definition of Latin square of finite set
Topics
About: set
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of set.
Target Context
- The reader will have a definition of Latin square of finite set.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( S\): \(= \{a_1, ..., a_n\}\), \(\in \{\text{ the sets }\}\)
\(*M\): \(\in \{\text{ n x n matrixes }\}\), whose components are of \(S\)
//
Conditions:
Each row of \(M\) is a permutation of \((a_1, ..., a_n)\)
\(\land\)
Each column of \(M\) is a permutation of \((a_1, ..., a_n)\)
//
2: Natural Language Description
For any set, \(S := \{a_1, ..., a_n\}\), any \(n x n\) 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 \((a_1, ..., a_n)\).
3: Note
There can be multiple Latin squares of \(S\): for example, when \(S = \{1, 2, 3\}\), \(\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{pmatrix}\) is a Latin square of \(S\) and \(\begin{pmatrix} 1 & 3 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{pmatrix}\) is another Latin square of \(S\).
