Tricomplex numbers (Trinions anyone?)

[EinsteinKreuz]

So what am I talking about? An extension of the complex numbers between ℂ¹ and the Quaternions.

A tricomplex number can be written as τ = {a + bi + cj | ∀(a,b,c)∈ℝ } where:

i² = j² = i×j = -1 = -(j×i)

Thus of course, j×i = +1
What is remarkable is that such objects are closed under muliplication and produce of a tricomplex number with its conjugate, τ×τ^* = (a+bi+cj)(a-bi-cj)

= a² + b² + c². You can verify this yourself using the multiplication rules I listed but I will do it in a follow up post.

Now surely there's another name for such things as it seems far too likely that someone discovered them before but if so, what exactly are they called?

Using the Cayley-Dickinson construction method as well as the properties of free groups I tried to created a Cayley table for 5-dimensional complex numbers(1 real basis and 4 imaginary bases) but it didn't work since each row and each column had redundant entries unlike the Quaternion and Octonion Cayley tables.

 

[Orodruin]

Unlike the quaternions, your suggested extension is not a division ring as it does not have a unique multiplicative inverse. For example $ix = -1$ is solved by both $x = i$ and $x = j$. It also not associative as, for example, $(i^2)j = -1j = -j$ and $i(ij) = i(-1) = -i$. So in short, it is missing several of the properties which make the quaternions useful.

 

[fresh_42]

An anecdote tells that Hamilton searched nearly 10 years to find a field extension of ℝ of degree 3 before he was convinced that there is none. And even the quaternions come to a prize: the commutative property.

 

[EinsteinKreuz]

Unlike the quaternions, your suggested extension is not a division ring as it does not have a unique multiplicative inverse. For example $ix = -1$ is solved by both $x = i$ and $x = j$. It also not associative as, for example, $(i^2)j = -1j = -j$ and $i(ij) = i(-1) = -i$. So in short, it is missing several of the properties which make the quaternions useful.

Good point. But the multiplicative identity here is +1. The equation $ix = 1$ has 2 solutions: $x=-i$,$x=-j$. I noticed the same problem when trying to construct 5-dimensional complex numbers.

That said, what is the term for a group-like algebraic structure that has Closure, Identity, and Inverse under a binary operation but is non-associative?

 

[fresh_42]

That said, what is the term for a group-like algebraic structure that has Closure, Identity, and Inverse under a binary operation but is non-associative?

If you have two operations, an additive group plus a distributive multiplication, then it's an algebra. For "group-like" structures that are not associative there is no name, as far as I (don't) know. Even halfgroups without inverse require the associative property.

EDIT: I think there is a 3 dimensional associative structure over ℝ. Just forgot what exactly it was, $ℝ^3$ with the ×-product or so.

 

[micromass]

If you have two operations, an additive group plus a distributive multiplication, then it's an algebra.

Do you mean a $\mathbb{Z}$-algebra?

 

[fresh_42]

Do you mean a $\mathbb{Z}$-algebra?

You are right, I withheld the field as it all began with the reals. I just wanted to emphasize that there exist non associative algebras but no non associative named structures with only one binary operation I knew of.

 

[micromass]

You are right, I withheld the field as it all began with the reals. I just wanted to emphasize that there exist non associative algebras but no non associative named structures with only one binary operation I knew of.

Oh, there are plenty. For example, loops: https://en.wikipedia.org/wiki/Quasigroup

 

[fresh_42]

That's very interesting. I've only met the right path of them (fig. on Wiki Link) plus algebras of all kind. Maybe I've forgotten the term. Thank you, something learned today.

 

[suremarc]

That said, what is the term for a group-like algebraic structure that has Closure, Identity, and Inverse under a binary operation but is non-associative?

It's worth pointing out that unital $\mathbb{R}$-algebras of odd dimension always have zero divisors, associative or not. Consider the left-multiplication map $L_a: x\rightarrow ax$ on a unital $\mathbb{R}$-algebra $A$ of odd dimension. Its characteristic polynomial has odd degree and therefore has a linear factor, yielding a real eigenvalue $\lambda$. We then may obtain the zero divisor $\lambda1_A-a$.

 

[EinsteinKreuz]

If you have two operations, an additive group plus a distributive multiplication, then it's an algebra. For "group-like" structures that are not associative there is no name, as far as I (don't) know. Even halfgroups without inverse require the associative property.

EDIT: I think there is a 3 dimensional associative structure over ℝ. Just forgot what exactly it was, $ℝ^3$ with the ×-product or so.

An example of a non-associative algebra under multiplication: The Octionions

 

[mathwonk]

i guess this is clear, but if it had been associative, it would seem to be a vector space over the usual complex numbers, hence of even real vector dimension.