Lectures on Boolean Algebras pp 115-118 | Cite as

# Products of algebras

## Abstract

A familiar way of making one new structure out of two old ones is to form their Cartesian product and, in case the structure involves some algebraic operations, to define the requisite operations coordinate-wise. Boolean algebras furnish an instance of this procedure. Since nothing is gained by restricting attention to two algebras at a time, we proceed at once to discuss arbitrary families. By the *product* of a family *A*_{ i } of Boolean algebras we shall understand their Cartesian product Π
_{ i } *A*_{ i }, construed as a Boolean algebra with respect to the coordinate-wise operations. This means that, for instance, 0 in Π
_{ i } *A*_{ i }
is defined by 0
_{ i }
= 0 for all *i*, and *p* ∨ *q* in Π
_{ i } *A*_{ i }
is defined by (*p* ∨ *q*)_{ i }
= *p*_{ i } V *q*_{ i } for all *i*. We shall indicate the product of finite or infinite sequences of Boolean algebras by such obvious and customary modifications of the symbolism as Π
_{i =1} ^{ n } *A*_{ i }. For sequences of length two (and sometimes even for longer ones) we use the multiplication cross, so that Π
_{i =1} ^{2} *A*_{ i }= *A*_{l} × *A*_{2}. It is an immediate consequence of the definition that the product of a family of *σ*-algebras is a *σ*-algebra, and, similarly, the product of a family of complete algebras is complete.

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