# Explanation: Operation Table

(related to Part: Algebraic Structures - Overview)

If the set $X$ in an algebraic structure is finite, then it is possible to write down the operation table. This is a tabular schema in which all elements $x_1,\ldots,x_n\in X$ are the headers of columns and rows and in which the entry in the $i$-th row and the $j$-th column is the element $x_{ij}\in X$ which fulfills the property $x_ij=x_i\ast x_j$.

$$\begin{array}{c|ccccc} \ast&x_1&\ldots&x_j&\ldots &x_n\\ \hline x_1&x_1\ast x_1&\ldots&x_1\ast x_j&\ldots& x_1\ast x_n\\ \vdots&\vdots&&\vdots&&\vdots\\ x_i&x_i\ast x_1&\ldots&x_i\ast x_j&\ldots& x_i\ast x_n\\ \vdots&\vdots&&\vdots&&\vdots\\ x_n&x_n\ast x_1&\ldots&x_n\ast x_j&\ldots& x_n\ast x_n\\ \end{array}$$

In some cases it is possible to recognize the properties of the binary operation "$\ast$" looking at the operation table:

• The operation table is symmetric, if "$\ast$" is commutative,
• The element $x_i$ is left-neutral, if the $i$-th row corresponds to the column headers of the operation table.
• The element $x_j$ is right-neutral, if the $j$-th column corresponds to the row headers of the operation table.
• If a neutral element exist and has been identified, whenever it occures as an entry in the operation table, the corresponding row and column headers are inverse to each other.

Motivations: 1

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### References

#### Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001