Some Algebraic Definitions and Constructions

Definition 1. A monoid is a set M with an element e and an associative multiplication M×M −→ M for which e is a two-sided identity element: em = m = me for all m ∈ M . A group is a monoid in which each element m has an inverse element m, so that mm = e = mm. A homomorphism f : M −→ N of monoids is a function f such that f(mn) = f(m)f(n) and f(eM ) = eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n ∈ M , we often write the product as +, the identity element as 0, and the inverse of m as −m. As a convention, it is convenient to say that a commutative monoid is “Abelian” when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1.