Partial Latin Squares Having a Santilli’s Autotopism in their Autotopism Groups

Abstract

An autotopism Θ = (α, β, γ) of a partial Latin square P = (pi, j ) with symbols in N = {0, 1, …, n − 1} is said to be a Santilli’s autotopism of P if there exist jalpha;, jβ,jγ ∈ N such that α(i) = pi, jα , β(i) = pi, jβ and γ(i) = pi, jγ , for all i ∈ N. In this paper, we are interested in the set SPLS (Θ) of partial Latin squares having Θ as a Santilli’s autotopism. Specifically, the cardinality of the set of minimal elements of SPLS(Θ) is obtained and a classification of Latin squares Santilli’s isotopisms of order ≤ 5 fixing at least one symbol is given.