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∪a = a||a∩a = a||idempotent law|
|a∪b = b∪a||a∩b = b∩a||commutative 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 = a||partial 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 = 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∪0 = a||a∪1 = 1|
|a∩0 = 0||a∩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