direct product of algebras

In this entry, let $O$ be a fixed operator set. All algebraic systems have the same type (they are all $O$-algebras).

Let $\{A_i\mid i\in I\}$ 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

$${\omega }_A(a_1,\mathrm{\dots },a_n)(i)={\omega }_{A_i}(a_1(i),\mathrm{\dots },a_n(i))\mathit{\hspace{1em}}\text{ for all }i\in I.$$

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 direct product of $A_i$. Each $A_i$ is called a 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 projection given by ${\pi }_i(a)=a(i)$. It is a homomorphism because ${\pi }_i({\omega }_A(a_1,\mathrm{\dots },a_n))={\omega }_A(a_1,\mathrm{\dots },a_n)(i)={\omega }_{A_i}(a_1(i),\mathrm{\dots },a_n(i))={\omega }_{A_i}({\pi }_i(a_1),\mathrm{\dots },{\pi }_i(a_n))$.

Remark. The direct product of a single algebraic system is the algebraic system itself. An empty direct product is defined to be a trivial algebraic system (one-element algebra).

Title	direct product of algebras
Canonical name	DirectProductOfAlgebras
Date of creation	2013-03-22 16:44:35
Last modified on	2013-03-22 16:44:35
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	9
Author	CWoo (3771)
Entry type	Definition
Classification	msc 08A05
Classification	msc 08A62
Defines	direct product
Defines	direct factor
Defines	direct power
Defines	projection
Defines	empty direct product