31. This is true for any commutative monoid . 32. This so-called period-doubling monoid is a subset of the modular group. 33. The monoid is then presented as the quotient of the free monoid by these relations. 34. The monoid is then presented as the quotient of the free monoid by these relations. 35. To form the quotient monoid , these relations are extended to monoid congruences as follows. 36. To form the quotient monoid, these relations are extended to monoid congruences as follows. 37. An " M "-act is closely related to a transformation monoid . 38. The first four properties listed above for multiplication say that under multiplication is a commutative monoid . 39. This means that the cancellative elements of any commutative monoid can be extended to a group. 40. For example, a quotient of a category with one object is just a quotient monoid .
