Multiplication of Sets

I may think of two contexts in which the notions of product or multiplication of sets is routinely used. One is the set intersection and another is known as the direct product. Both have counterparts (set union and direct sum, respectively) sheer existence of which makes the terminology quite arbitrary. For, as I have already mentioned, as far as the abstract definition is concerned, the only difference between operations of addition and multiplication is notational.

Intersection and Union of sets

A space on which two operations are defined in a way that reminds us of the intersection and union of sets is known as a lattice. Various notations are used to denote the two operation. To draw on analogy with the algebra of sets, I'll use the set-theoretic ∪ and . The lattice axioms are indeed quite symmetric. More accurately, for all a, b, c (elements of the lattice)

 a∪a = aa∩a = aidempotent law
 a∪b = b∪aa∩b = b∩acommutative law
 (a∪b)∪c = a∪(b∪c)(a∩b)∩c = a∩(b∩c)associative law
 (a∪b)∩c = (a∩c)∪(b∩c)(a∩b)∪c = (a∪c)∩(b∪c)distributive law
 (a∪b)∩a = a(a∩b)∪a = apartial order law

The partial order law is used to introduce the following asymmetrical notations:

  a ≤ b iff a∪b = b

This is equivalent to requesting that a∩b = a. Indeed, assume a ≤ b as defined, then

  a∩b = a∩(a∪b) = (a∩a)∪(a∩b) = a∪(a∩b) = a

The reverse is shown similarly. The order is complete if for any a and b either a ≤ b or b ≤ a. Often existence of the smallest element 0 and the largest element 1 is also stipulated. This are defined by

 a∪0 = aa∪1 = 1

or equivalently

 a∩0 = 0a∩1 = a

for all a.

Lattices have been introduced by the German mathematician J.W.R.Dedekind(1831-1916) along with his invention of ideals in rings. The word "lattice" was first circulated by the american G.D.Birkhoff (1884-1944) in 1930s. The definition is fantastically broad. In addition to set theory and ideals, numbers (integer and real) form a lattice if a∪b is defined as max(a,b) and the intersection of two numbers is set to be the minimum of the two.

Now, returning to the product of two sets. As is well known, the frequently used notation for the intersection of two sets A nd B is plain AB. Regardless of the notations, it's a semigroup operation.

Direct Sums and Products

We met direct sums when talking of Boolean Algebras. But the approach is more general. Given two sets A and B, their direct product A×B is the set of pairs (a, b) with a∈A and b∈B. Now, the definition is actually confusing because as often as not the same set of pairs is called the direct product of the two sets A and B. Some distinction is drawn when we consider direct sums and products of an infinite numbers of sets. For example, direct product of countably many sets R of real numbers is the set of all sequences {(a1, a2, a3,...)} while the direct sum of countably many copies of R consists only of such sequences in which only a finite number of terms differ from 0. This must not be confused with a product of finite number of factors. The definition never says which coordinates must be 0, only that there is a finite number of non-zero components.

What Can Be Multiplied?


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