What is "monoid"

n. (context mathematics English) A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as categories with a single object. Thus, they capture the idea of function composition within a set. Monoids are also commonly used in computer science, both in its foundational aspects and in practical programming. The set of strings built from a given set of characters is a free monoid. The transition monoid and syntactic monoid are used in describing finite state machines, whereas trace monoids and history monoids provide a foundation for process calculi and concurrent computing. Some of the more important results in the study of monoids are the Krohn–Rhodes theorem and the star height problem. The history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups.

In category theory, a monoid (or monoid object) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms

• μ: M ⊗ M → M called multiplication,
• η: I → M called unit,

such that the pentagon diagram

and the unitor diagram

commute. In the above notations, I is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C.

Dually, a comonoid in a monoidal category C is a monoid in the dual category C.

Suppose that the monoidal category C has a symmetry γ. A monoid M in C is commutative when μ o γ = μ.

Monoid may refer to:

• Monoid, an algebraic structure
• Monoid (category theory), a mathematical structure used in category theory
• Monoid (data structure), a data structure found in functional programming languages, such as Haskell