# 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 |