The Type A Half-Absorption Monoid

We give an explicit bijection between the full-support Green’s R-classes of the type A half-absorption monoid of rank K and permutations of K letters. The position of the first saturated Lehmer digit, minus one, equals the last descent of the corresponding permutation. The monoid shares the length enumerator of the type A 0-Hecke monoid. For ranks at least three, it lies strictly between the opposite free tree monoid and the Catalan monoid in the quotient chain. The bijection is constructed from recursive labels for the strongly connected components inside the Catalan fibres. For each support, the number of principal right ideals equals the number of idempotents. The supplementary files include a Python verification script tested through rank seven and an extract containing the cited sections of Atlas, Volume 5.