# ExtAlgBasis¶

A domain used in the construction of the exterior algebra on a set X over a ring R. This domain represents the set of all ordered subsets of the set X, assumed to be in correspondance with {1, 2, 3, ...}. The ordered subsets are themselves ordered lexicographically and are in bijective correspondance with an ordered basis of the exterior algebra. In this domain we are dealing strictly with the exponents of basis elements which can only be 0 or 1. The multiplicative identity element of the exterior algebra corresponds to the empty subset of X. A coerce from List Integer to an ordered basis element is provided to allow the convenient input of expressions. Another exported function forgets the ordered structure and simply returns the list corresponding to an ordered subset.

<: (%, %) -> Boolean
from PartialOrder
<=: (%, %) -> Boolean
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean
from PartialOrder
>=: (%, %) -> Boolean
from PartialOrder
~=: (%, %) -> Boolean
from BasicType
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: List Integer -> %
coerce(l) converts a list of 0's and 1's into a basis element, where 1 (respectively 0) designates that the variable of the corresponding index of l is (respectively, is not) present. Error: if an element of l is not 0 or 1.
degree: % -> NonNegativeInteger
degree(x) gives the numbers of 1's in x, i.e. the number of non-zero exponents in the basis element that x represents.
exponents: % -> List Integer
exponents(x) converts a domain element into a list of zeros and ones corresponding to the exponents in the basis element that x represents.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
max: (%, %) -> %
from OrderedSet
min: (%, %) -> %
from OrderedSet
Nul: NonNegativeInteger -> %
Nul() gives the basis element 1 for the algebra generated by n generators.
smaller?: (%, %) -> Boolean
from Comparable

BasicType

Comparable

OrderedSet

PartialOrder

SetCategory