An Introduction to Abstract Algebra

Part Of: Algebra sequence
Content Summary: 1200 words, 12 min read

A Brief Prelude

Recall that a set is a collection of distinct objects, and a function f: A \rightarrow B is a mapping from the elements of one set to another. Further, in number theory we can express numbers as infinite sets:

  • The natural numbers \mathbb{N} = \left\{ 0, 1, 2, 3, \ldots \right\}.
  • The integers \mathbb{Z} = \left\{ \dots, -2, -1, 0, -1, -2, \ldots \right\}.
  • The rational numbers \mathbb{Q} = \left\{ x | x = p/q, p \in \mathbb{Z}, q \in \mathbb{Z}, q \neq 0 \right\}.
  • The real numbers \mathbb{R}.

The Axioms of Addition and Multiplication

In elementary school you learned that a+b = b+a, for any two integers. In fact there exist five such axioms:

  • Closure. \forall a, b \in \mathbb{Z}: a + b \in \mathbb{Z}.
  • Associativity\forall a, b, c \in \mathbb{Z}: (a + b) + c = a + (b+c).
  • Identity. There exists an element 0 such that, \forall a \in \mathbb{Z}: 0 + a = a + 0 = a.
  • Inverse. \forall a \in \mathbb{Z} there exists an element \boldsymbol{-a} such that a + (-a) = (-a) + a = 0.
  • Commutativity. \forall a, b \in \mathbb{Z}: a + b = b + a.

These axioms encapsulate all of integer addition. We can represent “integer addition” more formally as a set-operator pair: (\mathbb{Z}, +)

Likewise, you have surely learned that a \times b = b \times a. Multiplication too can be described with five axioms:

  • Closure. \forall a, b \in \mathbb{Z}: a \times b \in \mathbb{Z}.
  • Associativity\forall a, b, c \in \mathbb{Z}: (a \times b) \times c = a \times (b \times c).
  • Identity. There exists an element 1 such that, \forall a \in \mathbb{Z}: 1 \times a = a \times 1 = a.
  • Inverse. \forall a \in \mathbb{Z} there exists an element \frac{1}{a} such that a \times \frac{1}{a} = \frac{1}{a} \times a = 1.
  • Commutativity. \forall a, b \in \mathbb{Z}: a \times b = b \times a.

These axioms encapsulate all of integer multiplication. We can represent “integer multiplication” more formally as a set-operator pair: (\mathbb{Z}, \times)

Towards Algebraic Structure

Did the above section feel redundant? A lesson from software engineering: if you notice yourself copy-pasting, you should consolidate the logic into a single block of code.

Let’s build an abstraction that captures the commonalities above.

Definition 1. A binary operation is a function that takes two arguments. Since functions can only map between two sets, we write f : A \times A \rightarrow A.

Examples of binary operations include +, \times, \text{etc}. Note that a \times b is just shorthand for the more formal \times(a, b). Note that the operation symbol \times is just a name: we could just as easily rename the above function to be f(a, b), as long as the underlying mapping doesn’t change.

Definition 2. Let arity denote the number of arguments to an operation. A binary operation has arity-2. A unary operation (e.g., sin(x)) has arity-1. A finitary operation has arity-n.

Definition 3. An algebraic structure is the conjunction of a set with some number of finitary operations, and may be subject to certain axioms. For each operation in an algebraic structure, the following axioms may apply:

  • Closure. \forall a, b \in \mathbb{A}: a \bullet b \in \mathbb{A}.
  • Associativity\forall a, b, c \in \mathbb{A}: (a \bullet b) \bullet c = a \bullet (b \bullet c).
  • Identity. There exists the element \boldsymbol{e} such that, \forall a \in \mathbb{A}: \boldsymbol{e} \bullet a = a \bullet \boldsymbol{e} = a.
  • Inverse. \forall a \in \mathbb{A} there exists an element \boldsymbol{a^{-1}} such that a \bullet \boldsymbol{a^{-1}} = \boldsymbol{a^{-1}} \bullet a = \boldsymbol{e}.
  • Commutativity. \forall a, b \in \mathbb{A}: a \bullet b = b \bullet a.

Algebraic structures are a generalization of  integer addition and integer multiplication. Our  (\mathbb{Z}, +) and (\mathbb{Z}, \times) tuples actually comprise parameters that specify an algebraic structure.

As soon as we define algebraic structures, we begin to recognize these objects strewn across the mathematical landscape. But before we begin, a word about axioms!

The Axiomatic Landscape

Consider algebraic structures that exhibit one binary operation. These structures may honor different combinations of axioms. We can classify these axiom-combinations. Here then, are five kinds algebraic structures (“Abelian” means commutative):

Abstract Algebra- Structure Names

Of course, more esoteric options are available, including:

Abstract Algebra- Other Structure Names (1)

Of all these structures, groups are the most well-studied. In fact, it is easy to find it is not uncommon to of people conflating groups vs algebraic structures.

Definition 4. An algebraic structure is group-like if it contains one 2-ary operation. If it has more than one operation, or operation(s) with a different arity, it is not group-like.

All of our examples today count as group-like algebraic structures. There is also a large body of research studying algebraic structures with two operations, including ring-, lattice-, module-, and algebra-like structures. We will meet these structures another day.

Examples of Group-Like Structures

We saw above that the integers under addition (\mathbb{Z}, +) and multiplication (\mathbb{Z}, \times) are abelian groups. A similar finding occurs when you switch to the reals, or rationals, or natural numbers.

But addition and multiplication are not the only possible binary operations. What about subtraction (\mathbb{Z}, +)? Well, that is only a magma. Closure is satisfied, but all other axioms are violated (e.g., associativity (4 - 2) - 2 \neq 4 - (2-2)) and commutativity (4 - 2 \neq 2 - 4). Likewise, the natural numbers under subtraction are not even a magma: 2 - 4 \not\in \mathbb{N}.

All of our examples so far have groups encapsulating sets of numbers. But groups can contain sets of anything! Let’s switch to linear algebra. What about the set of all n \times n matrices under matrix multiplication?

  • Does it have closure? Yes. Matrix multiplication yields another n \times n matrix.
  • Does it have associativity? Yes. Matrix multiplication is associative.
  • Does it have identity? Yes. The identity element is the matrix I = [ \begin{smallmatrix}1 & 0\\0 & 1\end{smallmatrix}].
  • Does it have inverse? No!  Some n \times n matrices have determinants of 0. Thus, not all members of our set are invertible.

We can now identify this algebraic structure. The set of all n \times n matrices under matrix multiplication is a monoid.

But what if we limit our set to be all n \times n matrices with non-zero determinants? Well, that is a group (the inverse exists for all members). More formally, that set forms the basis of the general linear group GL_{n}(\mathbb{R}). Why isn’t it abelian? Because matrix multiplication is not commutative.

These five examples provide a glimpse into the landscape of algebraic structures. Our recipe is simple:

Take any set and operation that you care about. Classify the resultant algebraic structure by examining which axioms hold.

With these tools, we can begin to build a map of algebraic structures:

group_disc

Takeaways

  • Multiplication and addition share a remarkable number of properties, including closure, associativity, identity, inverse, and commutativity.
  • An algebraic structure (set-operation pair) generalizes the similarities in the above examples.
  • Algebraic structures can have more than one operation. Group-like structures are those with only one (binary) operation.
  • Once you can know about algebraic structures, you can find examples of them strewn across the mathematical landscape.

Until next time.

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