<html><head><title>[ref] 50 Monoids</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href="../index.htm">Top</a>] [<a href = "chapters.htm">Up</a>] [<a href ="CHAP049.htm">Previous</a>] [<a href ="CHAP051.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>50 Monoids</h1><p> <p> This chapter describes functions for monoids. Currently there are only few of them. More general functions for magmas and semigroups can be found in Chapters <a href="CHAP033.htm">Magmas</a> and <a href="CHAP049.htm">Semigroups</a>. <p> <a name = ""></a> <li><code>IsMonoid( </code><var>D</var><code> ) P</code> <p> A <strong>monoid</strong> is a magma-with-one (see <a href="CHAP033.htm">Magmas</a>) with associative multiplication. <p> <a name = ""></a> <li><code>Monoid( </code><var>gen1</var><code>, </code><var>gen2</var><code> ... ) F</code> <a name = ""></a> <li><code>Monoid( </code><var>gens</var><code> ) F</code> <a name = ""></a> <li><code>Monoid( </code><var>gens</var><code>, </code><var>id</var><code> ) F</code> <p> In the first form, <code>Monoid</code> returns the monoid generated by the arguments <var>gen1</var>, <var>gen2</var> ..., that is, the closure of these elements under multiplication and taking the 0-th power. In the second form, <code>Monoid</code> returns the monoid generated by the elements in the homogeneous list <var>gens</var>; a square matrix as only argument is treated as one generator, not as a list of generators. In the third form, <code>Monoid</code> returns the monoid generated by the elements in the homogeneous list <var>gens</var>, with identity <var>id</var>. <p> It is <strong>not</strong> checked whether the underlying multiplication is associative, use <code>MagmaWithOne</code> (see <a href="CHAP033.htm#SSEC002.2">MagmaWithOne</a>) and <code>IsAssociative</code> (see <a href="CHAP033.htm#SSEC004.7">IsAssociative</a>) if you want to check whether a magma-with-one is in fact a monoid. <p> <a name = ""></a> <li><code>Submonoid( </code><var>M</var><code>, </code><var>gens</var><code> ) F</code> <a name = ""></a> <li><code>SubmonoidNC( </code><var>M</var><code>, </code><var>gens</var><code> ) F</code> <p> are just synonyms of <code>SubmagmaWithOne</code> and <code>SubmagmaWithOneNC</code>, respectively (see <a href="CHAP033.htm#SSEC002.8">SubmagmaWithOne</a>). <p> <a name = ""></a> <li><code>MonoidByGenerators( </code><var>gens</var><code> ) O</code> <a name = ""></a> <li><code>MonoidByGenerators( </code><var>gens</var><code>, </code><var>one</var><code> ) O</code> <p> is the underlying operation of <code>Monoid</code> (see <a href="CHAP050.htm">Monoid</a>). <p> <a name = ""></a> <li><code>AsMonoid( </code><var>C</var><code> ) A</code> <p> If <var>C</var> is a collection whose elements form a monoid (see <a href="CHAP050.htm">IsMonoid</a>) then <code>AsMonoid</code> returns this monoid. Otherwise <code>fail</code> is returned. <p> <a name = ""></a> <li><code>AsSubmonoid( </code><var>D</var><code>, </code><var>C</var><code> ) O</code> <p> Let <var>D</var> be a domain and <var>C</var> a collection. If <var>C</var> is a subset of <var>D</var> that forms a monoid then <code>AsSubmonoid</code> returns this monoid, with parent <var>D</var>. Otherwise <code>fail</code> is returned. <p> <a name = ""></a> <li><code>GeneratorsOfMonoid( </code><var>M</var><code> ) A</code> <p> Monoid generators of a monoid <var>M</var> are the same as magma-with-one generators (see <a href="CHAP033.htm#SSEC004.2">GeneratorsOfMagmaWithOne</a>). <p> <a name = ""></a> <li><code>TrivialSubmonoid( </code><var>M</var><code> ) A</code> <p> is just a synonym for <code>TrivialSubmagmaWithOne</code> (see <a href="CHAP033.htm#SSEC004.14">TrivialSubmagmaWithOne</a>). <p> <a name = ""></a> <li><code>FreeMonoid( [</code><var>wfilt</var><code>, ]</code><var>rank</var><code> ) F</code> <a name = ""></a> <li><code>FreeMonoid( [</code><var>wfilt</var><code>, ]</code><var>rank</var><code>, </code><var>name</var><code> ) F</code> <a name = ""></a> <li><code>FreeMonoid( [</code><var>wfilt</var><code>, ]</code><var>name1</var><code>, </code><var>name2</var><code>, ... ) F</code> <a name = ""></a> <li><code>FreeMonoid( [</code><var>wfilt</var><code>, ]</code><var>names</var><code> ) F</code> <a name = ""></a> <li><code>FreeMonoid( [</code><var>wfilt</var><code>, ]infinity, </code><var>name</var><code>, </code><var>init</var><code> ) F</code> <p> Called in the first form, <code>FreeMonoid</code> returns a free monoid on <var>rank</var> generators. Called in the second form, <code>FreeMonoid</code> returns a free monoid on <var>rank</var> generators, printed as <code></code><var>name</var><code>1</code>, <code></code><var>name</var><code>2</code> etc., that is, each name is the concatenation of the string <var>name</var> and an integer from <code>1</code> to <var>range</var>. Called in the third form, <code>FreeMonoid</code> returns a free monoid on as many generators as arguments, printed as <var>name1</var>, <var>name2</var> etc. Called in the fourth form, <code>FreeMonoid</code> returns a free monoid on as many generators as the length of the list <var>names</var>, the <i>i</i>-th generator being printed as <code></code><var>names</var><code>[<i>i</i>]</code>. Called in the fifth form, <code>FreeMonoid</code> returns a free monoid on infinitely many generators, where the first generators are printed by the names in the list <var>init</var>, and the other generators by <var>name</var> and an appended number. <p> If the extra argument <var>wfilt</var> is given, it must be either <code>IsSyllableWordsFamily</code> or <code>IsLetterWordsFamily</code> or <code>IsWLetterWordsFamily</code> or <code>IsBLetterWordsFamily</code>. The filter then specifies the representation used for the elements of the free group (see <a href="CHAP035.htm#SECT006">Representations for Associative Words</a>). If no such filter is given, a letter representation is used. <p> <a name = ""></a> <li><code>MonoidByMultiplicationTable( </code><var>A</var><code> ) F</code> <p> returns the monoid whose multiplication is defined by the square matrix <var>A</var> (see <a href="CHAP033.htm#SSEC003.1">MagmaByMultiplicationTable</a>) if such a monoid exists. Otherwise <code>fail</code> is returned. <p> <p> [<a href="../index.htm">Top</a>] [<a href = "chapters.htm">Up</a>] [<a href ="CHAP049.htm">Previous</a>] [<a href ="CHAP051.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>
