Three-equipped posets and their representations and corepresentations (inseparable case)

We define three-equipped posets, i.e. partially ordered sets equipped with three kinds of binary relations, and their representations and corepresentations over an inseparable cubic field extension F⊂G, in characteristic 3. These representations and corepresentations lead to some matrix problems of mixed type over the pair of fields (F, G). Through these problems of linear algebra, we completely describe in evident matrix form the indecomposables for some critical three-equipped posets, reducing the task for the rest of the critical sets to some matrix problems containing the pseudolinear pencil problem as a subproblem.