Characterize posets for which all totally compatible structures are proper

Characterize the finite posets X such that every totally compatible associative bilinear product on the Jacobson radical J(I(X,K)) of the incidence algebra I(X,K) over a field K is proper, meaning expressible as the sum of a structure determined by a centroid element and an annihilator-valued structure.

Background

The paper completely describes totally compatible structures on J(I(X,K)) as sums of annihilator-valued structures and linear combinations of structures indexed by equivalence classes of chains of length three. It also defines proper totally compatible structures as those that can be written as a centroid-determined structure plus an annihilator-valued structure.

A sufficient condition is established under which every totally compatible structure is proper, and examples show both that non-proper structures can occur and that the sufficient condition is not necessary. The paper further proves that all totally compatible structures are annihilator-valued exactly when the poset has length at most two. The remaining problem is to give a complete characterization of the finite posets for which all totally compatible structures are proper.