<p>This chapter describes categories of <em>words</em> and <em>nonassociative words</em>, and operations for them. For information about <em>associative words</em>, which occur for example as elements in free groups, see Chapter <a href="chap37_mj.html#X78C56A0A87CE380E"><span class="RefLink">37</span></a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWord</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWordWithOne</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWordWithInverse</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Given a free multiplicative structure <span class="SimpleMath">\(M\)</span> that is freely generated by a subset <span class="SimpleMath">\(X\)</span>, any expression of an element in <span class="SimpleMath">\(M\)</span> as an iterated product of elements in <span class="SimpleMath">\(X\)</span> is called a <em>word</em> over <span class="SimpleMath">\(X\)</span>.</p>
<p>Interesting cases of free multiplicative structures are those of free semigroups, free monoids, and free groups, where the multiplication is associative (see <code class="func">IsAssociative</code> (<a href="chap35_mj.html#X7C83B5A47FD18FB7"><span class="RefLink">35.4-7</span></a>)), which are described in Chapter <a href="chap37_mj.html#X78C56A0A87CE380E"><span class="RefLink">37</span></a>, and also the case of free magmas, where the multiplication is nonassociative (see <code class="func">IsNonassocWord</code> (<a href="chap36_mj.html#X808FA6F97E16502F"><span class="RefLink">36.1-3</span></a>)).</p>
<p>Elements in free magmas (see <code class="func">FreeMagma</code> (<a href="chap36_mj.html#X7CFFD9027DDD1555"><span class="RefLink">36.4-1</span></a>)) lie in the category <code class="func">IsWord</code>; similarly, elements in free magmas-with-one (see <code class="func">FreeMagmaWithOne</code> (<a href="chap36_mj.html#X86DB748080B4A9B9"><span class="RefLink">36.4-2</span></a>)) lie in the category <code class="func">IsWordWithOne</code>, and so on.</p>
<p><code class="func">IsWord</code> is mainly a <q>common roof</q> for the two <em>disjoint</em> categories <code class="func">IsAssocWord</code> (<a href="chap37_mj.html#X7FA8DA728773BA89"><span class="RefLink">37.1-1</span></a>) and <code class="func">IsNonassocWord</code> (<a href="chap36_mj.html#X808FA6F97E16502F"><span class="RefLink">36.1-3</span></a>) of associative and nonassociative words. This means that associative words are <em>not</em> regarded as special cases of nonassociative words. The main reason for this setup is that we are interested in different external representations for associative and nonassociative words (see <a href="chap36_mj.html#X84C2F9037EEE9CED"><span class="RefLink">36.5</span></a> and <a href="chap37_mj.html#X7934D3D5797102EC"><span class="RefLink">37.7</span></a>).</p>
<p>Note that elements in finitely presented groups and also elements in polycyclic groups in <strong class="pkg">GAP</strong> are <em>not</em> in <code class="func">IsWord</code> although they are usually called words, see Chapters <a href="chap47_mj.html#X7AA982637E90B35A"><span class="RefLink">47</span></a> and <a href="chap46_mj.html#X7EAD57C97EBF7E67"><span class="RefLink">46</span></a>.</p>
<p>Words are <em>constants</em> (see <a href="chap12_mj.html#X7F0C119682196D65"><span class="RefLink">12.6</span></a>), that is, they are not copyable and not mutable.</p>
<p>The usual way to create words is to form them as products of known words, starting from <em>generators</em> of a free structure such as a free magma or a free group (see <code class="func">FreeMagma</code> (<a href="chap36_mj.html#X7CFFD9027DDD1555"><span class="RefLink">36.4-1</span></a>), <code class="func">FreeGroup</code> (<a href="chap37_mj.html#X8215999E835290F0"><span class="RefLink">37.2-1</span></a>)).</p>
<p>Words are also used to implement free algebras, in the same way as group elements are used to implement group algebras (see <a href="chap62_mj.html#X7A7B00127DC9DD40"><span class="RefLink">62.3</span></a> and Chapter <a href="chap65_mj.html#X825897DC7A16E07D"><span class="RefLink">65</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNonassocWord</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNonassocWordWithOne</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A <em>nonassociative word</em> in <strong class="pkg">GAP</strong> is an element in a free magma or a free magma-with-one (see <a href="chap36_mj.html#X7F51B17983019D3E"><span class="RefLink">36.4</span></a>).</p>
<p>The default methods for <code class="func">ViewObj</code> (<a href="chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) and <code class="func">PrintObj</code> (<a href="chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) show nonassociative words as products of letters, where the succession of multiplications is determined by round brackets.</p>
<p>In this sense each nonassociative word describes a <q>program</q> to form a product of generators. (Also associative words can be interpreted as such programs, except that the exact succession of multiplications is not prescribed due to the associativity.) The function <code class="func">MappedWord</code> (<a href="chap36_mj.html#X7EC17930781D104A"><span class="RefLink">36.3-1</span></a>) implements a way to apply such a program. A more general way is provided by straight line programs (see <a href="chap37_mj.html#X7DC99E4284093FBB"><span class="RefLink">37.8</span></a>).</p>
<p>Note that associative words (see Chapter <a href="chap37_mj.html#X78C56A0A87CE380E"><spanclass="RefLink">37</span></a>) are <em>not</em> regarded as special cases of nonassociative words (see <code class="func">IsWord</code> (<a href="chap36_mj.html#X843F5C3A82239398"><span class="RefLink">36.1-1</span></a>)).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \=</code>( <var class="Arg">w1</var>, <var class="Arg">w2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Two words are equal if and only if they are words over the same alphabet and with equal external representations (see <a href="chap36_mj.html#X84C2F9037EEE9CED"><span class="RefLink">36.5</span></a> and <a href="chap37_mj.html#X7934D3D5797102EC"><span class="RefLink">37.7</span></a>). For nonassociative words, the latter means that the words arise from the letters of the alphabet by the same sequence of multiplications.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \<</code>( <var class="Arg">w1</var>, <var class="Arg">w2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Words are ordered according to their external representation. More precisely, two words can be compared if they are words over the same alphabet, and the word with smaller external representation is smaller. For nonassociative words, the ordering is defined in <a href="chap36_mj.html#X84C2F9037EEE9CED"><span class="RefLink">36.5</span></a>; associative words are ordered by the shortlex ordering via <code class="code"><</code> (see <a href="chap37_mj.html#X7934D3D5797102EC"><span class="RefLink">37.7</span></a>).</p>
<p>Note that the alphabet of a word is determined by its family (see <a href="chap13_mj.html#X846063757EC05986"><span class="RefLink">13.1</span></a>), and that the result of each call to <code class="func">FreeMagma</code> (<a href="chap36_mj.html#X7CFFD9027DDD1555"><span class="RefLink">36.4-1</span></a>), <code class="func">FreeGroup</code> (<a href="chap37_mj.html#X8215999E835290F0"><span class="RefLink">37.2-1</span></a>) etc. consists of words over a new alphabet. In particular, there is no <q>universal</q> empty word, every families of words in <code class="func">IsWordWithOne</code> (<a href="chap36_mj.html#X843F5C3A82239398"><span class="RefLink">36.1-1</span></a>) has its own empty word.</p>
<h4>36.3 <span class="Heading">Operations for Words</span></h4>
<p>Two words can be multiplied via <code class="code">*</code> only if they are words over the same alphabet (see <a href="chap36_mj.html#X852C815F85DBE4BD"><span class="RefLink">36.2</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MappedWord</code>( <var class="Arg">w</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">MappedWord</code> returns the object that is obtained by replacing each occurrence in the word <var class="Arg">w</var> of a generator in the list <var class="Arg">gens</var> by the corresponding object in the list <var class="Arg">imgs</var>. The lists <var class="Arg">gens</var> and <var class="Arg">imgs</var> must of course have the same length.</p>
<p><code class="func">MappedWord</code> needs to do some preprocessing to get internal generator numbers etc. When mapping many (several thousand) words, a dedicated loop might be faster.</p>
<p>For example, if the elements in <var class="Arg">imgs</var> are all <em>associative words</em> (see Chapter <a href="chap37_mj.html#X78C56A0A87CE380E"><span class="RefLink">37</span></a>) in the same family as the elements in <var class="Arg">gens</var>, and some of them are equal to the corresponding generators in <var class="Arg">gens</var>, then those may be omitted from <var class="Arg">gens</var> and <var class="Arg">imgs</var>. In this situation, the special case that the lists <var class="Arg">gens</var> and <var class="Arg">imgs</var> have only length <span class="SimpleMath">\(1\)</span> is handled more efficiently by <code class="func">EliminatedWord</code> (<a href="chap37_mj.html#X8486BFE1844CFE59"><span class="RefLink">37.4-6</span></a>).</p>
<p>If the word is from a free group, it is permitted to give inverses of (some) of the generators as extra generators. This can speed up the execution by removing the need to calculate inverses anew.</p>
<p>The easiest way to create a family of words is to construct the free object generated by these words. Each such free object defines a unique alphabet, and its generators are simply the words of length one over this alphabet; These generators can be accessed via <code class="func">GeneratorsOfMagma</code> (<a href="chap35_mj.html#X872E05B478EC20CA"><span class="RefLink">35.4-1</span></a>) in the case of a free magma, and via <code class="func">GeneratorsOfMagmaWithOne</code> (<a href="chap35_mj.html#X87DD93EC8061DD81"><span class="RefLink">35.4-2</span></a>) in the case of a free magma-with-one.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMagma</code>( <var class="Arg">rank</var>[, <var class="Arg">name</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMagma</code>( <var class="Arg">name1</var>[, <var class="Arg">name2</var>[, <var class="Arg">...</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMagma</code>( <var class="Arg">names</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMagma</code>( <var class="Arg">infinity</var>[, <var class="Arg">name</var>][, <var class="Arg">init</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">FreeMagma</code> returns a free magma. The number of generators, and the labels given to the generators, can be specified in several different ways. Warning: the labels of generators are only an aid for printing, and do not necessarily distinguish generators; see the examples at the end of <code class="func">FreeSemigroup</code> (<a href="chap51_mj.html#X7C72E4747BF642BB"><span class="RefLink">51.1-10</span></a>) for more information.</p>
<dl>
<dt><strong class="Mark">
1: For a given rank, and an optional generator name prefix
</strong></dt>
<dd><p>Called with a positive integer <var class="Arg">rank</var>, <code class="func">FreeMagma</code> returns a free magma on <var class="Arg">rank</var> generators. The optional argument <var class="Arg">name</var> must be a string; its default value is <code class="code">"x"</code>.</p>
<p>If <var class="Arg">name</var> is not given but the <code class="code">generatorNames</code> option is, then this option is respected as described in Section <a href="chap50_mj.html#X7D0FFDA4793995FC"><span class="RefLink">50.1-16</span></a>.</p>
<p>Otherwise, the generators of the returned free magma are labelled <var class="Arg">name</var><code class="code">1</code>, ..., <var class="Arg">name</var><code class="code">k</code>, where <code class="code">k</code> is the value of <var class="Arg">rank</var>.</p>
</dd>
<dt><strong class="Mark">2: For given generator names</strong></dt>
<dd><p>Called with various (at least one) nonempty strings, <code class="func">FreeMagma</code> returns a free magma on as many generators as arguments, which are labelled <var class="Arg">name1</var>, <var class="Arg">name2</var>, etc.</p>
</dd>
<dt><strong class="Mark">3: For a given list of generator names</strong></dt>
<dd><p>Called with a finite nonempty list <var class="Arg">names</var> of nonempty strings, <code class="func">FreeMagma</code> returns a free magma on <code class="code">Length(<var class="Arg">names</var>)</code> generators, whose <code class="code">i</code>-th generator is labelled <var class="Arg">names</var><code class="code">[i]</code>.</p>
</dd>
<dt><strong class="Mark">
4: For the rank <code class="keyw">infinity</code>,
an optional default generator name prefix,
and an optional finite list of generator names
</strong></dt>
<dd><p>Called in the fourth form, <code class="func">FreeMagma</code> returns a free magma on infinitely many generators. The optional argument <var class="Arg">name</var> must be a string; its default value is <code class="code">"x"</code>, and the optional argument <var class="Arg">init</var> must be a finite list of nonempty strings; its default value is an empty list. The generators are initially labelled according to the list <var class="Arg">init</var>, followed by <var class="Arg">name</var><code class="code">i</code> for each <code class="code">i</code> in the range from <code class="code">Length(<var class="Arg">init</var>)+1</code> to <code class="keyw">infinity</code>.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagma( 4 );</span>
<free magma on the generators [ x1, x2, x3, x4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagma( 3, "a" );</span>
<free magma on the generators [ a1, a2, a3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagma( "a", "b" );</span>
<free magma on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagma( [ "a", "b" ] );</span>
<free magma on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagma( infinity );</span>
<free magma with infinity generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeMagma( infinity, "gen" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMagma( F ){[ 1 .. 4 ]};</span>
[ gen1, gen2, gen3, gen4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeMagma( infinity, [ "z", "a" ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMagma( F ){[ 1 .. 3 ]};</span>
[ z, a, x3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeMagma( infinity, "y", [ "z", "a" ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMagma( F ){[ 1 .. 4 ]};</span>
[ z, a, y3, y4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagma( 3 : generatorNames := "elt" );</span>
<free magma on the generators [ elt1, elt2, elt3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagma( 2 : generatorNames := [ "u", "v", "w" ] );</span>
<free magma on the generators [ u, v ]>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMagmaWithOne</code>( <var class="Arg">rank</var>[, <var class="Arg">name</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMagmaWithOne</code>( [<var class="Arg">name1</var>[, <var class="Arg">name2</var>[, <var class="Arg">...</var>]]] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMagmaWithOne</code>( <var class="Arg">names</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMagmaWithOne</code>( <var class="Arg">infinity</var>[, <var class="Arg">name</var>][, <var class="Arg">init</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">FreeMagmaWithOne</code> returns a free magma-with-one. The number of generators, and the labels given to the generators, can be specified in several different ways. Warning: the labels of generators are only an aid for printing, and do not necessarily distinguish generators; see the examples at the end of <code class="func">FreeSemigroup</code> (<a href="chap51_mj.html#X7C72E4747BF642BB"><span class="RefLink">51.1-10</span></a>) for more information.</p>
<dl>
<dt><strong class="Mark">
1: For a given rank, and an optional generator name prefix
</strong></dt>
<dd><p>Called with a nonnegative integer <var class="Arg">rank</var>, <code class="func">FreeMagmaWithOne</code> returns a free magma-with-one on <var class="Arg">rank</var> generators. The optional argument <var class="Arg">name</var> must be a string; its default value is <code class="code">"x"</code>.</p>
<p>If <var class="Arg">name</var> is not given but the <code class="code">generatorNames</code> option is, then this option is respected as described in Section <a href="chap50_mj.html#X7D0FFDA4793995FC"><span class="RefLink">50.1-16</span></a>.</p>
<p>Otherwise, the generators of the returned free magma-with-one are labelled <var class="Arg">name</var><code class="code">1</code>, ..., <var class="Arg">name</var><code class="code">k</code>, where <code class="code">k</code> is the value of <var class="Arg">rank</var>.</p>
</dd>
<dt><strong class="Mark">2: For given generator names</strong></dt>
<dd><p>Called with various nonempty strings, <code class="func">FreeMagmaWithOne</code> returns a free magma-with-one on as many generators as arguments, which are labelled <var class="Arg">name1</var>, <var class="Arg">name2</var>, etc.</p>
</dd>
<dt><strong class="Mark">3: For a given list of generator names</strong></dt>
<dd><p>Called with a finite list <var class="Arg">names</var> of nonempty strings, <code class="func">FreeMagmaWithOne</code> returns a free magma-with-one on <code class="code">Length(<var class="Arg">names</var>)</code> generators, whose <code class="code">i</code>-th generator is labelled <var class="Arg">names</var><code class="code">[i]</code>.</p>
</dd>
<dt><strong class="Mark">
4: For the rank <code class="keyw">infinity</code>,
an optional default generator name prefix,
and an optional finite list of generator names
</strong></dt>
<dd><p>Called in the fourth form, <code class="func">FreeMagmaWithOne</code> returns a free magma-with-one on infinitely many generators. The optional argument <var class="Arg">name</var> must be a string; its default value is <code class="code">"x"</code>, and the optional argument <var class="Arg">init</var> must be a finite list of nonempty strings; its default value is an empty list. The generators are initially labelled according to the list <var class="Arg">init</var>, followed by <var class="Arg">name</var><code class="code">i</code> for each <code class="code">i</code> in the range from <code class="code">Length(<var class="Arg">init</var>)+1</code> to <code class="keyw">infinity</code>.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagmaWithOne( 4 );</span>
<free magma-with-one on the generators [ x1, x2, x3, x4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagmaWithOne( 3, "a" );</span>
<free magma-with-one on the generators [ a1, a2, a3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagmaWithOne( "a", "b" );</span>
<free magma-with-one on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagmaWithOne( [ "a", "b" ] );</span>
<free magma-with-one on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagmaWithOne( infinity );</span>
<free magma-with-one with infinity generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeMagmaWithOne( infinity, "gen" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMagmaWithOne( F ){[ 1 .. 4 ]};</span>
[ gen1, gen2, gen3, gen4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeMagmaWithOne( infinity, [ "z", "a" ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMagmaWithOne( F ){[ 1 .. 3 ]};</span>
[ z, a, x3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeMagmaWithOne( infinity, "y", [ "z", "a" ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMagmaWithOne( F ){[ 1 .. 4 ]};</span>
[ z, a, y3, y4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagmaWithOne( 0 );</span>
<free group of rank zero>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagmaWithOne( 3 : generatorNames := "elt" );</span>
<free magma-with-one on the generators [ elt1, elt2, elt3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMagmaWithOne( 2 : generatorNames := [ "u", "v", "w" ] );</span>
<free magma-with-one on the generators [ u, v ]>
</pre></div>
<h4>36.5 <span class="Heading">External Representation for Nonassociative Words</span></h4>
<p>The external representation of nonassociative words is defined as follows. The <span class="SimpleMath">\(i\)</span>-th generator of the family of elements in question has external representation <span class="SimpleMath">\(i\)</span>, the identity (if exists) has external representation <span class="SimpleMath">\(0\)</span>, the inverse of the <span class="SimpleMath">\(i\)</span>-th generator (if exists) has external representation <span class="SimpleMath">\(-i\)</span>. If <span class="SimpleMath">\(v\)</span> and <span class="SimpleMath">\(w\)</span> are nonassociative words with external representations <span class="SimpleMath">\(e_v\)</span> and <span class="SimpleMath">\(e_w\)</span>, respectively then the product <span class="SimpleMath">\(v * w\)</span> has external representation <span class="SimpleMath">\([ e_v, e_w ]\)</span>. So the external representation of any nonassociative word is either an integer or a nested list of integers and lists, where each list has length two.</p>
<p>One can create a nonassociative word from a family of words and the external representation of a nonassociative word using <code class="func">ObjByExtRep</code> (<a href="chap79_mj.html#X8542B32A8206118C"><span class="RefLink">79.8-1</span></a>).</p>
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