Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

CHAPTER
TWENTYTHREE
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: sage.structure.sage_object.SageObject
An 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, ...) and addition_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 pairs of elements from S. A natural source
of such functions is the Python operator module, and in particular operator.add() and
operator.mul(). This may also be a function defined with lambda or def.
•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 by column_keys(), padded to a common width with leading ‘a’s.
–’digits’ - base 10 representation of the elements’ positions in the list given by column_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 of S, 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
the S.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 it’s most basic use, the table needs a structure and an operation:
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