Converse from pre-adjunctions to semi-retractions

Determine whether every pre-adjunction between the ages of two locally finite structures, with embeddings as morphisms, induces a semi-retraction between the structures themselves.

Background

Semi-retractions between locally finite ordered structures are known to transfer the Ramsey property, and every semi-retraction induces a pre-adjunction between the corresponding ages. The converse direction is unresolved: although a pre-adjunction between the ages is given, it is not known in general whether one can recover a semi-retraction between the original structures. The paper identifies this as a question distinct from its own main result, which settles the ordered random-graph/atomless-Boolean-algebra instance.

References

Regarding this converse, however, some important questions remain open. While any semi-retraction between two locally finite structures is known to induce a pre-adjunction between their ages (with embeddings as morphisms), the inverse statement is not.

— The random ordered graph is a semi-retract of the canonically ordered atomless Boolean algebra  (2507.12078 - Pinsker et al., 16 Jul 2025) in Section 1, subsection “A question of Bartošová and Scow”