Monoid

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 naturally 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 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.

Algebraic structures
Group-like structures Semigroup and Monoid
Quasigroup and Loop
Abelian group
Ring-like structures Semiring
Near-ring
Ring
Commutative ring
Integral domain
Field
Lattice-like structures Semilattice
Lattice
Map of lattices
Module-like structures Group with operators
Module
Vector space
Algebra-like structures Algebra
Associative algebra
Non-associative algebra
Graded algebra
Bialgebra

Read more about Monoid:  Definition, Examples, Properties, Acts and Operator Monoids, Monoid Homomorphisms, Equational Presentation, Relation To Category Theory, Monoids in Computer Science