[index] [finite-set] Algebra::Map

Algebra::Map

Map Class

The class which represents maps.

File Name:

SuperClass:

Included Module:

Class Methods:

::[[x0 => y0, [x1 => y1, [x2 => y2, ...]]]]

Returns the map which has values y0 at x0, y1 at x1, y2 at x2...

Exampel: 

require "finite-map"
include Algebra
p Map[0 => 10, 1 => 11, 2 => 12]         #=> {0 => 10, 1 => 11, 2 => 12}
p Map[{0 => 10, 1 => 11, 2 => 12}]       #=> {0 => 10, 1 => 11, 2 => 12}
p Map.new(0 => 10, 1 => 11, 2 => 12)     #=> {0 => 10, 1 => 11, 2 => 12}
p Map.new({0 => 10, 1 => 11, 2 => 12})   #=> {0 => 10, 1 => 11, 2 => 12}
p Map.new_a([0, 10, 1, 11, 2, 12])       #=> {0 => 10, 1 => 11, 2 => 12}
::new([x0 => y0, [x1 => y1, [x2 => y2, ...]]])
Returns the map which has values y0 at x0, y1 at x1, y2 at x2... This is the same as ::[].
::new_a(a)
Returns the map defined so that the even-th elements are in domain and the odd-th elements are their image.
::phi([t])

Return the empty map (having empty set as domain). If t is given, target= is set by t.

Exampel: 

p Map.phi #=> {}
p Map.phi(Set[0, 1]) #=> {}
p Map.phi(Set[0, 1]).target #=> {0, 1}
::empty_set
Alias of ::phi.
::singleton(x, y)
Returns the map defined over singleton {x} and having value y on it.

Methods:

target=(s)
Sets the codomain s. This is used for surjective? and so on.
codomain=(s)
Alias of target=.
target
Returns the codomain of self
codomain
Alias of target.
source

Returns the domain of self.

Example: 

require "finite-map"
include Algebra
m = Map[0 => 10, 1 => 11, 2 => 12]
p m.source #=> {0, 1, 2}
p m.target #=> nil
m.target = Set[10, 11, 12, 13]
p m.target #=> {10, 11, 12, 13}
domain
Alias of source.
phi([t])
Returns the empty map. If t given, target= t is done.
empty_set
null
Alias of phi.
call(x)
Return the value of x.
act
[]
Alias of call.
each

Iterates for each [point, image] of the map.

Example: 

require "finite-map"
include Algebra
Map[0 => 10, 1 => 11, 2 => 12].each do |x, y|
  p [x, y] #=> [1, 11], [0, 10], [2, 12]
end
compose(other)

Returns the composition map of self and other.

Example: 

require "finite-map"
include Algebra
f = Map.new(0 => 10, 1 => 11, 2 => 12)
g = Map.new(10 => 20, 11 => 21, 12 => 22)
p g * f #=> {0 => 20, 1 => 21, 2 => 22}
*
Alias of compose.
dup
Duplicates self. target is also duplicated if it is set.
append!(x, y)

Let the value of x be y.

Example: 

require "finite-map"
include Algebra
m = Map[0 => 10, 1 => 11]
m.append!(2, 12)
p m #=> {0 => 10, 1 => 11, 2 => 12}
[x] = y
Alias of append!.
append(x, y)
dup and append!(x, y).
include?(x)
Returns true if x is in the domain.
contains?(x)
Alias of include?.
member?(a)

Returns true if there is [x, y] s.t. the value of x is y and [x, y] == a.

Example: 

require "finite-map"
include Algebra
m = Map[0 => 10, 1 => 11]
p m.include?(1)  #=> true
p m.member?([1, 11]) #=> true
has?
Alias of member?
image([s])
Returns the image of self. If s is indicated, returns the image of s by self.
inv_image(s)
Return the inverse image of s by self.
map_s

Returns the set given by evaluation of the block, iterating over each pair of [point, image].

Example: 

require "finite-map"
include Algebra
p Map.new(0 => 10, 1 => 11, 2 => 12).map_s{|x, y| y - 2*x}
#=> Set[10, 9, 8]
map_m

Returns the map given by evaluation of the block, iterating over each pair of [point, image]. The value of the block must be the two-dimensional array [x, y].

Example: 

require "finite-map"
include Algebra
p Map.new(0 => 10, 1 => 11, 2 => 12).map_m{|x, y| [y, x]}
#=> {10 => 0, 11 => 1, 12 => 2}
inverse
Returns the inverse map of self.
inv_coset

Reterns the map corresponding each point to its set of inverse images.

Examepe: 

require "finite-map"
include Algebra
m = Map[0=>0, 1=>0, 2=>2, 3=>2, 4=>0]
p m.inv_coset #=> {0=>{0, 1, 4}, 2=>{2, 3}}
m.codomain = Set[0, 1, 2, 3]
p m.inv_coset #=> {0=>{0, 1, 4}, 1=>{}, 2=>{2, 3}, 3=>{}}
identity?
Returns true if self is the identity map.
surjective?
Returns true if self is surjective. target should be set before using this.
injective?
Returns true if self is injective.
bijective?
Returns true if self is bijective. target should be set before using this.