monoid Monoid 2001-10-19 11:36:28 2008-05-13 01:46:18 Definition monoid homomorphism \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} 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 \emph{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.