Arrow relations in lattices of integer partitions
Asma Ibrahim Almazaydeh, Mike Behrisch, Edith Vargas-García, Andreas Wachtel · International Journal of Approximate Reasoning · 2024
We give a complete characterisation of the single and double arrow relations of the standard context K ( L n ) of the lattice L n of partitions of any positive integer n under the dominance order, thereby addressing an open question of Ganter, 2020/2022. • Type D partitions allow up-arrows, which fail to be down-arrows, to all types I–IV. • Type IV partitions allow down-arrows, which fail to be up-arrows, to all types A–D. • Type D partitions have double arrows only to those of type IV and vice versa. • Partitions of types A–C are double arrow related only to types I–III and vice versa. • For n ≥ 3 there are exactly 2 n − 4 one-generated arrow-closed ( 1 × 1 ) -subcontexts of K ( L n ) .