RESEARCH ARTICLE

The classification of serial posets with the non-negative quadratic Tits form being principal

Vitalij M. Bondarenko^a, Marina V. Styopochkina<sup>b

a</sup> Institute of Mathematics, Tereshchenkivska str., 3, 01024 Kyiv, Ukraine
^b Zhytomyr National Agroecological Univ., Staryi Boulevard, 7, 10008 Zhytomyr, Ukraine

Abstract: Using (introduced by the first author) the method of (min, max)-equivalence, we classify all serial principal posets, i.e. the posets $S$ satisfying the following conditions: (1) the quadratic Tits form $q_S(z)\colon\mathbb{Z}^{|S|+1}\to\mathbb{Z}$ of $S$ is non-negative; (2) $\operatorname{Ker}q_S(z):=\{t\mid q_S(t)=0\}$ is an infinite cyclic group (equivalently, the corank of the symmetric matrix of $q_S(z)$ is equal to $1$); (3) for any $m\in\mathbb{N}$, there is a poset $S(m)\supset S$ such that $S(m)$ satisfies (1), (2) and $|S(m)\setminus S|=m$.

Keywords: quiver, serial poset, principal poset, quadratic Tits form, semichain, minimax equivalence, one-side and two-side sums, minimax sum.

MSC: 15B33, 15A30

Received: 14.03.2019

Language: English