<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>
<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>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> asks for the resulting algebra to be generated by vector elements, see example below. The generators must of course be finite-state.</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"><span class="RefLink">7.1-4</span></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>
<p>This function returns the <em>nucleus</em> of the contracting algebra <var class="Arg">a</var>, i.e. the smallest subspace <code class="code">N</code> of <var class="Arg">a</var> such that the <code class="func">LimitStates</code> (<a href="chap4.html#X8303B36C83371FB3"><span class="RefLink">4.2-11</span></a>) of every element of <var class="Arg">a</var> belong to <code class="code">N</code>.</p>
<p>This function returns <code class="keyw">fail</code> if no such <code class="code">N</code> exists. Usually, it loops forever without being able to decide whether <code class="code">N</code> is finite or infinite. It succeeds precisely when <code class="code">IsContracting(g)</code> succeeds.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">> a := GrigorchukThinnedAlgebra(2);</span>
<self-similar algebra-with-one on alphabet GF(2)^2 with 4 generators, of dimension infinity>
<span class="GAPprompt">gap></span> <span class="GAPinput">NucleusOfFRAlgebra(a);</span>
<vector space over GF(2), with 4 generators>
</pre></div>
<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"><span class="RefLink">6.1-7</span></a>). The second functon returns an algebra homomorphism from <var class="Arg">a</var> to the matrix algebra.</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>
<p>The first command computes the nillity of <var class="Arg">x</var>, i.e. the smallest <code class="code">n</code> such that <span class="SimpleMath">x^n=0</span>. The command is not guaranteed to terminate.</p>
<p>The second command returns whether <var class="Arg">x</var> is nil, that is, whether its nillity is finite.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreeOfHomogeneousElement</code>( <var class="Arg">x</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHomogeneousElement</code>( <var class="Arg">x</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: The degree of <var class="Arg">x</var> in its parent.</p>
<p>If <var class="Arg">x</var> belongs to a graded algebra <code class="code">A</code>, then the second command returns whether <var class="Arg">x</var> belongs to a homogeneous component of <code class="code">Grading(A)</code>, and the first command returns the degree of that component (or <code class="keyw">fail</code> if no such component exists).</p>
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