✖

Bodies

You don't need any special math skills or knowledge to understand the following explanation. You just need to understand the very basics of high school mathematics and have a desire to learn or relearn something new.

A bubble named algebraic solid

If the very name algebraic solid scares you, know that there really is nothing to fear. Such a solid is actually a very simple thing if you look at it from the right angle.

Analgebraic solid is any arbitrary set plus two binary operations satisfying certain conditions. That's all. Nothing more or less. As you can see, certainly nothing dangerous or scary. And once we know that, we can go a little closer without fear.

The many elements of the body

We'll label our arbitrary set T and call its elements the elements of the solid. Believe me, my friends, the set of elements of a solid can be completely arbitrary. It can be the set of all even numbers, the set of all the socks you don't have a mate for, or even your collection of stuffed elephants. Perhaps the only condition is that the set T must not be empty. Because that wouldn't be much fun.

Binary operations

If it's not immediately obvious to you what a binary operation actually is, you can think of it as a little grinder. You put two (hence the binary in the name) elements of a set into it, turn the crank slightly, and it drops out some other element from your set. A nice binary operation like that is division. Wondering how much is 10 / 5? Take the binary operation of division (mill), where your set will be the set of real numbers. Put first a ten and then a five in the mill, turn the crank and, world wonder, a two falls out.

We'll call our two operations addition and multiplication and label them $\oplus$ and $\otimes$. Don't be fooled by their names. These operations may not have much in common with the addition and multiplication operations we normally use. This is precisely because the elements of a solid can be any objects. And "our" addition and multiplication cannot be applied to stuffed elephants or socks.

Operations on a solid

The operations $\oplus$ and $\otimes$ must satisfy certain conditions. We cannot introduce them arbitrarily. In fact, the point of introducing these operations is to generalize the ordinary addition and multiplication we use on numbers. We want to be able to introduce these operations so that we can apply them to elements of a solid, which, as we already know, can be completely arbitrary objects, not just numbers. But we still want the $\oplus$ and $\otimes$ operations to retain some basic sense of addition and multiplication. We ensure this by introducing ten conditions that these operations must satisfy for our structure to be a solid. We will call these conditions the axioms of a solid. Here they are:

  1. Commutativity of addition - For any two elements a, b of the solid T , $a \oplus b = b \oplus a$ must hold.
  • Associativity of addition - For any three elements a, b, c of the solid T , $(a \oplus b) \oplus c = a \oplus (b \oplus c)$ must hold.
  • Existence of a zero element - There must exist an element from the solid T, denoted 0 (note, not to be confused with the number 0), which must have the property that for any element a from the solid T, $a \oplus 0 = a$ holds.
  • Existence of the opposite element - There must exist an element from the body of T, let's denote it by -a, which must have the property that for any element a from the body of T, $a \oplus -a = 0$.
  • Commutativity of multiplication - For any two elements a, b from the solid T , $a \otimes b = b \otimes a$ must hold.
  • Associativity of multiplication - For any three elements a, b, c of the solid T , $(a \otimes b) \otimes c = a \otimes (b \otimes c)$ must hold.
  • Existence of a unit element - There must exist an element from the solid T, denoted 1 (note, not to be confused with the number 1), which must have the property that for any element a from the solid T, $a \otimes 1 = a$ holds.
  • Existence of an inverse element - There must exist an element from the body T, let us denote it by a-1 , which must have the property that for any element a different from 0 from the body T, $a \otimes a^{-1} = 1$ holds.
  • Distributivity - For any three elements a, b, c from the body T , $a \otimes ( b \oplus c ) = ( a \otimes b) \oplus (a \otimes c)$ must hold.
  • Nontriviality - The zero and unit elements must not be one and the same. Thus $0 \neq 1$

For those who would like to practice mathematical formalism, I have written here the axioms of the solid as they are written in mathematical parlance.

  1. $ \forall a,b \in T \quad a \oplus b = b \oplus a$
  • $ \forall a,b,c \in T \quad (a \oplus b) \oplus c = a \oplus (b \oplus c)$
  • $ \exists 0 \in T : \forall a \in T \quad a \oplus 0 = a$
  • $ \forall a \in T : \exists -a \in T \quad a \oplus -a = 0$
  • $ \forall a,b \in T \quad a \otimes b = b \otimes a$
  • $ \forall a,b,c \in T \quad (a \otimes b) \otimes c = a \otimes (b \otimes c)$
  • $ \exists 1 \in T : \forall a \in T \quad a \otimes 1 = a$
  • $ \forall a \in T : \exists a^{-1} \in T, a \neq 0 \quad a \otimes a^{-1} = 1$
  • $ \forall a,b,c \in T \quad a \otimes ( b \oplus c ) = ( a \otimes b) \oplus (a \otimes c)$
  • $0 \neq 1$

Now, you may be thinking that there are an awful lot of axioms. But they are all easy to understand when we realize what they mean. The first four describe the usual properties of addition, and the next four describe the usual properties of multiplication, which are also very similar to the first four. The ninth axiom tells how addition and multiplication are related. Only the tenth is a rather technical one, which ensures that the solid can't have any strange properties.

Notice how nicely we avoided introducing subtraction and division. These operations are introduced by adding the inverse element and multiplying by the inverse element. Thus, for example, 4 − 2 is just a shortcut for 4 + (−2) and 4 / 2 is just a shortcut for 4 · 2−1.

Parade of solids

We now have formally introduced the notion of an algebraic solid. If we want to express intuitively what a solid is, we can say that it is a kind of structure that allows us to add, subtract, multiply and divide arbitrary objects on which we can define these operations, in a way similar to how we perform these operations in elementary arithmetic.

Our first solid

Now that we've thought up and described solids so nicely, let's construct some real solids. First, let's choose a set of elements for the solid. For now, we won't experiment too much with stuffed elephants or socks, and we'll choose natural numbers as the elements of the solid. To keep things really simple, we'll just take the numbers 0, 1, 2, 3, 4. So our set T will look like this:

$$T = \left\{0, 1, 2, 3, 4\right\}$$

Now all that's left is to somehow cleverly define the operations $\oplus$ and $\otimes$. Here we'll use the addition and multiplication operations we already know from arithmetic. But we will modify them a bit to make them work on our solid.

  • The operations $\oplus$The addition of two elements $a \oplus b$ is done by calculating the remainder after dividing five by the sum of a + b . The addition operations in our solid will look like this:$ 1 \oplus 1 = 2 \qquad 1 \oplus 3 = 4 \qquad 2 \oplus 3 = 0 \qquad 4 \oplus 2 = 1$ etc.
  • The operation $\otimes$The product of two elements $a \otimes b$ is done by calculating the remainder after dividing five by the product ab . The multiplications in our solid will therefore look like this: $ 1 \otimes 1 = 1 \qquad 2 \otimes 2 = 4 \qquad 2 \otimes 3 = 1 \qquad 4 \otimes 2 = 3 \qquad$ etc.

And that's it. We now have a full-fledged solid made. We would still need to check the validity of all the axioms to be sure that we have made a real solid and not some kind of gossip.

  • The commutativity and associativity of addition and multiplication you are surely able to verify yourself, as well as distributivity.
  • It also follows from the way we have defined addition and multiplication that there is a zero element (in our case, it just happens to be the number 0) and a unit element (in our case, 1) that are different from each other.
  • So now all we have to do is verify the existence of the opposite and inverse elements. Surely we can all see that to every number in the set {0, 1, 2, 3, 4} there is some number in the same set such that the result of their sum is five (i.e., 0 in our solid).
  • Similarly, this is true for multiplication and the result is 6 (1 in our solid), except for the number 4 which has an inverse element of 4 because 4 · 4 = 16 and after dividing by 5 we have a remainder of 1. Try to find the other inverse elements!

Examples of other solids

A while ago we made a solid that has five elements. It turns out that other solids can be made in the same way, where the number of elements is prime. Such solids are called solids of residue classes and are denoted by Zp, where p is the number of elements of the solid, and they are used a lot in mathematics or computer science, and of course in various writing examples.

Can solids with other than a prime number of elements be made? Yes they can, but not in the way I demonstrated a moment ago. Why? Imagine a solid made using the procedure described above that has only four elements. In such a solid, $2 \otimes 2 = 0$. Thus, the product of two non-zero numbers is zero. This cannot happen in a solid. It is not directly forbidden in the axioms of the solid, but it is quite easy to deduce from them. It turns out, however, that if polynomials are used instead of numbers as elements of the solid, one can produce solids whose number of elements is a power of the prime number. Thus a four-element solid exists because four is a power of two.

Of course, solids with an infinite number of elements are also possible. So, for example, the set of real numbers plus our usual operations of addition and multiplication also form a solid. Of course, this applies not only to real numbers, but also to natural numbers, integers, rational numbers, and complex numbers.

And the best part at the end. Of course, no one is prescribing that solids must be made up of numbers, polynomials, or any mathematical objects at all. As I said, you can make the solid of socks, stuffed elephants, etc. I guess the only problem would be how to define addition and multiplication on elephants. Here's the interesting thing, you can never make a solid that contains six socks. Because six is neither a prime number nor a power of a prime number. But you can easily make a solid that has seventeen socks, number each one (a white sock would be 0, a black one would be 1, a green one would be 2, etc.), and then add them up the same way we did in the solid of the residual classes.

And what is all this for?

If you've read this far, you might be wondering why on earth do we need such a crazy structure as an algebraic solid? Well, solids are very important, both from a theoretical point of view, since they describe and, more importantly, generalize what one might call counting, i.e. arithmetic, and from a practical point of view, since many mathematical or computer science problems are easier to solve if we think of them as counting in solids. Finite solids, for example, are of great importance for codes on CDs or DVDs. Otherwise, as far as linear algebra itself is concerned, solids are the cornerstone for other, more complex and thus far more useful and interesting structures, such as vector spaces. These form a kind of foundation for the whole of linear algebra.

Did you like this article? Did it help you in your studies? Or do you have reservations or ideas for improving it? Write to me at lishaak[zavinac]matfyz.cz.

The author of this article is Lishaak.