Some Relations on Paratopisms and An Intuitive Interpretation on the Parastrophes of a Latin Square

TL;DR

This paper introduces an intuitive approach to understanding and generating parastrophes of Latin squares, simplifying computations and relations among transformations, thus enhancing efficiency in related combinatorial problems.

Contribution

It provides a direct method to generate parastrophes from a Latin square and clarifies the relations between isotopisms and parastrophe transformations, improving computational efficiency.

Findings

Direct generation of parastrophes without orthogonal arrays

Simplified relations between isotopisms and parastrophe transformations

Enhanced computational efficiency for Latin square classification

Abstract

This paper will present some intuitive interpretation of the parastrophe transformations of arbitrary Latin square. With this trick, we can generate the parastrophes of arbitrary Latin square directly from the original one without generating the orthogonal array. The relations of isotopisms and parastrophe transformations in composition will also be shown. It will solve the problem that when F1*I1=I2*F2 how can we obtain I2 and F2 from I1 and F1, where I1 and I2 are isotopisms while F1 and F2 are parastrophe transformations and "{*}" is the composition of transformations. These methods could distinctly simplify the computation on a computer for the issues related to main classes of Latin squares. This will improve the efficiency apparently in computation for some related problems.