If is a set, a *binary operation* on is a function , where we generally write for . We say that this binary operation is *associative* if for every we have that . A pair where is a set and an associative binary operation on is then called a *semigroup*. For example, is a semigroup.

If is a semigroup and there exists an such that for every we have that , then we say that is an *identity* (of ) and we call a *monoid*. The natural numbers together with zero, which we’ll denote , is then a monoid under addition.

Generally we’ll let juxtaposition denote the binary operation, writing for . In cases where is the “natural” symbol to use for the binary operation, however, (e.g., the natural numbers with addition), then we’ll still use .

When and (where still denotes the identity), then we say that is the inverse of (and vice versa). We will normally write to mean the inverse of , though when is the operation we’ll write and write to mean . A monoid where every element has an inverse is called a *group*. The integers under addition form the group . To check this, simply note that addition of integers is associative, addition of two integers yields another integer, zero is the identity element, and for any we have that is its inverse.

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