The nearring library contains all nearrings up to order 15 and all
nearrings with identity up to order 31. All nearrings in the library are
nearrings constructed via ExplicitMultiplicationNearRingNC
, so all functions
for these nearrings are applicable to LibraryNearRing
s.
LibraryNearRing(
G,
num )
LibraryNearRing
retrieves a nearring from the nearrings library files.
G must be a group of order ≤ 15. num must be an integer which
indicates the number of the class of nearrings on the specified group.
(Remark: due to the large number of nearrings on D_{12}, make sure that you have enough main memory - say at least 32 MB - available if you want to get a library nearring on D_{12}).
If G is given as a TWGroup
, then a nearring is returned whose group reduct
is equal to G. Otherwise the result is a nearring whose group reduct is
isomorphic to G, and a warning is issued.
The number of nearrings definable on a certain group G can be accessed via
NumberLibraryNearRings(
G )
AllLibraryNearRings(
G )
returns a list of all nearrings (in the library) that have the group G as group reduct.
gap> l := AllLibraryNearRings( GTW3_1 ); [ LibraryNearRing(3/1, 1), LibraryNearRing(3/1, 2), LibraryNearRing(3/1, 3), LibraryNearRing(3/1, 4), LibraryNearRing(3/1, 5) ] gap> Filtered( l, IsNearField ); [ LibraryNearRing(3/1, 3) ] gap> NumberLibraryNearRings( GTW14_2 ); 1821 gap> LN14_2_1234 := LibraryNearRing( GTW14_2, 1234 ); LibraryNearRing(14/2, 1234)
LibraryNearRingWithOne(
G,
num )
LibraryNearRingWithOne
retrieves a nearring from the nearrings library
files.
G must be one of the predefined groups of order ≤ 31.
num must be an integer which indicates the number of the class of
nearrings with identity on the specified group.
The number of nearrings with identity definable on a certain group G can be accessed via
NumberLibraryNearRingsWithOne(
G )
AllLibraryNearRingsWithOne(
G )
returns a list of all nearrings with identity (in the library) that have the group G as group reduct.
gap> NumberLibraryNearRingsWithOne( GTW24_6 ); 0 gap> NumberLibraryNearRingsWithOne( GTW24_4 ); 10 gap> LNwI24_4_8 := LibraryNearRingWithOne( GTW24_4, 8 ); LibraryNearRingWithOne(24/4, 8) gap> AllLibraryNearRingsWithOne( GTW24_6 ); [ ]
IdLibraryNearRing(
nr )
The function IdLibraryNearRing
returns a pair [G, n] such that the
nearring nr is isomorphic to the nth library nearring on the group G.
gap> p := PolynomialNearRing( GTW4_2 ); PolynomialNearRing( 4/2 ) gap> IdLibraryNearRing( p ); [ 8/3, 833 ] gap> n := LibraryNearRing( GTW3_1, 4 ); LibraryNearRing(3/1, 4) gap> d := DirectProductNearRing( n, n ); DirectProductNearRing( LibraryNearRing(3/1, 4), LibraryNearRing(3/1, 4)\ ) gap> IdLibraryNearRing( d ); [ 9/2, 220 ]
IdLibraryNearRingWithOne(
nr )
The function IdLibraryNearRingWithOne
returns a pair [G, n] such
that the nearring nr is isomorphic to the nth library nearring with
identity on the group G. This function can only be applied to nearrings
which have an identity.
gap> l := LibraryNearRingWithOne( GTW12_3, 1 ); LibraryNearRingWithOne(12/3, 1) gap> IdLibraryNearRing( l ); #this command requires time and memory!!! [ 12/3, 37984 ] gap> IdLibraryNearRingWithOne( l ); [ 12/3, 1 ]
IsLibraryNearRing(
nr )
The function IsLibraryNearRing
returns true
if the nearring nr has been
read from the nearring library and false
otherwise.
gap> IsLibraryNearRing( LNwI24_4_8 ); true
LibraryNearRingInfo(
group,
list,
string )
This function provides information about the specified library nearrings
in a way similar to how nearrings are presented in the appendix of
[Pil??].
The parameter group specifies a predefined group; valid
names are exactly those which are also valid for the function
LibraryNearrings
(cf. Section LibraryNearRing).
The parameter list must be a non-empty list of integers defining the classes of nearrings to be printed. Naturally, these integers must all fit into the ranges described in Section LibraryNearRing for the according groups.
The third parameter string is optional. string must be a string
consisting of one or more (or all) of the following characters:
l
, m
, i
, v
, s
, e
, a
.
Per default, (i.e. if this parameter is not specified), the output is
minimal. According to each specified character, the following is added:
a
c
e
g
i
l
m
r
s
v
Examples:
LibraryNearRingInfo( GTW3_1, [ 3 ], "lmivsea" )
will print all
available information about the third class of nearrings on the group
Z_{3}.
LibraryNearRingInfo( GTW4_1, [ 1..12 ] )
will provide a minimal output
for all classes of nearrings on Z_{4}.
LibraryNearRingInfo( GTW6_2, [ 5, 18, 21 ], "mi" )
will print
the minimal information plus the multiplication tables plus the ideals for
the classes 5, 18, and 21 of nearrings on the group S_{3}.
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