# real scator algebra

Submitted by mfg on Fri, 03/11/2017 - 14:00

The hyperbolic scator set is represented with a "+" in the following way, $\mathbb{E}_{+}^{1+n}$. These numbers have an algebra of their own. A relevant feature of this algebra is that the product is not distributive over addition. In the additive representation a scator is written as

\overset{o}\varphi=f_{0} + \sum_{j=1}^{n}{f_{j}\hat{\mathbf{e}}_{j} } ,
with $f_{0}, f_{j} \in \mathbb{R}$ for all $j:[1,n]$ y $\hat{\mathbf{e}}_{j} \notin \mathbb{R}$. Los escatores se componen de una parte escalar, y $n$ partes directoras. Una particularidad de los escatores reales, similar a los números hiperbólicos, radica en que el producto $\hat{\mathbf{e}}_{j} \hat{\mathbf{e}}_{l}=\delta_{j,l}$, a diferencia de los números escatores imaginarios en los cuales el signo de la delta es negativo.

## real scator sum

Let two scators be $\overset{o}\alpha=a_{0} + \sum_{j=1}^{n}{a_{j}\hat{\mathbf{e}}_{j}}$ and $\overset{o}\beta=b_{0} + \sum_{j=1}^{n}{b_{j}\hat{\mathbf{e}}_{j}}$. Their sum is performed component by component,

\overset{o}\gamma = g_{0} + \sum_{j=1}^{n}{g_{j}\hat{\mathbf{e}}_{j}} = \overset{o}\alpha + \overset{o}\beta = (a_{0} + b_{0}) + \sum_{j=1}^{n}({a_{j}+ b_{j})\hat{\mathbf{e}}_{j}}.
We can see that the scalar part of the sum is the sum of the scalar components of scators  $\overset{o}\alpha$ y $\overset{o}\beta$. In a similar fashion, the director part of the sum is equal to the sum of the director coefficients with equal unit director component $\hat{\mathbf{e}}_{j}$.

The hyperbolic scator set is represented with a "+" in the following way, $\mathbb{E}_{+}^{1+n}$. These numbers have an algebra of their own. A relevant feature of this algebra is that the product is not distributive over addition. In the additive representation a scator is written as

\overset{o}\varphi=f_{0} + \sum_{j=1}^{n}{f_{j}\hat{\mathbf{e}}_{j} } ,
with $f_{0}, f_{j} \in \mathbb{R}$ for all $j:[1,n]$ y $\hat{\mathbf{e}}_{j} \notin \mathbb{R}$. Los escatores se componen de una parte escalar, y $n$ partes directoras. Una particularidad de los escatores reales, similar a los números hiperbólicos, radica en que el producto $\hat{\mathbf{e}}_{j} \hat{\mathbf{e}}_{l}=\delta_{j,l}$, a diferencia de los números escatores imaginarios en los cuales el signo de la delta es negativo.

## real scator product

The product definition between two scators can be divided into three cases:

### Case 1

If $\overset{o}\alpha$ and $\overset{o}\beta$ two scators such that $a_{0}b_{0}\neq 0$ then, the product between them, i.e. $\overset{o}\gamma=\overset{o}\alpha\overset{o}\beta$ is defined as follows:$$\overset { o }\gamma ={ a }_{ 0 }{ b }_{ 0 }\prod _{ k=1 }^{ n }{ \left( 1+\frac { { a }_{ k }{ b }_{ k } }{ { a }_{ 0 }{ b }_{ 0 } } \right) +{ a }_{ 0 }{ b }_{ 0 }\sum _{ j=1 }^{ n }{ \left[ \prod _{ k\neq j }^{ n }{ \left( 1+\frac { { a }_{ k }{ b }_{ k } }{ { a }_{ 0 }{ b }_{ 0 } } \right) \left( \frac { { a }_{ j } }{ { a }_{ 0 } } +\frac { { b }_{ j } }{ { b }_{ 0 } } \right) } \right] } } { \hat {\mathbf{ e } }_{ j }}.$$

### Case 2

If one of the scators involved in the product, only has one direction component non zero  $a_{0}=0, b_{0}\neq 0$, i.e. $\overset{o}\alpha={a}_{l} { \hat {\mathbf{ e } }_{ l }}$ and the remanning scator such that $\overset{o}\beta=b_{0} + \sum_{j=1}^{n}{b_{j}}{\hat{\mathbf{e}}_{j}}$, then, the product beween them is defined as follows :$$\overset { o }{ \gamma } =\left( { a }_{ l }{ b }_{ l } \right) +\left( { a }_{ l }{ b }_{ 0 } \right) { \hat { \mathbf{ e } }_{ l }}+\sum _{ j\neq l }^{ n }{ \left( \frac { { a }_{ l }{ b }_{ l }{ b }_{ j } }{ { b }_{ 0 } } \right) } { \hat {\mathbf{ e } }_{ j }}.$$

### Case 3

Finally, when both scators, have scalar component equal zero, $a_{0}=0, b_{0}= 0$ namely, $\overset{o}\alpha={a}_{l} { \hat {\mathbf{ e } }_{ l }}$ and $\overset{o}\beta={b_{j}\hat{\mathbf{e}}_{j}}$. Then the product between these scators is defined as follows: $$\overset { o }{ \gamma } =({a}_{l} { \hat {\mathbf{ e } }_{ l }})({b_{j}\hat{\mathbf{e}}_{j}})={a}_{l} b_{j} \delta_{lj},$$

It is possible to obtain the following case from the previous cases:

Two scators such that, both have scalar component and only have one direction component non zero, namely:$\overset{o}\alpha={a}_{0}+{a}_{l} { \hat {\mathbf{ e } }_{ l }}$ and $\overset{o}\beta={b}_{0}+{b}_{j} { \hat {\mathbf{e} }_{ j }}$ then, the product between them is defined as follows:$$\overset { o }{ \gamma } ={ a }_{ 0 }{ b }_{ 0 }+{ a }_{ l }{ b }_{ j }{ \delta }_{ lj }+{ b }_{ 0 }{ a }_{ l }{ \hat { \mathbf{ e } }_{ l }}+{ a }_{ 0 }{ b }_{ j }{ \hat {\mathbf{ e } }_{ j }}.$$

If the scator  $\overset{o}\alpha$  has a scalar component non zero and all his direction components are equals zero, $\overset{o}\alpha={a}_{0}$, from the first case you get

$${a}_{0}\overset{o}\beta={a}_{0}b_{0} + \sum_{j=1}^{n}{a}_{0}{b_{j}\hat{\mathbf{e}}_{j}}$$

then  $\overset{o}\alpha={a}_{0}$ is a multiple for the scator  $\overset{o}\beta$. Hence the first component  is called the scalar component.