- #1
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Suppose we are given a binary operation on a finite set of abstract symbols in terms of a multiplication table such as:
[tex]
\begin{array}{c|ccc}
* & A & B & C \\ \hline
A & A & B & C \\
B & B & A & B \\
C & C & B & A \\
\end{array}
[/tex]
Suppose we want to represent the operation in some concrete way as a binary operation on some fairly simple mathematical objects. What are some good ways to do this? For example, are there simple binary operations on elements of a set or a group that are versatile enough to implement any multiplication table?
Since the abstract binary operation need not be associative, commutative, have an identity etc, we need a concrete binary operation that need not be any of those things. But we would want to leave open the possibility that on a particular set of objects, the operation might have those properties since some multiplication tables have them.
[tex]
\begin{array}{c|ccc}
* & A & B & C \\ \hline
A & A & B & C \\
B & B & A & B \\
C & C & B & A \\
\end{array}
[/tex]
Suppose we want to represent the operation in some concrete way as a binary operation on some fairly simple mathematical objects. What are some good ways to do this? For example, are there simple binary operations on elements of a set or a group that are versatile enough to implement any multiplication table?
Since the abstract binary operation need not be associative, commutative, have an identity etc, we need a concrete binary operation that need not be any of those things. But we would want to leave open the possibility that on a particular set of objects, the operation might have those properties since some multiplication tables have them.