The Algebra Conundrum
I was reading a paper which mentioned We may view [a specific] ring as an algebra over the real or over the complex numbers, and I realised I did not know formally what an Algebra was. So thought I’ll clear it up for myself. Turns out the word is a bit overused.
Algebraic Structures
An Algebraic Structure is a set \(A\), along with some operations (functions) on \(A\) of finite arity and a finite set of axioms that the operations must satisfy.
Examples of algebraic structures are structures of types Groups, Rings, Lattices, Modules and [wait for it…] Algebras.
Algebra over a field
That last type of Algebraic Structure, known as an Algebra is an Algebra over a field (which, it turns out, the paper I was reading actually meant). It is defined as follows:
Let \(\mathcal{F}\) be a field and let \(A\) be a vector space over \(\mathcal{F}\) along with a binary operation \(\cdot : A \times A \to A\). \(A\) is an algebra over \(\mathcal{F}\) if \(\forall x, y, z \in A, \forall a,b \in \mathcal{F}\),
- Right Distributivity of \(\cdot\) – \((x + y) \cdot z = x \cdot z + y \cdot z\)
- Left Distributivity of \(\cdot\) – \(z \cdot (x + y) = z \cdot x + z \cdot y\)
- Compatibility with Scalars – \((ax)\cdot (by) = (ab)(x \cdot y)\)
The above three axioms basically mean that the binary operation \(\cdot\) is bilinear.
It turns out these aren’t the only contexts in which the term is used. Here is a text snippet from Wikipedia:
The properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. The language of category theory is used to express and study relationships between different classes of algebraic and non-algebraic objects.
Mathematicians are clearly great at naming things. And confusing me.