##  [Monoid](/monoid) 

  ##  [Monoid](https://puremath.quantumdictionary.io/monoid-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A set with an associative binary operation that has an identity element, but whose elements need not have inverses.

 

 

 

 

 





 

 



 ##  [Monoid](https://algebra.quantumdictionary.io/monoid-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A semigroup equipped with an identity element; a monoid is a set with an associative binary operation and a neutral element 1 such that 1·a = a·1 = a for all a.