This is to announce a new package. Comments/ideas/criticisms are welcome.
Module Elements β elements of free modules.
A ModuleElt{K,V}
represents an element of a free module where basis
elements are of type K
and coefficients of type V
. Usually you want
objects of type V
to be elements of a ring, but it could also be useful
if they just belong to an abelian group. This is similar to the SageMath
CombinatorialFreeModule.
This basic data structure is used in my packages as an efficient
representation at many places. For example, the Monomial
type
representing multivariate monomials is a ModuleElt{Symbol,Int}
:
x^2y^3
is represented by ModuleElt(:x=>2,:y=>3)
And multivariate polynomials are represented by a ModuleElt{Monomial,C}
where C
is the type of the coefficients:
x*yz^2
is represented by ``ModuleElt(x*y=>1,z^2=>1)
ModuleElts
are also used for cyclotomics, CycPols, elements of Hecke
algebras, etcβ¦
A ModuleElt{K,V}
is essentially a list of Pairs{K,V}
. The constructor
takes as argument a list of pairs, or a variable number of pair arguments,
or a generator of pairs.
We provide two implementations:

HModuleElt
, an implementation byDict
s
This requires that the type K
is hashable. It is a very simple
implementation since the interface of the type is close to that of dicts;
the only difference is weeding out keys which have a zero cofficient β
which is necessary since for testing equality of module elements one needs
a canonical form for each element.
 a faster implementation
ModuleElt
is obtained by keeping the list of
pairs sorted by key. This demands that the typeK
has aisless
method.
This implementation is two to four times faster than theDict
one and
requires half the memory.
Both implementations have the same methods, which are mostly the same
methods as a Dict
(haskey
, getindex
, keys
, values
. pairs
,
first
, iterate
, length
, eltype
), with some exceptions. Adding
elements is implemented as merge(+,...)
which is a variation on merge
for Dict
s where keys with zero value are deleted after the operation
(here +
can be replaced by any operation op
with the property that
op(0,x)=op(x,0)=x
).
A module element can also be negated, or multiplied or divided (/
or //
)
by some element (acting on coefficients) if the method is defined between
type V
and that element; there are also zero
and iszero
methods.
ModuleElt
s have methods cmp
and isless
which HModuleElt
s donβt
have. There is also ModuleElts.merge2
which does the same as merge but is
valid for more general operations (I use it with min
and max
which
implement gcd
and lcm
for Monomial
s and CycPol
s).
Here is an example where basis elements are Symbol
s and coefficients are
Int
. As you can see in the examples, at the REPL (or in Jupyter or Pluto,
when IO
has the :limit
attribute) the show
method shows the
coefficients (bracketed if necessary, which is when they have inner
occurences of +*/
), followed by showing the basis elements. The repr
method gives a representation which can be read back in julia:
julia> a=ModuleElt(:xy=>1,:yx=>1)
:xy:yx
julia> repr(a)
"ModuleElt([:xy => 1, :yx => 1])"
julia> ModuleElt([:xy=>1//2,:yx=>1])
(1//2):xy+(1//1):yx
Setting the IO
property :showbasis
to a custom printing function
changes how the basis elements are printed.
julia> show(IOContext(stdout,:showbasis=>(io,s)>string("<",s,">")),a)
3<xy>+2<yx>
We illustrate basic operations on ModuleElt
s:
julia> aa
0
julia> a*99
99:xy99:yx
julia> a//2
(1//2):xy+(1//2):yx
julia> a/2
0.5:xy0.5:yx
julia> a+ModuleElt(:yx=>1)
:xy
julia> a[:xy] # indexing by a basis element finds the coefficient
1
julia> a[:xx] # the coefficient of an absent basis element is zero.
0
julia> haskey(a,:xx)
false
julia> first(a)
:xy => 1
julia> collect(a)
2element Vector{Pair{Symbol, Int64}}:
:xy => 1
:yx => 1
julia> collect(keys(a))
2element Vector{Symbol}:
:xy
:yx
julia> collect(values(a))
2element Vector{Int64}:
1
1
julia> length(a)
2
julia> eltype(a)
Pair{Symbol, Int64}
In both implementations the constructor normalizes the constructed element,
removing zero coefficients and merging duplicate basis elements, adding the
corresponding coefficients (and sorting the basis in the default
implementation). If you know this normalisation is unnecessary, to get
maximum speed you can disable this by giving the keyword check=false
to
the constructor.
julia> a=ModuleElt(:yy=>1, :yx=>2, :xy=>3, :yy=>1;check=false)
:yy+2:yx+3:xy:yy
julia> a=ModuleElt(:yy=>1, :yx=>2, :xy=>3, :yy=>1)
3:xy+2:yx
Adding or subtracting ModuleElt
s does promotion on the type of the keys
and the coefficients if needed:
julia> a+ModuleElt([:z=>1.0])
3.0:xy+2.0:yx+1.0:z