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[parent] direct product of algebras (Definition)

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$:

$\displaystyle A:=\prod_{i\in I} A_i.$
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

$\displaystyle \omega_A(a_1,\ldots,a_n)(i)=\omega_{A_i}(a_1(i),\ldots,a_n(i))$    for all $\displaystyle i\in I.$
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,\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))$.

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).



"direct product of algebras" is owned by CWoo.
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Also defines:  direct product, direct factor, direct power, projection, empty direct product

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subdirect product of algebraic systems (Definition) by CWoo
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Cross-references: algebra, trivial algebraic system, homomorphism, associate, isomorphic, well-defined, operator, arity, function, Cartesian product, indexed by, type, algebraic systems, operator set, fixed
There are 45 references to this entry.

This is version 6 of direct product of algebras, born on 2007-02-23, modified 2007-08-31.
Object id is 8967, canonical name is DirectProductOfAlgebras.
Accessed 2584 times total.

Classification:
AMS MSC: 08A62 (General algebraic systems :: Algebraic structures :: Finitary algebras)
 08A05 (General algebraic systems :: Algebraic structures :: Structure theory)
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