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Exploring the Math:
Magic Squares: 'Multiplication' in a new context.
How can we call the result a 'product'?
Suzanne's Magic Squares ||
Multiplying Magic Squares: Contents
The first magic square, which will be called A, is 3x3, and the second one, which will be called B, is 4x4. The 'product' of A and B, denoted by A*B, will be a 12x12 magic square.
How can this be called a product?
The words product and multiplication are intended to convey the notion that two magic squares are being combined to get another magic square. Since we are not dealing with numbers, however, we cannot claim that the word 'multiplication' already means something here; the use of the word in this new context is an invention.
Having used the word, it is reasonable to ask whether it has anything else in common with the usual notion of multiplication for numbers besides the fact that things get combined.
What must be true to have the operation qualify as multiplication?
The operation * , which we defined for magic squares, satisfies the associative law.
In other words, if A,B,C are magic squares, then A*(B*C)=(A*B)*C.
There is an identity element for this multiplication, namely the 1x1 magic square whose only entry is 1:
One way this is different from multiplication of numbers is that the commutative law does NOT hold.
For example, with the 3x3 square A and the 4x4 square B, the two 12x12 squares A*B and B*A are not equal.
However, that doesn't mean that the word multiplication is not appropriate. For example, we also speak of multiplication of matrices, and that notion of multiplication is not commutative.
Another way this is different from multiplication is that there is no corresponding notion of addition, but that is not a serious problem.
For example, the set of all powers of 2 is closed under ordinary multiplication, even though it is not closed under addition. It still makes sense to call the operation multiplication even though the set has no addition. In the case of the powers of 2, we can enlarge the set in various ways (by putting in other numbers) to get something where the multiplication still makes sense and where we also have addition, so addition is still there in the background and we don't need to have it explicitly before us. Similarly, it is possible to enlarge the set of magic squares to a larger set on which the multiplication makes sense and in which one also has addition.
Once we accept that this is a somewhat reasonable notion of multiplication, we can study its other properties. In multiplication of positive numbers, we have the notion of prime versus composite, with the 'unit' 1
being excluded from these two categories.
In the same way, we can define a magic square to be composite if it is the product of two strictly smaller magic squares.
Thus, the 12x12 magic square A*B is composite. On the other hand, the 4x4 magic square B is not composite, because the only way it could be the product of two strictly smaller magic squares would be if both were 2x2, and it is easy to see that there are no 2x2 magic squares.
A magic square will be called prime if it is not composite and is not the 'unit', i.e. the 1x1 magic square whose only entry is 1.
One of the remarkable facts that justifies this use of the words prime and composite is that every magic square can be written in one and only one way as a product of prime magic squares. (When we say only one way, this includes the ordering of the factors, since the multiplication is not commutative.)
So not only do we have some general properties such as associativity and identity element, we even have a property analogous to unique factorization of numbers, and this above all justifies the use of the word 'multiplication'. Furthermore, the proof that this property holds is strikingly similar to the proof of unique factorization of numbers into primes in ordinary arithmetic.
Definition of Terms:
A set M with an associative operation * and an identity element 1 for that operation is called a monoid.
Numbers under multiplication form a monoid, and under our extended concept of multiplication, magic squares form a monoid.
Let (M,*,1) be a monoid and let P be a subset of M. We say that M is freely generated by P if the unique factorization theorem holds with respect to P, i.e. every element of M can be written in one and only one way as a product (in the sense of *) of elements of P, including order.
The set of all magic squares is freely generated by the set of all prime magic squares.
We say a monoid (M,*,1) is free if there is a subset P of M such that M is freely generated by P, so this fundamental theorem of arithmetic for magic squares also says that
The set of all magic squares is a free monoid.
The analogous result holds for magic cubes, magic tessaracts, and higher dimensional analogues, and the same applies to the multiplication itself.
Unique factorization of numbers:
Since the unique factorization of magic squares has this interpretation, it is interesting and useful to see what the corresponding interpretation is for unique factorization of numbers.
It is basically the same, but instead of using the notion of monoid, we use the notion of commutative monoid. Thus, a commutative monoid is a monoid (M,*,1) in which the operation * is commutative. If P is a subset of M, we say that the commutative monoid M is freely generated by P if every element of M can be written in one and only one way as a product (in the sense of *) of elements of P. Here, when we say 'one and only one way', we don't care about the ordering of the factors since we are assuming commutativity. If (M,*,1) is a commutative monoid and if M has a subset P such that M is freely generated by P as a commutative monoid, we say that M is a free commutative monoid.
So the usual unique factorization theorem for positive integers can be expressed by saying that the set of all positive integers, under multiplication, is a free commutative monoid.