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<p><a id="X7DDBF6F47A2E021C" name="X7DDBF6F47A2E021C"></a></p>
<div class="ChapSects"><a href="chap8.html#X7DDBF6F47A2E021C">8 <span class="Heading">Algebras</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X842EE9427C63F92E">8.1 <span class="Heading">Creators for FR algebras</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X812FEA6778152E49">8.1-1 FRAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X844B890F7BF56236">8.1-2 SCAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X81D8D0E886C8E143">8.1-3 BranchingIdeal</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X7EFB4F2E7E908B9F">8.2 <span class="Heading">Operations for FR algebras</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8115B018871FD364">8.2-1 MatrixQuotient</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8150FC4E84D208C6">8.2-2 ThinnedAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8572DCAE7F888DDA">8.2-3 Nillity</a></span>
</div>
</div>
<h3>8 <span class="Heading">Algebras</span></h3>
<p>Self-similar algebras and algebras with one (below <em>FR algebras</em>) are simply algebras [with one] whose elements are linear FR machines. They naturally act on the alphabet of their elements, which is a vector space.</p>
<p>Elements may be added, subtracted and multiplied. They can be vector or algebra linear elements; the vector elements are in general preferable, for efficiency reasons.</p>
<p>Finite-dimensional approximations of self-similar algebras can be computed; they are given as matrix algebras.</p>
<p><a id="X842EE9427C63F92E" name="X842EE9427C63F92E"></a></p>
<h4>8.1 <span class="Heading">Creators for FR algebras</span></h4>
<p>The most straightforward creation method for FR algebras is <code class="code">Algebra()</code>, applied with linear FR elements as arguments. There are shortcuts to this somewhat tedious method:</p>
<p><a id="X812FEA6778152E49" name="X812FEA6778152E49"></a></p>
<h5>8.1-1 FRAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> FRAlgebra</code>( <var class="Arg">ring, {definition, }</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">> FRAlgebraWithOne</code>( <var class="Arg">ring, {definition, }</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>A new self-similar algebra [with one].</p>
<p>This function constructs a new FR algebra [with one], generated by linear FR elements. It receives as argument any number of strings, each of which represents a generator of the object to be constructed.</p>
<p><var class="Arg">ring</var> is the acting domain of the vector space on which the algebra will act.</p>
<p>Each <var class="Arg">definition</var> is of the form <code class="code">"name=[[...],...,[...]]"</code> or of the form <code class="code">"name=[[...],...,[...]]:out"</code>, namely a matrix whose entries are algebraic expressions in the <code class="code">names</code>, possibly using <code class="code">0,1</code>, optionally followed by a scalar. The matrix entries specify the decomposition of the element being defined, and the optional scalar specifies the output of that element, by default assumed to be one.</p>
<p>The option <code class="code">IsVectorElement</code>, passed e.g. as in <code class="code">FRAlgebra(Rationals,"a=[[a,1],[a,0]]":IsVectorElement)</code>, asks for the resulting algebra to be generated by vector elements. The generators must of course be finite-state. Their names ("a",...) are not remembered in the constructing algebra (but can be set using <code class="func">SetName</code> (<a href="/usr/local/src/4.0/doc/ref/chap12.html#s8ss1"><b>Reference: SetName</b></a>)).</p>
<table class="example">
<tr><td><pre>
gap> m := FRAlgebra(Rationals,"a=[[1,a],[a,0]]");
<self-similar algebra on alphabet Rationals^2 with 1 generator>
gap> Display(Activity(m.1,2));
[ [ 1, 0, 1, 1 ],
[ 0, 1, 1, 0 ],
[ 1, 1, 0, 0 ],
[ 1, 0, 0, 0 ] ]
</pre></td></tr></table>
<p><a id="X844B890F7BF56236" name="X844B890F7BF56236"></a></p>
<h5>8.1-2 SCAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SCAlgebra</code>( <var class="Arg">m</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">> SCAlgebraWithOne</code>( <var class="Arg">m</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">> SCAlgebraNC</code>( <var class="Arg">m</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">> SCAlgebraWithOneNC</code>( <var class="Arg">m</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>The state-closed algebra [with one] generated by the machine <var class="Arg">m</var>.</p>
<p>This function constructs a new FR algebra [vith one] <code class="code">a</code>, generated by all the states of the FR machine <var class="Arg">m</var>. There is a bijective correspondence between <code class="code">GeneratorsOfFRMachine(m)</code> and the generators of <code class="code">a</code>, which is accessible via <code class="code">Correspondence(a)</code> (See <code class="func">Correspondence</code> (<a href="chap7.html#X7F15D57A7959FEF6"><b>7.1-3</b></a>)); it is a homomorphism from the stateset of <var class="Arg">m</var> to <code class="code">a</code>, or a list indicating for each state of <var class="Arg">m</var> a corresponding generator index in the generators of <code class="code">a</code> (with 0 for identity).</p>
<p>In the non-<code class="code">NC</code> forms, redundant (equal, zero or one) states are removed from the generating set of <code class="code">a</code>.</p>
<table class="example">
<tr><td><pre>
gap> a := SCAlgebra(AsLinearMachine(Rationals,I4Machine));
<self-similar algebra on alphabet Rationals^2 with 3 generators>
gap> a.1 = AsLinearElement(Rationals,I4Monoid.1);
true
</pre></td></tr></table>
<p><a id="X81D8D0E886C8E143" name="X81D8D0E886C8E143"></a></p>
<h5>8.1-3 BranchingIdeal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> BranchingIdeal</code>( <var class="Arg">A</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>An ideal I that contains matrices over itself.</p>
<table class="example">
<tr><td><pre>
gap> !!!
</pre></td></tr></table>
<p><a id="X7EFB4F2E7E908B9F" name="X7EFB4F2E7E908B9F"></a></p>
<h4>8.2 <span class="Heading">Operations for FR algebras</span></h4>
<p><a id="X8115B018871FD364" name="X8115B018871FD364"></a></p>
<h5>8.2-1 MatrixQuotient</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MatrixQuotient</code>( <var class="Arg">a, l</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">> EpimorphismMatrixQuotient</code>( <var class="Arg">a, l</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>The matrix algebra of <var class="Arg">a</var>'s action on level <var class="Arg">l</var>.</p>
<p>The first function returns the matrix algebra generated by the activities of <var class="Arg">a</var> on level <var class="Arg">l</var> (see the examples in <a href="chap6.html#X7FCEE3BF86B02CC6"><b>6.1-6</b></a>). The second functon returns an algebra homomorphism from <var class="Arg">a</var> to the matrix algebra.</p>
<table class="example">
<tr><td><pre>
gap> a := ThinnedAlgebraWithOne(GF(2),GrigorchukGroup);
<self-similar algebra-with-one on alphabet GF(2)^2 with 4 generators>
gap> List([0..4],i->Dimension(MatrixQuotient(a,i)));
[ 1, 2, 6, 22, 78 ]
</pre></td></tr></table>
<p><a id="X8150FC4E84D208C6" name="X8150FC4E84D208C6"></a></p>
<h5>8.2-2 ThinnedAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ThinnedAlgebra</code>( <var class="Arg">r, g</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">> ThinnedAlgebraWithOne</code>( <var class="Arg">r, g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>The thinned algebra [with one] associated with <var class="Arg">g</var>.</p>
<p>The first function returns the thinned algebra of a FR group/monoid/semigroup <var class="Arg">g</var>, over the domain <var class="Arg">r</var>. This is the linear envelope of <var class="Arg">g</var> in its natural action on sequences.</p>
<p>The embedding of <var class="Arg">g</var> in its thinned algebra is returned by <code class="code">Embedding(g,a)</code>.</p>
<table class="example">
<tr><td><pre>
gap> a := ThinnedAlgebraWithOne(GF(2),GrigorchukGroup);
<self-similar algebra on alphabet GF(2)^2 with 5 generators>
gap> a.1 = GrigorchukGroup.1^Embedding(GrigorchukGroup,a);
true
gap> Dimension(VectorSpace(GF(2),[One(a),a.2,a.3,a.4]));
3
</pre></td></tr></table>
<p><a id="X8572DCAE7F888DDA" name="X8572DCAE7F888DDA"></a></p>
<h5>8.2-3 Nillity</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> Nillity</code>( <var class="Arg">x</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>The smallest <code class="code">n</code> such that x^n=0.</p>
<p>This command computes the nillity of <var class="Arg">x</var>, i.e. the smallest <code class="code">n</code> such that x^n=0. The command is of course not guaranteed to terminate.</p>
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