053.htm#SECT001">(Near-)Additive Magma Categories</a>). <p> <p> <h2><a name="SECT001">53.1 (Near-)Additive Magma Categories</a></h2> <p><p> <a name = "SSEC001.1"></a> <li><code>IsNearAdditiveMagma( </code><var>obj</var><code> ) C</code> <p> A <strong>near-additive magma</strong> in <font face="Gill Sans,Helvetica,Arial">GAP</font> is a domain <i>A</i> with an associative but not necessarily commutative addition <code>+</code>: <i>A</i> ×<i>A</i> → <i>A</i>. <p> <a name = "SSEC001.2"></a> <li><code>IsNearAdditiveMagmaWithZero( </code><var>obj</var><code> ) C</code> <p> A <strong>near-additive magma-with-zero</strong> in <font face="Gill Sans,Helvetica,Arial">GAP</font> is a near-additive magma <i>A</i> with an operation <code>0*</code> (or <code>Zero</code>) that yields the zero of <i>A</i>. <p> So a near-additive magma-with-zero <var>A</var> does always contain a unique additively neutral element <i>z</i>, i.e., <i>z</i> + <i>a</i> = <i>a</i> = <i>a</i> + <i>z</i> holds for all <i>a</i> ∈ <i>A</i> (see <a href="CHAP053.htm#SSEC003.5">AdditiveNeutralElement</a>). This element <i>z</i> can be computed with the operation <code>Zero</code> (see <a href="CHAP030.htm#SSEC010.3">Zero</a>) as <code>Zero( </code><var>A</var><code> )</code>, and <i>z</i> is also equal to <code>Zero( </code><var>elm</var><code> )</code> and to <code>0*</code><var>elm</var><code></code> for each element <var>elm</var> in <var>A</var>. <p> <strong>Note</strong> that a near-additive magma containing a zero may <strong>not</strong> lie in the category <code>IsNearAdditiveMagmaWithZero</code> (see <a href="CHAP030.htm#SECT006">Domain Categories</a>). <p> <a name = "SSEC001.3"></a> <li><code>IsNearAdditiveGroup( </code><var>obj</var><code> ) C</code> <a name = "SSEC001.3"></a> <li><code>IsNearAdditiveMagmaWithInverses( </code><var>obj</var><code> ) C</code> <p> A <strong>near-additive group</strong> in <font face="Gill Sans,Helvetica,Arial">GAP</font> is a near-additive magma-with-zero <i>A</i> with an operation <code>-1*</code>: <i>A</i> → <i>A</i> that maps each element <var>a</var> of <i>A</i> to its additive inverse <code>-1*</code><var>a</var><code></code> (or <code>AdditiveInverse( </code><var>a</var><code> )</code>, see <a href="CHAP030.htm#SSEC010.9">AdditiveInverse</a>). <p> The addition <code>+</code> of <i>A</i> is assumed to be associative, so a near-additive group is not more than a <strong>near-additive magma-with-inverses</strong>. <code>IsNearAdditiveMagmaWithInverses</code> is just a synonym for <code>IsNearAdditiveGroup</code>, and can be used alternatively in all function names involving <code>NearAdditiveGroup</code>. <p> Note that not every trivial near-additive magma is a near-additive magma-with-zero, but every trivial near-additive magma-with-zero is a near-additive group. <p> <a name = "SSEC001.4"></a> <li><code>IsAdditiveMagma( </code><var>obj</var><code> ) C</code> <p> An <strong>additive magma</strong> in <font face="Gill Sans,Helvetica,Arial">GAP</font> is a domain <i>A</i> with an associative and commutative addition <code>+</code>: <i>A</i> ×<i>A</i> → <i>A</i>, see <a href="CHAP053.htm#SSEC001.1">IsNearAdditiveMagma</a> and <a href="CHAP053.htm#SSEC003.1">IsAdditivelyCommutative</a>. <p> <a name = "SSEC001.5"></a> <li><code>IsAdditiveMagmaWithZero( </code><var>obj</var><code> ) C</code> <p> An <strong>additive magma-with-zero</strong> in <font face="Gill Sans,Helvetica,Arial">GAP</font> is an additive magma <i>A</i> with an operation <code>0*</code> (or <code>Zero</code>) that yields the zero of <i>A</i>. <p> So an additive magma-with-zero <var>A</var> does always contain a unique additively neutral element <i>z</i>, i.e., <i>z</i> + <i>a</i> = <i>a</i> = <i>a</i> + <i>z</i> holds for all <i>a</i> ∈ <i>A</i> (see <a href="CHAP053.htm#SSEC003.5">AdditiveNeutralElement</a>). This element <i>z</i> can be computed with the operation <code>Zero</code> (see <a href="CHAP030.htm#SSEC010.3">Zero</a>) as <code>Zero( </code><var>A</var><code> )</code>, and <i>z</i> is also equal to <code>Zero( </code><var>elm</var><code> )</code> and to <code>0*</code><var>elm</var><code></code> for each element <var>elm</var> in <var>A</var>. <p> <strong>Note</strong> that an additive magma containing a zero may <strong>not</strong> lie in the category <code>IsAdditiveMagmaWithZero</code> (see <a href="CHAP030.htm#SECT006">Domain Categories</a>). <p> <a name = "SSEC001.6"></a> <li><code>IsAdditiveGroup( </code><var>obj</var><code> ) C</code> <a name = "SSEC001.6"></a> <li><code>IsAdditiveMagmaWithInverses( </code><var>obj</var><code> ) C</code> <p> An <strong>additive group</strong> in <font face="Gill Sans,Helvetica,Arial">GAP</font> is an additive magma-with-zero <i>A</i> with an operation <code>-1*</code>: <i>A</i> → <i>A</i> that maps each element <var>a</var> of <i>A</i> to its additive inverse <code>-1*</code><var>a</var><code></code> (or <code>AdditiveInverse( </code><var>a</var><code> )</code>, see <a href="CHAP030.htm#SSEC010.9">AdditiveInverse</a>). <p> The addition <code>+</code> of <i>A</i> is assumed to be commutative and associative, so an additive group is not more than an <strong>additive magma-with-inverses</strong>. <code>IsAdditiveMagmaWithInverses</code> is just a synonym for <code>IsAdditiveGroup</code>, and can be used alternatively in all function names involving <code>AdditiveGroup</code>. <p> Note that not every trivial additive magma is an additive magma-with-zero, but every trivial additive magma-with-zero is an additive group. <p> <p> <h2><a name="SECT002">53.2 (Near-)Additive Magma Generation</a></h2> <p><p> <a name = "SSEC002.1"></a> <li><code>NearAdditiveMagma( </code><var>gens</var><code> ) F</code> <li><code>NearAdditiveMagma( </code><var>Fam</var><code>, </code><var>gens</var><code> ) F</code> <p> returns the (near-)additive magma <i>A</i> that is generated by the elements in the list <var>gens</var>, that is, the closure of <var>gens</var> under addition <code>+</code>. The family <var>Fam</var> of <i>A</i> can be entered as first argument; this is obligatory if <var>gens</var> is empty (and hence also <i>A</i> is empty). <p> <a name = "SSEC002.2"></a> <li><code>NearAdditiveMagmaWithZero( </code><var>gens</var><code> ) F</code> <li><code>NearAdditiveMagmaWithZero( </code><var>Fam</var><code>, </code><var>gens</var><code> ) F</code> <p> returns the (near-)additive magma-with-zero <i>A</i> that is generated by the elements in the list <var>gens</var>, that is, the closure of <var>gens</var> under addition <code>+</code> and <code>Zero</code>. The family <var>Fam</var> of <i>A</i> can be entered as first argument; this is obligatory if <var>gens</var> is empty (and hence <i>A</i> is trivial). <p> <a name = "SSEC002.3"></a> <li><code>NearAdditiveGroup( </code><var>gens</var><code> ) F</code> <li><code>NearAdditiveGroup( </code><var>Fam</var><code>, </code><var>gens</var><code> ) F</code> <p> returns the (near-)additive group <i>A</i> that is generated by the elements in the list <var>gens</var>, that is, the closure of <var>gens</var> under addition <code>+</code>, <code>Zero</code>, and <code>AdditiveInverse</code>. The family <var>Fam</var> of <i>A</i> can be entered as first argument; this is obligatory if <var>gens</var> is empty (and hence <i>A</i> is trivial). <p> The underlying operations for which methods can be installed are the following. <p> <a name = "SSEC002.4"></a> <li><code>NearAdditiveMagmaByGenerators( </code><var>gens</var><code> ) O</code> <li><code>NearAdditiveMagmaByGenerators( </code><var>Fam</var><code>, </code><var>gens</var><code> ) O</code> <p> <a name = "SSEC002.5"></a> <li><code>NearAdditiveMagmaWithZeroByGenerators( </code><var>gens</var><code> ) O</code> <li><code>NearAdditiveMagmaWithZeroByGenerators( </code><var>Fam</var><code>, </code><var>gens</var><code> ) O</code> <p> <a name = "SSEC002.6"></a> <li><code>NearAdditiveGroupByGenerators( </code><var>gens</var><code> ) O</code> <li><code>NearAdditiveGroupByGenerators( </code><var>Fam</var><code>, </code><var>gens</var><code> ) O</code> <p> Substructures of an additive magma can be formed as follows. <p> <a name = "SSEC002.7"></a> <li><code>SubnearAdditiveMagma( </code><var>D</var><code>, </code><var>gens</var><code> ) F</code> <a name = "SSEC002.7"></a> <li><code>SubnearAdditiveMagmaNC( </code><var>D</var><code>, </code><var>gens</var><code> ) F</code> <p> <code>SubadditiveMagma</code> returns the near-additive magma generated by the elements in the list <var>gens</var>, with parent the domain <var>D</var>. <code>SubadditiveMagmaNC</code> does the same, except that it is not checked whether the elements of <var>gens</var> lie in <var>D</var>. <p> <a name = "SSEC002.8"></a> <li><code>SubnearAdditiveMagmaWithZero( </code><var>D</var><code>, </code><var>gens</var><code> ) F</code> <a name = "SSEC002.8"></a> <li><code>SubnearAdditiveMagmaWithZeroNC( </code><var>D</var><code>, </code><var>gens</var><code> ) F</code> <p> <code>SubadditiveMagmaWithZero</code> returns the near-additive magma-with-zero generated by the elements in the list <var>gens</var>, with parent the domain <var>D</var>. <code>SubadditiveMagmaWithZeroNC</code> does the same, except that it is not checked whether the elements of <var>gens</var> lie in <var>D</var>. <p> <a name = "SSEC002.9"></a> <li><code>SubnearAdditiveGroup( </code><var>D</var><code>, </code><var>gens</var><code> ) F</code> <a name = "SSEC002.9"></a> <li><code>SubnearAdditiveGroupNC( </code><var>D</var><code>, </code><var>gens</var><code> ) F</code> <p> <code>SubadditiveGroup</code> returns the near-additive group generated by the elements in the list <var>gens</var>, with parent the domain <var>D</var>. <code>SubadditiveGroupNC</code> does the same, except that it is not checked whether the elements of <var>gens</var> lie in <var>D</var>. <p> <p> <h2><a name="SECT003">53.3 Attributes and Properties for (Near-)Additive Magmas</a></h2> <p><p> <a name = "SSEC003.1"></a> <li><code>IsAdditivelyCommutative( </code><var>A</var><code> ) P</code> <p> A near-additive magma <var>A</var> in <font face="Gill Sans,Helvetica,Arial">GAP</font> is <strong>additively commutative</strong> if for all elements <i>a</i>, <i>b</i> ∈ <i>A</i> the equality <i>a</i> + <i>b</i> = <i>b</i> + <i>a</i> holds. <p> Note that the commutativity of the <strong>multiplication</strong> <code>*</code> in a multiplicative structure can be tested with <code>IsCommutative</code>, (see <a href="CHAP033.htm#SSEC004.9">IsCommutative</a>). <p> <a name = "SSEC003.2"></a> <li><code>GeneratorsOfNearAdditiveMagma( </code><var>A</var><code> ) A</code> <a name = "SSEC003.2"></a> <li><code>GeneratorsOfAdditiveMagma( </code><var>A</var><code> ) A</code> <p> is a list <var>gens</var> of elements of the near-additive magma <var>A</var> that generates <var>A</var> as a near-additive magma, that is, the closure of <var>gens</var> under addition is <var>A</var>. <p> <a name = "SSEC003.3"></a> <li><code>GeneratorsOfNearAdditiveMagmaWithZero( </code><var>A</var><code> ) A</code> <a name = "SSEC003.3"></a> <li><code>GeneratorsOfAdditiveMagmaWithZero( </code><var>A</var><code> ) A</code> <p> is a list <var>gens</var> of elements of the near-additive magma-with-zero <var>A</var> that generates <var>A</var> as a near-additive magma-with-zero, that is, the closure of <var>gens</var> under addition and <code>Zero</code> (see <a href="CHAP030.htm#SSEC010.3">Zero</a>) is <var>A</var>. <p> <a name = "SSEC003.4"></a> <li><code>GeneratorsOfNearAdditiveGroup( </code><var>A</var><code> ) A</code> <a name = "SSEC003.4"></a> <li><code>GeneratorsOfAdditiveGroup( </code><var>A</var><code> ) A</code> <p> is a list <var>gens</var> of elements of the near-additive group <var>A</var> that generates <var>A</var> as a near-additive group, that is, the closure of <var>gens</var> under addition, taking the zero element, and taking additive inverses (see <a href="CHAP030.htm#SSEC010.9">AdditiveInverse</a>) is <var>A</var>. <p> <a name = "SSEC003.5"></a> <li><code>AdditiveNeutralElement( </code><var>A</var><code> ) A</code> <p> returns the element <i>z</i> in the near-additive magma <var>A</var> with the property that <i>z</i> + <i>a</i> = <i>a</i> = <i>a</i> + <i>z</i> holds for all <i>a</i> ∈ <i>A</i> , if such an element exists. Otherwise <code>fail</code> is returned. <p> A near-additive magma that is not a near-additive magma-with-zero can have an additive neutral element <i>z</i>; in this case, <i>z</i> <strong>cannot</strong> be obtained as <code>Zero( </code><var>A</var><code> )</code> or as <code>0*</code><var>elm</var><code></code> for an element <var>elm</var> in <var>A</var>, see <a href="CHAP030.htm#SSEC010.3">Zero</a>. <p> <a name = "SSEC003.6"></a> <li><code>TrivialSubnearAdditiveMagmaWithZero( </code><var>A</var><code> ) A</code> <p> is the additive magma-with-zero that has the zero of the near-additive magma-with-zero <var>A</var> as only element. <p> <p> <h2><a name="SECT004">53.4 Operations for (Near-)Additive Magmas</a></h2> <p><p> <a name = "SSEC004.1"></a> <li><code>ClosureNearAdditiveGroup( </code><var>A</var><code>, </code><var>a</var><code> ) O</code> <li><code>ClosureNearAdditiveGroup( </code><var>A</var><code>, </code><var>B</var><code> ) O</code> <p> returns the closure of the near-additive magma <var>A</var> with the element <var>a</var> or the near-additive magma <var>B</var>, w.r.t. addition, taking the zero element, and taking additive inverses. <p> <p> [<a href="../index.htm">Top</a>] [<a href = "chapters.htm">Up</a>] [<a href ="CHAP052.htm">Previous</a>] [<a href ="CHAP054.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>
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