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

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