Suppose \({ ( A , f ) }\) is a semigroup and \({ x ∈ A }\). Then there is a function from the positive natural numbers to \({ A }\) obtained by taking the product of \({ n }\) copies of \({ x }\), with the convention discussed in an earlier section of using the word “product” to denote the result of repeated applications of the binary operation \({ f }\). We don’t need to specify the order because of the generalised associativity property proved in that section. Addition is an associative operation on the positive natural numbers, making them into a semigroup, and this function is a semigroup homomorphism. The proof of this depends on the generalised associativity property. This function is not generally written with functional notation but with exponential notation. The value of the function corresponding to a particular \({ x }\) at a positive natural number \({ n }\) is written \({ x ^ n }\). The property that the function is a semigroup homomorphism is, in this notation, \[ f ( x ^ m , x ^ n ) = x ^ { m + n } . \] This notation is confusing in some examples. If, for example, the semigroup in question is the natural numbers with the operation of addition then \({ x ^ n }\) is not in fact the number normally denoted by that expression but rather is \({ n · x }\). If the operation is the maximum then \({ x ^ n }\) is just \({ x }\). With sets and the operation of union or intersection \({ A ^ n }\) would just be \({ A }\), rather than the set of lists of length \({ n }\) of items in \({ A }\). Unfortunately the exponential notation is too well established to abolish entirely, but I’d suggest not using it where it conflicts with an established notation.
In a commutative semigroup it’s possible to prove, by induction on \({ n }\), that \({ f ( x ^ n , y ^ n ) = f ( x , y ) ^ n }\). This is not generally true in a noncommutative semigroup though.
If our semigroup is a monoid then we can extend the function described above from the positive natural numbers to all natural numbers by defining \({ x ^ 0 }\) to be the identity. The resulting extension is a monoid homomorphism. If the monoid is a group then we can extend it still further, by defining \({ x ^ { - n } }\) to by \({ y ^ n }\) where \({ y }\) is the inverse of \({ x }\). This extended function is a group homomorphism. In this case we have the useful relation \[ f ( x , y ) ^ { - 1 } = f ( y ^ { - 1 } , x ^ { - 1 } ) . \] Note the reversal of the order of the arguments. This identity was in fact proved earlier, but in a different notation, in the course of proving that the product of invertible elements is invertible.