Basic Algebra

• Ray Mines
• Fred Richman
• Wim Ruitenburg
Part of the Universitext book series (UTX)

Abstract

A monoid is a set G together with a function ϕ from G×G to G, usually written as ϕ(a, b) = ab, and a distinguished element of G, usually denoted by 1, such that for all a, b, c in G
1. (i)

(ab)c = a(bc), (associative law)

2. (ii)

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 a1a2a 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 aaa 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.

Nite

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© Springer Science+Business Media New York 1988

Authors and Affiliations

• Ray Mines
• 1
• Fred Richman
• 1
• Wim Ruitenburg
• 2
1. 1.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA
2. 2.Department of Mathematics, Statistics, and Computer ScienceMarquette UniversityMilwaukeeUSA