HomomorphismBetweenAlgebraicSystems 2006-05-28 22:37:25 2007-08-31 13:28:03 Definition algebraic system compatible function homomorphism monomorphism epimorphism endomorphism isomorphism automorphism homomorphic image \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % define commands here Let $(A,O),(B,O)$ be two algebraic systems with operator set $O$. Given operators $\omega_A$ on $A$ and $\omega_B$ on $B$, with $\omega\in O$ and $n=$ arity of $\omega$, a function $f:A\to B$ is said to be \emph{compatible} with $\omega$ if $$f(\omega_A(a_1,\ldots,a_n))=\omega_B(f(a_1),\ldots,f(a_n)).$$ Dropping the subscript, we now simply identify $\omega\in O$ as an operator for both algebras $A$ and $B$. If a function $f:A\to B$ is compatible with every operator $\omega\in O$, then we say that $f$ is a \emph{homomorphism} from $A$ to $B$. If $O$ contains a constant operator $\omega$ such that $a\in A$ and $b\in B$ are two constants assigned by $\omega$, then any homomorphism $f$ from $A$ to $B$ maps $a$ to $b$. \textbf{Examples}. \begin{enumerate} \item When $O$ is the empty set, any function from $A$ to $B$ is a homomorphism. \item When $O$ is a singleton consisting of a constant operator, a homomorphism is then a function $f$ from one pointed set $(A,p)$ to another $(B,q)$, such that $f(p)=q$. \item A homomorphism defined in any one of the well known algebraic systems, such as groups, modules, rings, and \PMlinkname{lattices}{Lattice} is consistent with the more general definition given here. The essential thing to remember is that a homomorphism preserves constants, so that between two rings with 1, both the additive identity 0 and the multiplicative identity 1 are preserved by this homomorphism. Similarly, a homomorphism between two \PMlinkname{bounded lattices}{BoundedLattice} is called a $\lbrace 0,1\rbrace$-\PMlinkname{lattice homomorphism}{LatticeHomomorphism} because it preserves both 0 and 1, the bottom and top elements of the lattices. \end{enumerate} \textbf{Remarks}. \begin{itemize} \item Like the familiar algebras, once a homomorphism is defined, special types of homomorphisms can now be named: \begin{itemize} \item a homomorphism that is one-to-one is a \emph{monomorphism}; \item an onto homomorphism is an \emph{epimorphism}; \item an \emph{isomorphism} is both a monomorphism and an epimorphism; \item a homomorphism such that its codomain is its domain is called an \emph{endomorphism}; \item finally, an \emph{automorphism} is an endomorphism that is also an isomorphism. \end{itemize} \item All trivial algebraic systems (of the same type) are isomorphic. \item If $f:A\to B$ is a homomorphism, then the image $f(A)$ is a subalgebra of $B$. If $\omega_B$ is an $n$-ary operator on $B$, and $c_1,\ldots,c_n\in f(A)$, then $\omega_B(c_1,\ldots,c_n)=\omega_B(f(a_1),\ldots,f(a_n))=f(\omega_A(a_1,\ldots, a_n))\in f(A)$. $f(A)$ is sometimes called the \emph{homomorphic image} of $f$ in $B$ to emphasize the fact that $f$ is a homomorphism. \end{itemize}