A Course in Constructive Algebra pp 35-77 | Cite as

# Basic Algebra

## Abstract

*a, b*) =

*ab*, and a distinguished element of G, usually denoted by 1, such that for all

*a, b, c*in G

- (i)
(

*ab*)*c*=*a*(*bc*), (associative law) - (ii)
1

*a*=*a*1 =*a*. (identity)

The function ϕ is called **multiplication** and the element 1 the **identity**. The associative law allows us to ignore parentheses in products *a*_{1}*a*_{2}⋯*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 a^{0} =1; in an additive monoid we write the _{n}-fold sum *a* + *a* + ⋯ + *a* as *na*, and set 0*a* = 0.

### Keywords

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