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