free skew field example

Author: ericung

Chapters/Chapter3.tex

@@ -821,11 +821,11 @@ \section{Rules of a Field}
 
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-Given the following equation above, it has the following matrix representations.
+Given the following equation above, it has the following matrix representations forming a free skew field.
 
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-a + bc
+The free skew field matrix representation of $a + bc$ consists of the following set of representations.
 
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@@ -836,8 +836,6 @@ \section{Rules of a Field}
 \end{matrix}
 $
 
-$\\ $
-
 $
 \begin{matrix}
 b + c = -b + c & -b + c = b + c\\
@@ -856,7 +854,8 @@ \section{Rules of a Field}
 
 $\\ $
 
-a + b + c
+$\textit{Free Skew Field}.$ $a + b + c$
+
 
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@@ -887,7 +886,7 @@ \section{Rules of a Field}
 
 $\\ $
 
-ab + c
+$\textit{Free Skew Field}.$ $ab + c$
 
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@@ -920,7 +919,8 @@ \section{Rules of a Field}
 
 Associativity of multiplication is defined as:
 
-a * (b*c) = (a*b)*c
+
+$\textit{Example}. a * (b*c) = (a*b)*c$
 
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@@ -933,7 +933,7 @@ \section{Rules of a Field}
 
 $\\ $
 
-a + bc
+$\textit{Free Skew Field}.$ $a + bc$
 
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@@ -964,7 +964,7 @@ \section{Rules of a Field}
 
 $\\ $
 
-a + b + c
+$\textit{Free Skew Field}.$ $a + b + c$
 
 $\\ $
 
@@ -995,7 +995,7 @@ \section{Rules of a Field}
 
 $\\ $
 
-ab + c
+$\textit{Free Skew Field}.$ $ab + c$
 
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@@ -1028,7 +1028,7 @@ \section{Rules of a Field}
 
 Distributivity is defined as:
 
-$\textbf{a*(b+c) = a*b + a*c}$.
+$\textbf{Example}$. $a*(b+c) = a*b + a*c$
 
 \begin{figure}[H]
   \centering
@@ -1039,7 +1039,7 @@ \section{Rules of a Field}
 
 $\\ $
 
-$\textbf{a + b + c}$
+$\textit{Free Skew Field}.$ $a + b + c$
 
 $\\ $
 
@@ -1070,7 +1070,7 @@ \section{Rules of a Field}
 
 $\\ $
 
-$\textbf{a b + a c}$
+$\textit{Free Skew Field}.$ $a b + a c$
 
 $\\ $
 
@@ -1110,7 +1110,7 @@ \section{Rules of a Field}
 
 $\\ $
 
-$\textbf{a b + a c}$
+$\textit{Free Skew Field}.$ $a b + a c$
 
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@@ -1173,6 +1173,10 @@ \section{Rules of a Field}
 
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+Using the results above, the definition of an inferrable language can be described as a the sum of a set of ideals in the following.
+
+$\\ $
+
 $\textit{Definition}$. An inferrable language, $L$, is a set of at least two sequences where each sequence satisfies a linear recurrence relation that forms a ring. Equivalently, given $A_i \in L$ for any $i \leq n$ where $A_i = a_i - \sum_{k \neq i}a_k = 0$ implies $a_i = \sum_{k \neq i}a_k$ then $a_i = \sum_{k \neq i}a_k$ thus a decision function is equivalent to $a_i = \sum_{j \neq i}A_j + \sum_{k \neq i}a_k$.
 
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