Tensor #19. Algebraic Structures I텐서 #19. 대수 구조 I

These are my notes on XylyXylyX’s What is Tensor lectures. This post covers lecture 19 of the playlist, Algebraic Structures I. The previous post finished the derivatives, and this time I change direction for a moment and go through, from the beginning, the vocabulary of algebraic structures that keeps popping up in physics books. It is mostly definitions, with hardly any proofs.

What an algebraic structure is

When we defined a vector space, we gave a set $V$ an addition and brought in another set, the scalars $F$. Operations are written with Cartesian products. Addition is $V \times V \to V$, scalar multiplication is $F \times V \to V$, and an inner product is $V \times V \to F$. Specifying the sets, the operations on them, and the properties those operations obey fixes one algebraic structure. A vector space is a fairly complicated structure, so today we build up from the simplest ones, which have only one set.

Magma and semigroup

The simplest structure is the magma (also called a groupoid). It is a set $G$ with one operation $G \times G \to G$, and nothing more. The only condition is closure: any two elements go in and the result stays in $G$. It asks for no commutativity, no associativity, and no identity. A pure magma is, for example, binary trees. Joining two trees under a new root is closed, but it is neither commutative nor associative and has no identity.

Add associativity $(ab)c = a(bc)$ to a magma and you get a semigroup. Concatenating strings of letters is a good example. Whichever pair you join first, the string is the same, but $ab$ and $ba$ differ, and if the empty string is not allowed there is no identity. The names come from adding one condition at a time, so every structure on an upper step is also a structure on every step below it.

Monoid and units

Add an identity $e$ to a semigroup and you get a monoid.

Adjoin the empty string to the string semigroup above and it becomes a monoid, and so do the natural numbers $0, 1, 2, \dots$ with addition. The latter is commutative as well, so it is better than needed as an example.

Some structures have an identity that works on one side only. If only $e a = a$ holds it is a left identity, if only $a e = a$ holds it is a right identity, and if both hold it is a two-sided identity. For commutative structures the distinction is unnecessary. When “left” and “right” start to appear in a text, it is usually a sign that it deals with something non-commutative.

With an identity we can talk about inverses. For an element $x$, if some $y$ satisfies $x y = y x = e$, then $x$ is called a unit and we write $y = x^{-1}$. A monoid may have units, but not everything has to be one. In the natural numbers, even though 1 plus $-1$ is 0, $-1$ is not a natural number, so 1 is not a unit.

Q. How do we see that the inverse of a unit is unique?
A. Suppose $y$ and $z$ are both inverses of $x$. Then $z = z e = z (x y) = (z x) y = e y = y$. The middle equality is associativity. Even a simple proof like this needs associativity, which is why it is demanded first, while commutativity is demanded much later.

Group

If every element of a monoid is a unit, it is a group. In sum, the definition of a group is a closed operation, associativity, an identity, and an inverse for every element, with no demand for commutativity. A commutative group is called an abelian group, and its operation is often written as addition. When every element has an inverse we can use forms like $a b^{-1}$ that resemble division. The integers under addition are the easiest example, and the Lie groups of general relativity, the groups of continuous transformations, live on top of this. The basic structures defined with one operation end here. There are other routes to a group with names like loop, but they hardly appear in textbooks, so the lecture skips them.

Ring

With two operations we get a ring. The addition and multiplication on a set $R$ must obey the following.

• It is an abelian group under addition. The identity is 0, and the inverse of $a$ is $-a$.
• It is a monoid under multiplication. The multiplicative identity is written 1, and commutativity is not demanded.
• Distributivity holds. $a(b + c) = ab + ac$, and since multiplication need not commute, $(b + c)a = ba + ca$ is demanded separately.

For example, $\mathbb{Z}_6 = \{0, 1, 2, 3, 4, 5\}$ with addition and multiplication taken modulo 6 is a ring. Under addition every element has an inverse, but under multiplication 0 has none. A more important example is the square matrices with real entries. Matrix addition is an abelian group, matrix multiplication is a non-commutative monoid, and its identity is the identity matrix.

The matrix ring also shows a strange phenomenon that happens in rings. The product of two nonzero elements can be 0.

Such elements are called zero divisors. Since multiplication does not commute, we speak of left zero divisors and right zero divisors separately. (The same phenomenon appears in $\mathbb{Z}_6$ as $2 \times 3 = 0$. This example was not covered in the lecture.)

There are also variants that drop conditions from a ring. Dropping additive inverses gives a semiring (rig, the name with “negative” removed), and dropping the multiplicative identity gives an rng (the name with “identity” removed). In physics rings themselves appear occasionally, and these variants are hardly used.

Division ring and field

If every nonzero element of a ring is a unit under multiplication, it is a division ring. Since 0 is the additive identity it cannot have a multiplicative inverse, so a ring can never be a group under multiplication, and only after removing 0 can it become one. Mathematically, the set of all units $R^\times$ of a ring always forms a group under multiplication. An example of a division ring is the quaternions.

Addition is done component by component, and multiplication is expanded with rules such as $i^2 = -1$. Multiplying both sides of $ijk = -1$ by $k$ gives $ij = k$, and the whole multiplication table follows the same way. Every nonzero quaternion has a multiplicative inverse, but $ij \neq ji$. A division ring whose multiplication commutes is a field, so the quaternions are not a field. The real and complex numbers are the standard fields. A field gets subtraction from addition and division from multiplication with 0 removed. So a field $F$ is an abelian group under addition, and the set $F$ with 0 removed is an abelian group under multiplication.

The scalars of a vector space come from a field. The old definition said a division ring is enough, but the modern definition almost always asks for a field, and the case with quaternion scalars is handled as a module.

Checking directly on a small structure

The ladder above does not come across well from the definitions alone. The toy below builds a small structure on $\{0, 1, \dots, n-1\}$ with operations taken modulo n, and checks the axioms one at a time. Addition is an abelian group, multiplication is a monoid, and subtraction stays at a magma because associativity breaks. In “ring” mode it shows distributivity, zero divisors, and units, and decides whether the result is a field. Change n from 4, 6, 12 to 5, 7 and you can watch the zero divisors disappear. (That the structure is a field when n is prime was not covered in the lecture.)

Q. What is the difference between a monoid and a group in a word?
A. Whether every element has an inverse. In multiplication mode, change n and you will see that only some of the elements are units. Every element must be a unit for it to be a group.

In the next post we go on to structures with more than one set, from modules up to algebras.