Operation Tables#
This module implements general operation tables, which are very matrix-like.
- class sage.matrix.operation_table.OperationTable(S, operation, names='letters', elements=None)#
Bases:
SageObjectAn object that represents a binary operation as a table.
Primarily this object is used to provide a
multiplication_table()for objects in the category of magmas (monoids, groups, …) andaddition_table()for objects in the category of commutative additive magmas (additive monoids, groups, …).INPUT:
S- a finite algebraic structure (or finite iterable)operation- a function of two variables that accepts pairsof elements from
S. A natural source of such functions is the Pythonoperatormodule, and in particularoperator.add()andoperator.mul(). This may also be a function defined withlambdaordef.
names- (default:'letters') The type of names used, values are:'letters'- lowercase ASCII letters are used for a base 26 representation of the elements’ positions in the list given bycolumn_keys(), padded to a common width with leading ‘a’s.'digits'- base 10 representation of the elements’ positions in the list given bycolumn_keys(), padded to a common width with leading zeros.'elements'- the string representations of the elements themselves.a list - a list of strings, where the length of the list equals the number of elements.
elements- (default:None) A list of elements ofS, in forms that can be coerced into the structure, eg. their string representations. This may be used to impose an alternate ordering on the elements of \(S`\), perhaps when this is used in the context of a particular structure. The default is to use whatever ordering theS.list()method returns. \(elements`\) can also be a subset which is closed under the operation, useful perhaps when the set is infinite.
OUTPUT: An object with methods that abstracts multiplication tables, addition tables, Cayley tables, etc. It should be general enough to be useful for any finite algebraic structure whose elements can be combined with a binary operation. This is not necessarily meant be constructed directly, but instead should be useful for constructing operation tables of various algebraic structures that have binary operations.
EXAMPLES:
In its most basic use, the table needs a structure and an operation:
sage: from sage.matrix.operation_table import OperationTable sage: G = SymmetricGroup(3) # needs sage.groups sage: OperationTable(G, operation=operator.mul) # needs sage.groups * a b c d e f +------------ a| a b c d e f b| b a d c f e c| c e a f b d d| d f b e a c e| e c f a d b f| f d e b c a
With two operations present, we can specify which operation we want:
sage: from sage.matrix.operation_table import OperationTable sage: R = Integers(6) sage: OperationTable(R, operation=operator.add) + a b c d e f +------------ a| a b c d e f b| b c d e f a c| c d e f a b d| d e f a b c e| e f a b c d f| f a b c d e
The default symbol set for elements is lowercase ASCII letters, which take on a base 26 flavor for structures with more than 26 elements.
sage: from sage.matrix.operation_table import OperationTable sage: G = DihedralGroup(14) # needs sage.groups sage: OperationTable(G, operator.mul, names='letters') # needs sage.groups * aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb +------------------------------------------------------------------------------------ aa| aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb ab| ab aa ad ac af ae ah ag aj ai al ak an am ap ao ar aq at as av au ax aw az ay bb ba ac| ac ba aa ae ad ag af ai ah ak aj am al ao an aq ap as ar au at aw av ay ax bb ab az ad| ad bb ab af ac ah ae aj ag al ai an ak ap am ar ao at aq av as ax au az aw ba aa ay ae| ae az ba ag aa ai ad ak af am ah ao aj aq al as an au ap aw ar ay at bb av ab ac ax af| af ay bb ah ab aj ac al ae an ag ap ai ar ak at am av ao ax aq az as ba au aa ad aw ag| ag ax az ai ba ak aa am ad ao af aq ah as aj au al aw an ay ap bb ar ab at ac ae av ah| ah aw ay aj bb al ab an ac ap ae ar ag at ai av ak ax am az ao ba aq aa as ad af au ai| ai av ax ak az am ba ao aa aq ad as af au ah aw aj ay al bb an ab ap ac ar ae ag at aj| aj au aw al ay an bb ap ab ar ac at ae av ag ax ai az ak ba am aa ao ad aq af ah as ak| ak at av am ax ao az aq ba as aa au ad aw af ay ah bb aj ab al ac an ae ap ag ai ar al| al as au an aw ap ay ar bb at ab av ac ax ae az ag ba ai aa ak ad am af ao ah aj aq am| am ar at ao av aq ax as az au ba aw aa ay ad bb af ab ah ac aj ae al ag an ai ak ap an| an aq as ap au ar aw at ay av bb ax ab az ac ba ae aa ag ad ai af ak ah am aj al ao ao| ao ap ar aq at as av au ax aw az ay ba bb aa ab ad ac af ae ah ag aj ai al ak am an ap| ap ao aq ar as at au av aw ax ay az bb ba ab aa ac ad ae af ag ah ai aj ak al an am aq| aq an ap as ar au at aw av ay ax bb az ab ba ac aa ae ad ag af ai ah ak aj am ao al ar| ar am ao at aq av as ax au az aw ba ay aa bb ad ab af ac ah ae aj ag al ai an ap ak as| as al an au ap aw ar ay at bb av ab ax ac az ae ba ag aa ai ad ak af am ah ao aq aj at| at ak am av ao ax aq az as ba au aa aw ad ay af bb ah ab aj ac al ae an ag ap ar ai au| au aj al aw an ay ap bb ar ab at ac av ae ax ag az ai ba ak aa am ad ao af aq as ah av| av ai ak ax am az ao ba aq aa as ad au af aw ah ay aj bb al ab an ac ap ae ar at ag aw| aw ah aj ay al bb an ab ap ac ar ae at ag av ai ax ak az am ba ao aa aq ad as au af ax| ax ag ai az ak ba am aa ao ad aq af as ah au aj aw al ay an bb ap ab ar ac at av ae ay| ay af ah bb aj ab al ac an ae ap ag ar ai at ak av am ax ao az aq ba as aa au aw ad az| az ae ag ba ai aa ak ad am af ao ah aq aj as al au an aw ap ay ar bb at ab av ax ac ba| ba ac ae aa ag ad ai af ak ah am aj ao al aq an as ap au ar aw at ay av bb ax az ab bb| bb ad af ab ah ac aj ae al ag an ai ap ak ar am at ao av aq ax as az au ba aw ay aa
Another symbol set is base 10 digits, padded with leading zeros to make a common width.
sage: from sage.matrix.operation_table import OperationTable sage: G = AlternatingGroup(4) # needs sage.groups sage: OperationTable(G, operator.mul, names='digits') # needs sage.groups * 00 01 02 03 04 05 06 07 08 09 10 11 +------------------------------------ 00| 00 01 02 03 04 05 06 07 08 09 10 11 01| 01 02 00 05 03 04 07 08 06 11 09 10 02| 02 00 01 04 05 03 08 06 07 10 11 09 03| 03 06 09 00 07 10 01 04 11 02 05 08 04| 04 08 10 02 06 11 00 05 09 01 03 07 05| 05 07 11 01 08 09 02 03 10 00 04 06 06| 06 09 03 10 00 07 04 11 01 08 02 05 07| 07 11 05 09 01 08 03 10 02 06 00 04 08| 08 10 04 11 02 06 05 09 00 07 01 03 09| 09 03 06 07 10 00 11 01 04 05 08 02 10| 10 04 08 06 11 02 09 00 05 03 07 01 11| 11 05 07 08 09 01 10 02 03 04 06 00
If the group’s elements are not too cumbersome, or the group is small, then the string representation of the elements can be used.
sage: from sage.matrix.operation_table import OperationTable sage: G = AlternatingGroup(3) # needs sage.groups sage: OperationTable(G, operator.mul, names='elements') # needs sage.groups * () (1,2,3) (1,3,2) +------------------------ ()| () (1,2,3) (1,3,2) (1,2,3)| (1,2,3) (1,3,2) () (1,3,2)| (1,3,2) () (1,2,3)
You can give the elements any names you like, but they need to be ordered in the same order as returned by the
column_keys()method.sage: # needs sage.groups sage: from sage.matrix.operation_table import OperationTable sage: G = QuaternionGroup() sage: T = OperationTable(G, operator.mul) sage: T.column_keys() ((), (1,2,3,4)(5,6,7,8), ..., (1,8,3,6)(2,7,4,5)) sage: names=['1', 'I', '-1', '-I', 'J', '-K', '-J', 'K'] sage: T.change_names(names=names) sage: sorted(T.translation().items()) [('-1', (1,3)(2,4)(5,7)(6,8)), ..., ('K', (1,8,3,6)(2,7,4,5))] sage: T * 1 I -1 -I J -K -J K +------------------------ 1| 1 I -1 -I J -K -J K I| I -1 -I 1 K J -K -J -1| -1 -I 1 I -J K J -K -I| -I 1 I -1 -K -J K J J| J -K -J K -1 -I 1 I -K| -K -J K J I -1 -I 1 -J| -J K J -K 1 I -1 -I K| K J -K -J -I 1 I -1
With the right functions and a list comprehension, custom names can be easier. A multiplication table for hex digits (without carries):
sage: from sage.matrix.operation_table import OperationTable sage: R = Integers(16) sage: names=['{:x}'.format(Integer(a)) for a in R] sage: OperationTable(R, operation=operator.mul, names=names) * 0 1 2 3 4 5 6 7 8 9 a b c d e f +-------------------------------- 0| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1| 0 1 2 3 4 5 6 7 8 9 a b c d e f 2| 0 2 4 6 8 a c e 0 2 4 6 8 a c e 3| 0 3 6 9 c f 2 5 8 b e 1 4 7 a d 4| 0 4 8 c 0 4 8 c 0 4 8 c 0 4 8 c 5| 0 5 a f 4 9 e 3 8 d 2 7 c 1 6 b 6| 0 6 c 2 8 e 4 a 0 6 c 2 8 e 4 a 7| 0 7 e 5 c 3 a 1 8 f 6 d 4 b 2 9 8| 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 9| 0 9 2 b 4 d 6 f 8 1 a 3 c 5 e 7 a| 0 a 4 e 8 2 c 6 0 a 4 e 8 2 c 6 b| 0 b 6 1 c 7 2 d 8 3 e 9 4 f a 5 c| 0 c 8 4 0 c 8 4 0 c 8 4 0 c 8 4 d| 0 d a 7 4 1 e b 8 5 2 f c 9 6 3 e| 0 e c a 8 6 4 2 0 e c a 8 6 4 2 f| 0 f e d c b a 9 8 7 6 5 4 3 2 1
This should be flexible enough to create a variety of such tables.
sage: from sage.matrix.operation_table import OperationTable sage: from operator import xor sage: T=OperationTable(ZZ, xor, elements=range(8)) sage: T . a b c d e f g h +---------------- a| a b c d e f g h b| b a d c f e h g c| c d a b g h e f d| d c b a h g f e e| e f g h a b c d f| f e h g b a d c g| g h e f c d a b h| h g f e d c b a sage: names=['000', '001','010','011','100','101','110','111'] sage: T.change_names(names) sage: T.set_print_symbols('^', '\\land') sage: T ^ 000 001 010 011 100 101 110 111 +-------------------------------- 000| 000 001 010 011 100 101 110 111 001| 001 000 011 010 101 100 111 110 010| 010 011 000 001 110 111 100 101 011| 011 010 001 000 111 110 101 100 100| 100 101 110 111 000 001 010 011 101| 101 100 111 110 001 000 011 010 110| 110 111 100 101 010 011 000 001 111| 111 110 101 100 011 010 001 000 sage: T = OperationTable([False, True], operator.or_, names = 'elements') sage: T . False True +------------ False| False True True| True True
AUTHORS:
Rob Beezer (2010-03-15)
Bruno Edwards (2022-10-31)
- change_names(names)#
For an existing operation table, change the names used for the elements.
INPUT:
names- the type of names used, values are:'letters'- lowercase ASCII letters are used for a base 26 representation of the elements’ positions in the list given bylist(), padded to a common width with leading ‘a’s.'digits'- base 10 representation of the elements’ positions in the list given bylist(), padded to a common width with leading zeros.'elements'- the string representations of the elements themselves.a list - a list of strings, where the length of the list equals the number of elements.
OUTPUT:
None. This method changes the table “in-place”, so any printed version will change and the output of thedict()will also change. So any items of interest about a particular table need to be copied/saved prior to calling this method.EXAMPLES:
More examples can be found in the documentation for
OperationTablesince creating a new operation table uses the same routine.sage: # needs sage.groups sage: from sage.matrix.operation_table import OperationTable sage: D = DihedralGroup(2) sage: T = OperationTable(D, operator.mul) sage: T * a b c d +-------- a| a b c d b| b a d c c| c d a b d| d c b a sage: T.translation()['c'] (1,2) sage: T.change_names('digits') sage: T * 0 1 2 3 +-------- 0| 0 1 2 3 1| 1 0 3 2 2| 2 3 0 1 3| 3 2 1 0 sage: T.translation()['2'] (1,2) sage: T.change_names('elements') sage: T * () (3,4) (1,2) (1,2)(3,4) +-------------------------------------------- ()| () (3,4) (1,2) (1,2)(3,4) (3,4)| (3,4) () (1,2)(3,4) (1,2) (1,2)| (1,2) (1,2)(3,4) () (3,4) (1,2)(3,4)| (1,2)(3,4) (1,2) (3,4) () sage: T.translation()['(1,2)'] (1,2) sage: T.change_names(['w', 'x', 'y', 'z']) sage: T * w x y z +-------- w| w x y z x| x w z y y| y z w x z| z y x w sage: T.translation()['y'] (1,2)
- color_table(element_names=True, cmap=None, **options)#
Return a graphic image as a square grid where entries are color coded.
INPUT:
element_names- (default :True) Whether to display text with element names on the imagecmap– (default:matplotlib.cm.gist_rainbow) color map for plot, seematplotlib.cm**options– passed on tomatrix_plot()
EXAMPLES:
sage: from sage.matrix.operation_table import OperationTable sage: OTa = OperationTable(SymmetricGroup(3), operation=operator.mul) # needs sage.groups sage: OTa.color_table() # needs sage.groups sage.plot Graphics object consisting of 37 graphics primitives
- column_keys()#
Returns a tuple of the elements used to build the table.
Note
column_keysandrow_keysare identical. Both list the elements in the order used to label the table.OUTPUT:
The elements of the algebraic structure used to build the table, as a list. But most importantly, elements are present in the list in the order which they appear in the table’s column headings.
EXAMPLES:
sage: from sage.matrix.operation_table import OperationTable sage: G = AlternatingGroup(3) # needs sage.groups sage: T = OperationTable(G, operator.mul) # needs sage.groups sage: T.column_keys() # needs sage.groups ((), (1,2,3), (1,3,2))
- gray_table(**options)#
Return a graphic image as a square grid where entries are displayed in grayscale.
INPUT:
element_names– (default:True) whether to display text with element names on the image**options– passed on tomatrix_plot()
EXAMPLES:
sage: from sage.matrix.operation_table import OperationTable sage: OTa = OperationTable(SymmetricGroup(3), operation=operator.mul) # needs sage.groups sage: OTa.gray_table() # needs sage.groups sage.plot Graphics object consisting of 37 graphics primitives
- matrix_of_variables()#
This method provides some backward compatibility for Cayley tables of groups, whose output was restricted to this single format.
EXAMPLES:
The output here is from the doctests for the old
cayley_table()method for permutation groups.sage: # needs sage.groups sage: from sage.matrix.operation_table import OperationTable sage: G = PermutationGroup(['(1,2,3)', '(2,3)']) sage: T = OperationTable(G, operator.mul) sage: T.matrix_of_variables() [x0 x1 x2 x3 x4 x5] [x1 x0 x3 x2 x5 x4] [x2 x4 x0 x5 x1 x3] [x3 x5 x1 x4 x0 x2] [x4 x2 x5 x0 x3 x1] [x5 x3 x4 x1 x2 x0] sage: T.column_keys()[2]*T.column_keys()[2] == T.column_keys()[0] True
- row_keys()#
Returns a tuple of the elements used to build the table.
Note
column_keysandrow_keysare identical. Both list the elements in the order used to label the table.OUTPUT:
The elements of the algebraic structure used to build the table, as a list. But most importantly, elements are present in the list in the order which they appear in the table’s column headings.
EXAMPLES:
sage: from sage.matrix.operation_table import OperationTable sage: G = AlternatingGroup(3) # needs sage.groups sage: T = OperationTable(G, operator.mul) # needs sage.groups sage: T.column_keys() # needs sage.groups ((), (1,2,3), (1,3,2))
- set_print_symbols(ascii, latex)#
Set the symbols used for text and LaTeX printing of operation tables.
INPUT:
ascii- a single character for text tablelatex- a string to represent an operation in LaTeX math mode. Note the need for double-backslashes to escape properly.
EXAMPLES:
sage: # needs sage.groups sage: from sage.matrix.operation_table import OperationTable sage: G = AlternatingGroup(3) sage: T = OperationTable(G, operator.mul) sage: T.set_print_symbols('@', '\\times') sage: T @ a b c +------ a| a b c b| b c a c| c a b sage: T._latex_() '{\\setlength{\\arraycolsep}{2ex}\n\\begin{array}{r|*{3}{r}}\n\\multicolumn{1}{c|}{\\times}&a&b&c\\\\\\hline\n{}a&a&b&c\\\\\n{}b&b&c&a\\\\\n{}c&c&a&b\\\\\n\\end{array}}'
- table()#
Returns the table as a list of lists, using integers to reference the elements.
OUTPUT: The rows of the table, as a list of rows, each row being a list of integer entries. The integers correspond to the order of the elements in the headings of the table and the order of the output of the
list()method.EXAMPLES:
sage: from sage.matrix.operation_table import OperationTable sage: C = CyclicPermutationGroup(3) # needs sage.groups sage: T=OperationTable(C, operator.mul) # needs sage.groups sage: T.table() # needs sage.groups [[0, 1, 2], [1, 2, 0], [2, 0, 1]]
- translation()#
Returns a dictionary associating names with elements.
OUTPUT: A dictionary whose keys are strings used as names for entries of the table and values that are the actual elements of the algebraic structure.
EXAMPLES:
sage: from sage.matrix.operation_table import OperationTable sage: G = AlternatingGroup(3) # needs sage.groups sage: T = OperationTable(G, operator.mul, names=['p','q','r']) # needs sage.groups sage: T.translation() # needs sage.groups {'p': (), 'q': (1,2,3), 'r': (1,3,2)}