monoid

A monoid is a semigroup $G$ which contains an identity element; that is, there exists an element $e\in G$ such that $e\cdot a=a\cdot e=a$ for all $a\in G$.

If $e$ and $f$ are identity elements of a monoid $G$, then $e=e\cdot f=f\cdot e=f$, so we may speak of “the” identity element of $G$.

A monoid homomorphism from monoids $G$ to $H$ is a semigroup homomorphism $f:G\to H$ such that $f(e_G)=e_H$, where $e_G,e_H$ are identity elements of $G$ and $H$ respectively.

Title	monoid
Canonical name	Monoid
Date of creation	2013-03-22 11:50:15
Last modified on	2013-03-22 11:50:15
Owner	djao (24)
Last modified by	djao (24)
Numerical id	9
Author	djao (24)
Entry type	Definition
Classification	msc 20M99
Classification	msc 34-01
Synonym	homomorphism
Related topic	Semigroup
Defines	monoid homomorphism