Iowa Ruby Brigade kata showing Roman numeral computation is a monoid.

Roman.rb

# Roman Numeral Evaluation is a Monoid
# Chad Brewbaker | Software Engineer | Telligen.org
# crb002@gmail.com
# Initial release January 15, 2013

class RomanMonoid

  attr_accessor :prefix, :prefix_size, :suffix, :suffix_size, :suffix_credit, :sum, :homo

	def initialize(val)
		#Monoid identity values
		@prefix ='I'
		@prefix_size = 0
		@suffix = 'I'
		@suffix_size =0
		@suffix_credit =1
		@sum = 0
		@homo = true

		if (not (val.nil?))
 			values = [1, 5, 10, 50, 100, 500, 1000]
			indexh = {'I' => 0, 'V'=> 1, 'X'=>2, 'L'=>3, 'C'=>4, 'D'=>5, 'M'=>6}
			@prefix=val
			@suffix=val
			@suffix_credit = 1
			@prefix_size=1
			@suffix_size=1
			@homo = true #When a Roman monoid stops being homogeneous the suffix credit value is known 
			@sum = values[indexh[val]]
		end

	end

	def add_right(elt)
		values = [1, 5, 10, 50, 100, 500, 1000]
		indexh = {'I' => 0, 'V'=> 1, 'X'=>2, 'L'=>3, 'C'=>4, 'D'=>5, 'M'=>6}
		same_bonus=0
				# Adding a hetero
				if elt.homo == false
					if(@prefix == elt.suffix) 
						if elt.suffix_credit < 0
							@sum = @sum - (@prefix_size * values[indexh[@prefix]]*2)
						end
					elsif (indexh[@prefix] < indexh[elt.suffix])
						@sum = @sum - (@prefix_size * values[indexh[@prefix]]*2)
						if @homo
							@suffix_credit = -1
						end
					end
					@homo =false
				# Adding a homo
				elsif elt.homo == true
					if(@prefix == elt.suffix)
						same_bonus = @prefix_size
					elsif (indexh[@prefix] < indexh[elt.suffix])
						@sum = @sum - (@prefix_size * values[indexh[@prefix]]*2)
						if(@homo)
							@suffix_credit = -1
						end
						@homo =false
					elsif (indexh[@prefix] > indexh[elt.suffix])
						@homo =false
					end
				end
				@prefix_size = elt.prefix_size + same_bonus
				@sum = @sum + elt.sum
				@prefix = elt.prefix		
	end
end 


def convert_monoid(input)
	tmparr = input.chars.to_a
	arr = Array.new(input.size) { |i| RomanMonoid.new(tmparr[i]) }
	last = RomanMonoid.new(nil)
	while (arr.size >0)
			v = arr.pop
			v.add_right(last)
			last = v
	end
	return last.sum
end

def convert_credit(input)
		# Rule 1) The last element is always a credit.
		# Rule 2) A repeat shares the debit/credit value of it's right neighbor.
		# Rule 3) A greater left elt makes this elt a credit.
		# Rule 4) A lesser left elt make this elt a debit.

		values = [1, 5, 10, 50, 100, 500, 1000]
		indexh = {'I' => 0, 'V'=> 1, 'X'=>2, 'L'=>3, 'C'=>4, 'D'=>5, 'M'=>6}
		
		arr = input.chars.to_a

		last_val = 'I'
		last_mult = 1
		sum = 0
		
		while (arr.size >0)
			v = arr.pop
			
			mult = last_mult
			if(indexh[v] < indexh[last_val])
				mult = -1
			end
			if(indexh[v] > indexh[last_val])
				mult = 1
			end

			sum = sum + (mult* values[indexh[v]])
			last_val = v
			last_mult = mult
		end

		return sum
end


puts "Should be 1999"
puts convert_credit("MDCCCCLXXXXVIIII")
puts convert_credit("MCMXCIX")
puts convert_credit("MIM")
puts "-----------------------------------"
puts convert_monoid("MDCCCCLXXXXVIIII")
puts convert_monoid("MCMXCIX")
puts convert_monoid("MIM")

I'm lost, how can roman numbers be a monoid if there's no identity element? There's no 0.

Look at initialize(). That struct of seven variables is the monoid identity. For complex monoids you have to do a lot of book-keeping variables.