We study the relationship between the (min, max)-equivalence of posets and properties of their quadratic Tits form related to nonnegative definiteness. In particular, we prove that the Tits form of a poset S is nonnegative definite if and only if the Tits form of any poset (min, max)-equivalent to S is weakly nonnegative.

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We describe all serial posets with positive-definite quadratic Tits form and prove that any poset of order greater than 7 with positive-definite Tits form is serial.Quadratic forms are encountered in the solution of various problems in algebra, geometry, the theory of differential and integral equations, operator theory, and other fields of mathematics (see, e.g., ). Among them, an important role is played by quadratic Tits forms for oriented graphs, posets, algebras, etc. In the present paper, we consider exactly these forms. Formulation of the Main ResultFirst, recall some definitions. Let S be a finite or an infinite poset. We say that S is the sum of its subsets A 1 , . . . , A s and writeIf all elements of different terms are always incomparable, then S is called the direct sum of the indicated subsets. Further, according to [27], the sum S = A 1 + . . . + A s is called one-sided if (up to enumeration of terms) one has i < j whenever there exist elements b ∈ A i and c ∈ A j for i = j such that b < c. According to [27], the sum S = A 1 + . . . + A s is called minimax if it follows from the relation x < y, where x and y belong to different terms, that x and y are, respectively, minimal and maximal elements of the set S. Formally, a direct sum is minimax. Nevertheless, considering minimax sums in what follows, we always assume for convenience that they are not direct.A subset of a poset S is understood as a complete partially ordered subset, i.e., a partial order on it is induced by a partial order on S.The form q S (z) : Z S∪0 0 → Z defined by the equality

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We consider the Bondarenko hypothesis about the structure of finite posets with positive-definite Tits quadratic form. We prove that the hypothesis holds for a fixed natural number k > 8 if it holds for the number k − 1.Quadratic forms are encountered in the solution of numerous problems in algebra, the theory of differential and integral equations, functional analysis, and other fields of mathematics (see, e.g., [1] and references therein; see also [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). Among quadratic forms, an important role is played by Tits quadratic forms for various objects (graphs, posets, algebras, etc.). In the present paper, we study exactly these forms.In [18], Gabriel associated a quiver (an oriented graph) with a certain quadratic form and called it a Tits quadratic form. This form plays an important role in the theory of finite-dimensional representations of graphs. In particular, it was shown in [18] that a graph has a finite type (i.e., it has a finite, up to an isomorphism, number of indecomposable representations) if and only if the corresponding Tits form is positive definite. This paper originated a new trend in algebra that studies the relationship between the properties of representations of various objects and the properties of the corresponding quadratic forms.The next step in this direction was made in the works of Brenner [19] and Drozd [20], where Tits quadratic forms were determined for quivers with relations and for posets, respectively. In the general case, the Tits form for matrix problems without relations was introduced by Kleiner and Roiter in [21].It was shown in [20] that a poset has a finite type if and only if its Tits form is weakly positive (representations of posets were introduced in [22]). Positive-definite forms are not distinguished in this case, but in the further investigation of these representations positive-definite Tits forms play an important role [23]. For the first time, these forms were described in [1] (see also [24]), where infinite posets were considered. In [25], Bondarenko considered the case of finite sets and formulated a hypothesis concerning positive-definite Tits forms. This hypothesis is considered in the present paper. Formulation of a HypothesisIn the present paper, we consider finite posets. A subset X of a poset S is always understood as a subset complete with respect to the partial-order condition (i.e., if a, b ∈ X, then a ≥ b in X if and only if a ≥ b in S).If a poset S is the union of its pairwise disjoint subsets A 1 , . . . , A s , s ≥ 1, then we say that S is the sum of these subsets and write S = A 1 + . . . + A s . If elements of different summands are always incomparable, then S is called the direct sum of these subsets.We also recall several definitions from [1]. Let a poset S be the sum of subsets A 1 , . . . , A s . This sum is called one-sided if (up to reenumeration of summands) one has i < j whenever there exist elements b ∈ A i and c ∈ A j , i = j, such that b < c. Further, the sum S = A 1 + . . . + A s is ...