DirectProductOfAlgebras 2007-02-23 19:23:51 2007-08-31 13:25:04 Definition algebraic system direct product direct factor direct power projection empty direct product \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} \usepackage{pst-plot} \usepackage{psfrag} % define commands here \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} In this entry, let $O$ be a fixed operator set. All algebraic systems have the same type (they are all $O$-algebras). Let $\lbrace A_i\mid i\in I\rbrace$ be a set of algebraic systems of the same type ($O$) indexed by $I$. Let us form the Cartesian product of the underlying sets and call it $A$: $$A:=\prod_{i\in I} A_i.$$ Recall that element $a$ of $A$ is a function from $I$ to $\bigcup A_i$ such that for each $i\in I$, $a(i)\in A_i$. For each $\omega\in O$ with arity $n$, let $\omega_{A_i}$ be the corresponding $n$-ary operator on $A_i$. Define $\omega_A: A^n\to A$ by $$\omega_A(a_1,\ldots,a_n)(i)=\omega_{A_i}(a_1(i),\ldots,a_n(i))\quad\mbox{ for all }i\in I.$$ One readily checks that $\omega_A$ is a well-defined $n$-ary operator on $A$. $A$ equipped with all $\omega_A$ on $A$ is an $O$-algebra, and is called the \emph{direct product} of $A_i$. Each $A_i$ is called a \emph{direct factor} of $A$. If each $A_i=B$, where $B$ is an $O$-algebra, then we call $A$ the direct power of $B$ and we write $A$ as $B^I$ (keep in mind the isomorphic identifications). If $A$ is the direct product of $A_i$, then for each $i\in I$ we can associate a homomorphism $\pi_i:A\to A_i$ called a \emph{projection} given by $\pi_i(a)=a(i)$. It is a homomorphism because $\pi_i(\omega_A(a_1,\ldots, a_n))=\omega_A(a_1,\ldots, a_n)(i)=\omega_{A_i}(a_1(i),\ldots,a_n(i))=\omega_{A_i}(\pi_i(a_1),\ldots, \pi_i(a_n))$. \textbf{Remark}. The direct product of a single algebraic system is the algebraic system itself. An \emph{empty direct product} is defined to be a trivial algebraic system (one-element algebra).