The Map, A New Method to Define Latin Square

Zhaoqi ZHANG


A Latin Square is generally defined as a matrix, in which each entry occurs exactly once in each row and exactly once in each column, and the range of the entries can be from 1 to the order of the matrix. In the past, the researches of Latin Squares were mainly focused on the 2-dimension case, because the definition as matrix limits the extension of the dimension. In this paper, the Latin Square is redefined by map and is renamed as Latin map. This new definition enables Latin map to describe high dimensional extended Latin square. By exploiting the properties of bijection, the propositions and theorems of Latin square are rebuilt to be algebraic results instead of the usual combinatoric results. The behaviors of Latin square are discussed when regarded as map such as the composition of Latin maps, which finally comes to the Extending Construction theorems.


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