Authors: Martin Erik Horn
Comments: 14 Pages. Poster presentation at AGACSE 2012 in La Rochelle
Quarks are described mathematically by (3 x 3) matrices. To include these quarkonian mathematical structures into Geometric algebra it is helpful to restate Geometric algebra in the mathematical language of (3 x 3) matrices. It will be shown in this paper how (3 x 3) permutation matrices can be interpreted as unit vectors. And as S3 permutation symmetry is flavour symmetry a unified flavour picture of Geometric algebra will emerge.
A new notion of special quasi dual numbers is introduced.
If a+bg is the special quasi dual number with a, b reals, g the new element is such that g^2 = - g.
The rich source of getting new elements of the form g^2 = - g is from Z_n, the ring of modulo integers.
For the first time we construct non associative structures using them.
We have proposed some research problems.
In this book we define x = a+bg to be a special dual like number, where a, b are reals and g is a new element such that g^2 = g.
The new element which is idempotent can be got from Z_n or from lattices or from linear operators.
Mixed dual numbers are constructed using dual numbers and special dual like numbers.
Neutrosophic numbers are a natural source of special dual like numbers, since they have the form a+bI, where I = indeterminate and I^2 = I.
The zero divisor graph of semigroups of finite modulo integers n under product is studied and characterized. If n is a non-prime, the zero divisor graph is not a tree.
We introduce the new notion of tree covering a pseudo lattice. When n is an even integer of the form 2p, p a prime, then the modulo integer zero divisor graph is a tree-covering pseudo lattice.