Counting all monoids with three elements

monoids_of_size_three.rb

class Table
  # 1 is an identity, so we know it's composition with other elements.
  ELTS = [1, :a, :b].freeze
  BASIC = {'11' => 1, '1a' => :a, '1b' => :b, 'a1' => :a, 'b1' => :b}.freeze

  # Iterate through all possible composition tables.
  def self.each
    return to_enum unless block_given?

    ELTS.product(ELTS, ELTS, ELTS) { |list| yield new(*list) }
  end

  # Composition table:
  # * | 1 a b
  # --+ -----
  # 1 | a a b
  # a | a ? ?
  # b | b ? ?

  # Define 4 compositions: aa, ab, ba, bb
  def initialize(aa, ab, ba, bb)
    @table = BASIC.merge('aa' => aa, 'ab' => ab, 'ba' => ba, 'bb' => bb)
  end

  # What is the composition of x and y?
  def [](x, y)
    @table.fetch("#{x}#{y}")
  end

  # Does composition table stay the same when :a and :b are swapped?
  def symmetric?
    [[[:a, :a], [:b, :b]], [[:a, :b], [:b, :a]]].all? do |one, two|
      one, two = self[*one], self[*two]
      (one == 1 && two == 1) || (one != 1 && two != 1 && one != two)
    end
  end

  # It is a monoid when composition is associative.
  def associative?
    ELTS.product(ELTS, ELTS).all? { |a, b, c| self[self[a, b], c] == self[a, self[b, c]] }
  end
end

count = Table.each.with_object(Hash.new(0)) do |t, cnt|
  next unless t.associative?
  symmetric = t.symmetric? ? :s : :ns
  cnt[symmetric] += 1
end

p count
# => {:ns=>8, :s=>3}
# 
# It means there are 4 different non-symmetric monoids and 3 symmetric, total: 7