The set of autotopisms of partial Latin squares

Symmetries of a partial Latin square are primarily determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism ΘΘ, the cardinality of the set PLSΘPLSΘ of partial Latin squares which are invariant under ΘΘ 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 were studied. It is also seen that the cycle structure of ΘΘ determines the possible sizes of the elements of PLSΘPLSΘ 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.