The set of autotopisms of partial Latin squares

TL;DR

This paper characterizes the cycle structures of autotopisms of partial Latin squares, linking these structures to the invariance properties and sizes of the partial Latin squares they stabilize, and generalizes Latin square completion.

Contribution

It provides a detailed characterization of autotopism cycle structures and their influence on invariant partial Latin squares, extending classical Latin square completion concepts.

Findings

Cycle structures of autotopisms are fully characterized.

The cycle structure determines possible sizes of invariant partial Latin squares.

The number of invariant partial Latin squares of a given size is linked to autotopism cycle structure.

Abstract

Symmetries of a partial Latin square are determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism $\Theta$, the cardinality of the set $\mathcal{PLS}_{\Theta}$ of partial Latin squares which are invariant under $\Theta$ only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms of partial Latin squares are characterized and several related properties studied. It is also seen that the cycle structure of $\Theta$ determines the possible sizes of the elements of $\mathcal{PLS}_{\Theta}$ and the number of those partial Latin squares of this set with a given size. Finally, it is generalized the traditional notion of partial Latin square completable to a Latin square.