Hedetniemi's conjecture

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Musean hypernumbers are an algebraic concept envisioned by Charles A. Musès (1919–2000) to form a complete, integrated, connected, and natural number system. Musès sketched certain fundamental types of hypernumbers and arranged them in ten "levels", each with its own associated arithmetic and geometry.

Mostly criticized for lack of mathematical rigor and unclear defining relations, Musean hypernumbers are often perceived as an unfounded mathematical speculation. This impression was not helped by Musès' outspoken confidence in applicability to fields far beyond what one might expect from a number system, including consciousness, religion, and metaphysics.

The term "M-algebra" was used by Musès for investigation into a subset of his hypernumber concept (the 16 dimensional conic sedenions and certain subalgebras thereof), which is at times confused with the Musean hypernumber level concept itself. The current article separates this well-understood "M-algebra" from the remaining controversial hypernumbers, and lists certain applications envisioned by the inventor.

"M-algebra" and "hypernumber levels"

Musès was convinced that the basic laws of arithmetic on the reals are in direct correspondence with a concept where numbers could be arranged in "levels", where fewer arithmetical laws would be applicable with increasing level number. However, this concept was not developed much further beyond the initial idea, and defining relations for most of these levels have not been constructed.

Higher dimensional numbers built on the first three levels were called "M-algebra" by Musès if they yielded a distributive multiplication, unit element, and multiplicative norm. It contains kinds of octonions and historical quaternions (except A. MacFarlane's hyperbolic quaternions) as subalgebras. A proof of completeness of M-algebra has not been provided.

Conic sedenions / "16 dimensional M-algebra"

The term "M-algebra" (after C. Musès) refers to number systems that are vector spaces over the reals, whose bases consist in roots of −1 or +1, and which possess a multiplicative modulus. While the idea of such numbers was far from new and contains many known isomorphic number systems (like e.g. split-complex numbers or tessarines), certain results from 16 dimensional (conic) sedenions were a novelty. Musès demonstrated the existence of a logarithm and real powers in number systems built to non-real roots of +1.

Multiplication table

The conic sedenions form an algebra with a non-commutative, non-associative, but alternative multiplication and a multiplicative modulus. It consists of one real axis (to basis $1$), eight imaginary axes (to bases $i_n$ with $i_n^2=-1$), and seven counterimaginary axes (to bases $\epsilon$ with $\epsilon {}_n^2=+1$).

The multiplication table is:

File:ConicSedenionsMultTable.png

Similar to unity (1), the imaginary basis $i_0$ is always commutative and associative under multiplication. Musès at times used the symbol ${\epsilon }_0:=1$ to highlight this similarity. In fact, conic sedenions are isomorphic to complex octonions, i.e. octonions with complex number coefficients. By examining ${\epsilon }_n$ as bases to real number coefficients, however, Musès was able to show certain algebraic relations, including power and logarithm of ${\epsilon }_n$.

Select findings

Musès showed that a countercomplex basis $\epsilon {}_n$ ($n=1,\dots ,7$) not only has an exponential function

$e^{\epsilon {}_n\alpha }=\cosh \alpha +\epsilon {}_n(\sinh \alpha )$

($\alpha$ real) but also possesses real powers:

$\epsilon {}_n^{\alpha }=\frac{1}{2}[(1-\epsilon {}_n)+(1+\epsilon {}_n)e^{-\pi i_n\alpha }]$

This is referred to as "power orbit" of $\epsilon {}_n$ by Musès. Also, a logarithm

$\ln \epsilon {}_n=\frac{\pi }{2}(i_0-i_n)$

is possible in this arithmetic. Their multiplicative modulus $\left|z\right|$ is

$\left|z\right|=|a+\sum b_n{i}_n+\sum c_n{\epsilon }_n+d|:=\sqrt[4]{(a^2+b_n^2-c_n^2-d^2{)}^2+4(ad-b_n{c}_n{)}^2}$

List of number types and their isomorphisms

Circular quaternions and octonions

Circular quaternions and octonions from the Musean hypernumbers are identical to quaternions and octonions from Cayley–Dickson construction. They are built on imaginary bases $i_n$ only.

Hyperbolic quaternions

Hyperbolic quaternions after Musès, to bases {$1,\epsilon {}_1,\epsilon {}_2,i_3$} are isomorphic to coquaternions (split-quaternions). They are different from Alexander Macfarlane's hyperbolic quaternions (first mention in 1891), which are not associative.

Conic quaternions

Conic quaternions are built on bases {$1,i,\epsilon ,i_0$} and form a commutative, associative, and distributive arithmetic. They contain non-trivial idempotents and zero divisors, but no nilpotents. Conic quaternions are isomorphic to tessarines, and also to bicomplex numbers (from the multicomplex numbers).

In contrast, circular and hyperbolic quaternions are not commutative, hyperbolic quaternions also contain nilpotents.

Hyperbolic octonions

Hyperbolic octonions are isomorphic to split-octonion algebra. They consist of one real, three imaginary ($\sqrt{-1}$), and four counterimaginary ($\epsilon$) bases, e.g. {$1,i_1,i_2,i_3,\epsilon {}_4,\epsilon {}_5,\epsilon {}_6,\epsilon {}_7$}.

Conic octonions

Conic octonions to bases $\{1,i_1,i_2,i_3,i_0,\epsilon {}_1,\epsilon {}_2,\epsilon {}_3\}$ form an associative, non-commutative octonionic number system. They are isomorphic to biquaternions.

External links

• Mention in zero-divisor analysis by R. de Marrais on arXiv.org
• Zero-divisor algebras on Tony Smith's personal home page (as of 12 Jan 2007)

The hypernumber "level" concept

In Musès paired certain fundamental laws of arithmetic with suggested number levels, where fewer of these laws would be applicable with increasing level number. Musès envisioned "... sensitivity to operational distinctions on the part of hypernumbers". In the absence of rigorous mathematical treatment, however, Musès' hypernumber level concept has only been adapted for metaphysical or religious ideas.

Providing defining relations for hypernumbers remains a fringe interest today, though it could benefit description of physical law that is based on the lower, well-understood levels.

The following lists an overview of the levels as envisioned by Musès.

Real, complex, and epsilon numbers

The first two levels in hypernumber arithmetic correspond to real and imaginary number arithmetic. The $\epsilon$ basis after Musès is identical to j from the split-complex numbers, and is a non-real root of $+1$. Epsilon numbers are assigned the 3rd level in the hypernumbers program.

w arithmetic

Beginning with w arithmetic, Musès envisioned hypernumber types that are increasingly unfamiliar and speculative. While providing certain rules on how to use these numbers, many open questions remain to date. w numbers are assigned the 4th level in the hypernumbers program.

In the two-dimensional (real, w) plane, the power orbit $w^{\alpha }$ (with $\alpha$ real) is periodic with $w^0=w^6=1$ and the following integral powers:

$w^1=w$

$w^2=-1+w$

$w^3=-1$

$w^4=-w$

$w^5=1-w.$

They offer a multiplicative modulus:

$\left|\right|a+bw\left|\right|=\sqrt{a^2+ab+b^2}$

If a and b are real number coefficients, the arithmetic <(1,w), +, *> is a field (in fact the complex numbers with basis 1 and a primitive sixth root of unity rather than the usual fourth). However, the dual base number to (w) is (-w), which is different from the conjugate of (w), which is 1-(w). This is in contrast to e.g. the imaginary base $i:=\sqrt{-1}$, for which both dual and conjugate are the same (-i). The resulting (-w) arithmetic is therefore distinct from -(w) arithmetic, while coexisting on the same number plane.

File:HypernumbersPowerOrbitW.gif

p and q numbers

So-called p and q numbers are assigned the 5th level in the hypernumbers program, and form a nearly dual system. Each being nilpotent ($p^2=q^2=0$), the arithmetic is envisioned to offer a multiplicative modulus, an argument, and a polar form.

The integral powers are:

$p^0=q^0=p^2=q^2=0$

$p^1=p$

$q^1=q$

$p^3=q$

$q^3=p$

In the {p, q} plane, both $p^{\alpha }$ and $q^{\alpha }$ (with $\alpha$ real) lie on a two-leaved rose, described through $ap+bq$ with

$(a^2+b^2{)}^2=(a+b)(a-b{)}^2.$

File:HypernumbersPowerOrbitPQ.gif

Note on (−p), p ⁻¹, 1/p

From:

"...Note that −p is generated via w, thus: $(qw{)}^3=(wq{)}^3=(w^3)(q^3)=(-1)p=-p$. It must be remembered that because p is nilpotent ($p^2=0,p\ne 0$), its zeroth power cannot be 1; in fact $p^0=0$. Hence also $p^{-1}\ne 1/p$, and since $(1/p)(1/p)=1/p^2=\mathrm{\infty }$, we see that $1/p$ is panpotent, i.e. a root of infinity. Compare $1/(1\pm \epsilon )$, which are a pair of divisors of infinity."

m numbers

The 6th level in the Musean hypernumbers is governed by cassinoids or Cassinian ovals, which geometrically describe their multiplication.

In the {real, m} plane, they offer the following relations:

$m^2=m$

$(\sqrt{2}m{)}^2=0$

$(\sqrt{3}m{)}^2=-1$

It is speculated that a number system like this would use coefficients such as $\sqrt{2}$ in the expression $\sqrt{2}m$, that are not actually real numbers. Instead, one would need to look at +1, -1, +m, and -m as units, and the coefficients as absolute numbers which are distinct from real numbers and are never negative.

File:HypernumbersPowerOrbitM.gif

The Cassinian ovals are described by:

$s^4:=(a^2+b^2{)}^2+2(a^2-b^2)+1$

The remaining levels

In the 7th level, Musès pictured a number $\mathrm{\Omega }$ where ${\mathrm{\Omega }}^n=\mathrm{\Omega }$ for any finite n, ${\mathrm{\Omega }}^{\mathrm{\infty }}=0$, but ${\mathrm{\Omega }}^{\mathrm{\infty }-n}$ would be a number of the form $a+b\mathrm{\Omega }$ (with a, b real).

The 8th level, $\upsilon$ is envisioned as unifying concept to allow to transition between all the lower hypernumber types.

The 9th level, $\sigma$ is envisioned as the creator of axes, and has somewhat the characteristic of an operator (rather than a number). The product $\sigma \upsilon$ is proposed to be the unit step function.

The 10th level consists of 0 and antinumbers. Antinumbers are envisioned to be numbers beyond positive and negative infinity. With use of $\upsilon$ one would be able to span entire spaces consisting of axes of zeros, and connect numbers beyond positive and negative infinity.

Visions of applicability

The range of applications envisioned by Musès of his hypernumber concept can be considered grandiose: A full and complete understanding of all laws of physics (in particular quantum mechanics), a description of consciousness in terms of physical formulations, spiritual growth, religious enlightenment, the solution of well-known mathematical problems (including the Riemann hypothesis), and the exploration of para-psychological phenomena (e.g.). Many of Musès' own writings combine mathematical content with one or more of these speculative projects,. The secondary literature on Musès devotes itself more to his speculative thought than to his mathematics.

See also

• Absolute infinite
• Biquaternion
• Hyperbolic quaternion (per A. MacFarlane)
• Hypercomplex number
• Nilpotents
• Octonion
• Quaternion
• Sedenion
• Split-complex number
• Split-octonions
• Split-quaternion / Coquaternion
• Tessarines
• Zero divisor

References

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External links

• Kevin Carmody's website on hypernumbers at the Wayback Machine