Vítor Hugo Fernandes

Let n be a positive integer and let [n]={1,2,…,n}. Let Γ_n denote the group of permutations on [n] whose restrictions to maximal proper subsets of [n] are even, let Σ_n denote the monoid of transformations on [n] whose injective restrictions to maximal proper subsets of [n] are even and let Δ_n denote the submonoid of Σ_n generated by transformations of rank at least n−1. In this paper, we present descriptions of Γ_n, Δ_n and Σ_n, determine their cardinalities and ranks, and provide minimal generating sets for each of them.

In this paper, we study permutations on n elements that are even on every subset of size t. We describe all groups of these permutations. Unexpectedly, these groups (except for some special cases) are either trivial, cyclic or dihedral. In this context, we define and study monoids that generalize both monoids of order-preserving mappings and monoids of orientation-preserving mappings.