(ab)c = a(bc), (associative law)
1a = a1 = a. (identity)
The function ϕ is called multiplication and the element 1 the identity. The associative law allows us to ignore parentheses in products a1a2⋯a n . The monoid is said to be abelian, or commutative, if ab = ba for all elements a and b. In an abelian monoid the function ϕ is often called addition and written as ϕ(a, b) = a + b; the identity element is then denoted by 0. In this case we speak of an additive monoid, as opposed to a multiplicative monoid. In a multiplicative monoid we write the n-fold product aa⋯a as a n for each positive integer n, and set a0 =1; in an additive monoid we write the n-fold sum a + a + ⋯ + a as na, and set 0a = 0.
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