The paper gives explicit descriptions, cardinalities, ranks, and minimal generating sets for the group Γ_n and the monoids Σ_n and Δ_n defined by evenness conditions on restrictions to maximal proper subsets.
Fernandes, Semigroups of order-preserving mappings on a finite chain: a new class of divisors, Semi- group Forum 54, 230–236 (1997)
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Groups of permutations that are even on maximal proper subsets, and related monoids
The paper gives explicit descriptions, cardinalities, ranks, and minimal generating sets for the group Γ_n and the monoids Σ_n and Δ_n defined by evenness conditions on restrictions to maximal proper subsets.