Poincare Project: Identities and Monoids

A set with an associative binary operation is called a semigroup. We'll learn what it takes to be a full group soon.

Consider a semigroup (S, #). If there is an element e in S such that:

e # x = x # e = x for all x in S

then e is referred to as an identity and the semigroup is called a monoid.

For example, the integers under addition is a monoid with identity 0. The integers under multiplication is a monoid with identity 1.

Note that our definition requires both e # x and x # e to be x even though we don't require x # y = y # x in general. It is possible to have so-called left-identities and right-identities for which only e # x = x or x # e = x respectively is required for all x. The unqualified term identity is taken to mean it is both a left-identity and right-identity and the definition of monoid requires this.

Note also that, because of our definition, the identity must be unique. The proof is straightforward. Imagine two identities e and f. Then e # f = f # e = e but also e # f = f # e = f. So e = f.

UPDATE: next post

The original post was in the category: poincare_project but I'm still in the process of migrating categories over.