Minimax isomorphism algorithm and primitive posets

The notion of minimax equivalence of posets, and a close notion of minimax isomorphism, introduced by the author are widely used in the study of quadratic Tits forms (in particular, for the description of $P$-critical and $P$-supercritical posets). In this paper, for an important special case, we modify an algorithm of classifying all posets minimax isomorphic to a given one (described earlier by the author together with M. V. Stepochkina) by introducing the concept of weak isomorphism.