Algebras

2020 James B. Wilson

Colorado State University

 

Objective

  • Define general algebraic structures.
  • Develop expected notation.
  • Explore examples.

Let  \(\sigma:\mathcal{O}\to\mathbb{N}\) be a function.

 

Definition.  a \(\sigma\)-algebraic structure, or \(\sigma\)-Algebra for short, is a type \(A\) together with operators \[(\ldots)_o:A^{\sigma(o)}\to A\]

for each \(o\in \mathcal{O}\).

 

E.g. \(+\in \mathcal{O}\) with \(\sigma(+)=2\) then \[(x,y)_+:= x+y\]

maps \(A^2\to A\).

Example. 

  1. \(\mathbb{N}\) is a \([+,0]\)-Algebra.
  2. \(\mathbb{Z}\) is a \([+,-,0]\)-Algebra.
  3. \(\mathbb{Q}\) is a \([*,1,+,-,0]\)-Algebra, recall inverses exclude 0 so they are not technically operators.

 

If \(\mathcal{O}\) is familiar, e.g. \(\{+,0\}\) and \(\sigma(+)=2, \sigma(0)=0\); then it is enough to state the symbols.

Example. 

  1. Every group is a \([2,1,0]\)-Algebra, usually with operators \([*,^{-},1]\).
  2. Every ring is a \([2,2,1,0,0]\)-Algebra, with operators \([*,+,-,0,1]\).

 

The converses are not true as there are no axioms, like "associative" so far.

Derived Operations

Common Setting: 

  • We have \(\sigma\)-Algebra
  • We define new operators \(\tau\) from those in \(\sigma\).
  • We switch to talk about a \(\upsilon\)-Algebra, for some \(\upsilon\subset \sigma\sqcup \tau\).

Example.

\(\mathbb{N}\) is a technically a \([1,0]\)-Algebra:  \[\vdash 0\in \mathbb{N}\qquad \frac{n\in \mathbb{N}}{S(n)\in \mathbb{N}}.\qquad(I_{\mathbb{N}})\]

So these two introduction rules give us

  1. a  nullary operator \(0:\mathbb{N}^0\to \mathbb{N}\), and
  2. a unary operator \(S:\mathbb{N}^1\to \mathbb{N}\) called successor.

Define a new operator:

\[n+m:=\left\{\begin{array}{cc} m & n=0\\ S(k+m) & n=S(k)\end{array}\right.\]

Now \(\mathbb{N}\) becomes a \([+,0]\)-Algebra, i.e. the familiar \([2,0]\)-Algebra.

Example.

\(G\) is a \([2,2]\)-Algebra with operators

  • \((x,y)\mapsto x*y:G^2\to G\) and
  • \((x,y)\mapsto x/y:G^2\to G\).

Define the left commutator operator:

\[[x,y] := (x*y)/(y*x)\]

So \(G\) becomes a \([2,2,2]\)-Algebra.

Example.

In a group \(G\) as a \([2,1,0]\)-Algebra with operators

  • \((x,y)\mapsto x*y:G^2\to G\),
  • \(x\mapsto x^{-1}:G^1\to G\), and
  • \(1:G^0\to G\)

\[x/y := x*(y^{-1})\]

\[[x,y] := (x*y)*(x^{-1}*y^{-1})\]

\[x^y := (x*y)*(y^{-1})\]

Example.

In an \([\cdot,1,+,-,0]\)-Algebra (e.g. a ring)

\([x,y]=xy-yx\)

\[(x,y,z) := (xy)z-x(yz)\]

\[\vdots\]

 

Leads to Sabinin algebras

Example.

A group \(G\) as a \([*,^-,1]\)-Algebra is also just as much a \([*]\)-Algebra and the identity and inverses can be recovered from the product.

 

Many books present groups that way.

Example.

A ring \(R\) is a natural \([*,+,-,1,0]\)-Algebra.  But we can replace this with a more interesting ternary operator like

\[(m,x,b) := m*x+b\]

This lead M. Hall Jr. to invent "ternary rings" and he used them to describe coordinates of new geometries.

 

Key idea, each line is like \(y=mx+b\) so each line becomes the result of a ternary product.