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 Magmas and Semigroups.
IsMonoid( D ) P
A monoid is a magma-with-one (see Magmas) with associative multiplication.
Monoid( gen1, gen2 ... ) F
Monoid( gens ) F
Monoid( gens, id ) F
In the first form, Monoid returns the monoid generated by the
arguments gen1, gen2 ...,
that is, the closure of these elements under multiplication and taking
the 0-th power.
In the second form, Monoid returns the monoid generated by the
elements in the homogeneous list gens;
a square matrix as only argument is treated as one generator,
not as a list of generators.
In the third form, Monoid returns the monoid generated by the
elements in the homogeneous list gens, with identity id.
It is not checked whether the underlying multiplication is associative,
use MagmaWithOne (see MagmaWithOne) and IsAssociative
(see IsAssociative)
if you want to check whether a magma-with-one is in fact a monoid.
Submonoid( M, gens ) F
SubmonoidNC( M, gens ) F
are just synonyms of SubmagmaWithOne and SubmagmaWithOneNC,
respectively (see SubmagmaWithOne).
MonoidByGenerators( gens ) O
MonoidByGenerators( gens, one ) O
is the underlying operation of Monoid (see Monoid).
AsMonoid( C ) A
If C is a collection whose elements form a monoid
(see IsMonoid) then AsMonoid returns this monoid.
Otherwise fail is returned.
AsSubmonoid( D, C ) O
Let D be a domain and C a collection.
If C is a subset of D that forms a monoid then AsSubmonoid
returns this monoid, with parent D.
Otherwise fail is returned.
GeneratorsOfMonoid( M ) A
Monoid generators of a monoid M are the same as magma-with-one generators (see GeneratorsOfMagmaWithOne).
TrivialSubmonoid( M ) A
is just a synonym for TrivialSubmagmaWithOne
(see TrivialSubmagmaWithOne).
FreeMonoid( [wfilt, ]rank ) F
FreeMonoid( [wfilt, ]rank, name ) F
FreeMonoid( [wfilt, ]name1, name2, ... ) F
FreeMonoid( [wfilt, ]names ) F
FreeMonoid( [wfilt, ]infinity, name, init ) F
Called in the first form, FreeMonoid returns a free monoid on
rank generators.
Called in the second form, FreeMonoid returns a free monoid on
rank generators, printed as name1, name2 etc.,
that is, each name is the concatenation of the string name and an
integer from 1 to range.
Called in the third form, FreeMonoid returns a free monoid on
as many generators as arguments, printed as name1, name2 etc.
Called in the fourth form, FreeMonoid returns a free monoid on
as many generators as the length of the list names, the i-th
generator being printed as names[i].
Called in the fifth form, FreeMonoid returns a free monoid on
infinitely many generators, where the first generators are printed
by the names in the list init, and the other generators by name
and an appended number.
If the extra argument wfilt is given, it must be either
IsSyllableWordsFamily or IsLetterWordsFamily or
IsWLetterWordsFamily or IsBLetterWordsFamily. The filter then
specifies the representation used for the elements of the free group
(see Representations for Associative Words). If no such filter is
given, a letter representation is used.
MonoidByMultiplicationTable( A ) F
returns the monoid whose multiplication is defined by the square
matrix A (see MagmaByMultiplicationTable) if such a monoid exists.
Otherwise fail is returned.
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GAP 4 manual
March 2006