732: Latin Square of Finite Set

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definition of Latin square of finite set

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of set.

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Target Context

• The reader will have a definition of Latin square of finite set.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$S$: $=\{a_1,...,a_n\}$, $\in \{\text{ the sets }\}$
$\ast M$: $\in \{\text{ n x n matrixes }\}$, whose components are of $S$
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Conditions:
Each row of $M$ is a permutation of $(a_1,...,a_n)$
$\wedge$
Each column of $M$ is a permutation of $(a_1,...,a_n)$
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2: Natural Language Description

For any set, $S:=\{a_1,...,a_n\}$, any $nxn$ matrix whose components are of $S$ such that each row of $M$ is a permutation of $(a_1,...,a_n)$ and each column of $M$ is a permutation of $(a_1,...,a_n)$.

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3: Note

There can be multiple Latin squares of $S$: for example, when $S=\{1,2,3\}$, $\left(\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 3& 1& 2\end{array}\right)$ is a Latin square of $S$ and $\left(\begin{array}{ccc}1& 3& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right)$ is another Latin square of $S$.

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References

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