Suppose we are given a binary operation on a finite set of abstract symbols in terms of a multiplication table such as:(adsbygoogle = window.adsbygoogle || []).push({});

[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 operationmighthave those properties since some multiplication tables have them.

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# What are some simple non-associative binary operations?

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