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.
- \(\mathbb{N}\) is a \([+,0]\)-Algebra.
- \(\mathbb{Z}\) is a \([+,-,0]\)-Algebra.
- \(\mathbb{Q}\) 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.
- Every group is a \([2,1,0]\)-Algebra, usually with operators \([*,^{-},1]\).
- 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
- a nullary operator \(0:\mathbb{N}^0\to \mathbb{N}\), and
- 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.
Algebras
By James Wilson
Algebras
Definitions of Algebraic structures
