Among the quadratic forms, playing an important role in modern mathematics, the Tits quadratic forms should be distinguished. Such quadratic forms were first introduced by P. Gabriel for any quiver in connection with the study of representations of quivers (also introduced by him). P. Gabriel proved that the connected quivers with positive Tits form coincide with the Dynkin quivers. This quadratic form is naturally generalized to a poset. The posets with positive quadratic Tits form (analogs of the Dynkin diagrams) were classified by the authors together with the P-critical posets (the smallest posets with non-positive quadratic Tits form). The quadratic Tits form of a P-critical poset is non-negative and corank of its symmetric matrix is 1. In this paper we study all posets with such two properties, which are called principal, related to equivalence of their quadratic Tits forms and those of Euclidean diagrams. In particular, one problem posted in 2014 is solved.

No abstract

We calculate the coefficients of transitiveness for all posets of MM-type (1, 2, 7) (i.e. posets, which are minimax equivalent to the poset (1, 2, 7)).

No abstract

In 2005, the authors described all introduced by them P-critical posets (minimal finite posets with the quadratic Tits form not being positive); up to isomorphism, their number is 132 (75 if duality is considered). Later (in 2014) A. Polak and D. Simson offered an alternative way of proving by using computer algebra tools. In doing this, they defined and described the Tits P-critical posets as a special case of the P-critical posets. In this paper we classify all the non-Tits P-critical posets without complex calculations and without using the list of all P-critical ones.