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