| 41. | The identity element of the monoid is the empty list, " nil ".
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| 42. | The notion of a semi-Thue system essentially coincides with the presentation of a monoid.
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| 43. | As such, the theorem is a deep and important result in semigroup / monoid theory.
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| 44. | The " monoid of words " over an alphabet is the free monoid over.
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| 45. | The " monoid of words " over an alphabet is the free monoid over.
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| 46. | Repeated function composition forms a monoid.
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| 47. | Furthermore, since a monoid is a semigroup, one can consider semigroup actions of monoids.
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| 48. | Every commutative monoid can be endowed with its " algebraic " preordering d ".
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| 49. | Any cancellative commutative monoid " M " can be embedded into an abelian group.
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| 50. | Consequently, the operation ?" acting on the nonnegative real numbers forms a commutative monoid.
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