On combinatorial properties of minimal posets with nonnegative Tits quadratic form
In this paper, we study combinatorial properties of finite posets connected with the negativity of their Tits quadratic form. We calculate the coefficients of transitivity for all minimal posets with nonnegative Tits quadratic form (such posets are called \(NP\)-critical and their number is 115 up t...
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| author | Styopochkina, Maryna V. |
| author_facet | Styopochkina, Maryna V. |
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{
"author": "Maryna V. Styopochkina",
"institution": null
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| description | In this paper, we study combinatorial properties of finite posets connected with the negativity of their Tits quadratic form. We calculate the coefficients of transitivity for all minimal posets with nonnegative Tits quadratic form (such posets are called \(NP\)-critical and their number is 115 up to isomorphism and duality). Some relationships between these coefficients and the heights of posets are established. |
| doi_str_mv | 10.12958/adm2490 |
| first_indexed | 2026-07-09T01:00:16Z |
| format | Article |
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© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 41 (2026). Number 2, pp. 260–268
DOI:10.12958/adm2490
On combinatorial properties of minimal posets
with nonnegative Tits quadratic form
Maryna V. Styopochkina
Communicated by A. Petravchuk
Abstract. In this paper, we study combinatorial properties
of finite posets connected with the negativity of their Tits quadratic
form. We calculate the coefficients of transitivity for all minimal
posets with nonnegative Tits quadratic form (such posets are called
NP -critical and their number is 115 up to isomorphism and dua-
lity). Some relationships between these coefficients and the heights
of posets are established.
1. Introduction
When studying the representations of quivers, P. Gabriel [1] introduced a
quadratic form qQ(z) = qQ(z1, . . . , zn) for any finite quiver Q = (Q0, Q1)
with the set of vertices Q0 and the set of arrows Q1:
qQ(z) :=
∑︂
i∈Q0
z2i −
∑︂
i→j
zizj ,
where n = |Q0| and i → j runs through Q1. This form was called the
Tits quadratic form of the quiver Q. P. Gabriel proved that a connected
quiver is of finite representation type over a field if and only if the corre-
sponding non-oriented graph is one of the Dynkin diagrams, and if and
only if its Tits quadratic form is positive. This Gabriel’s work laid the
2020 Mathematics Subject Classification: 06A07, 11E04.
Key words and phrases: height, neighboring elements, Hasse diagram, Dynkin
diagram, Tits quadratic form, NP -critical poset, coefficient of transitivity.
https://doi.org/10.12958/adm2490
M. V. Styopochkina 261
foundations of a new direction in the representation theory. The direc-
tion deals with the investigation of the relationships between properties
of representations of various objects and properties of quadratic forms
associated with these objects.
In [2], Yu. A. Drozd showed that a finite poset S is of finite represen-
tation type if and only if its Tits quadratic form
qS(z) = z20 +
∑︂
i∈S
z2i +
∑︂
i<j; i,j∈S
zizj − z0
∑︂
i∈S
zi
is weakly positive, i.e., positive on the nonzero vectors with nonnegative
coordinates (matrix representations of posets were introduced in [3]; see
also [4]–[10]). In contrast to quivers, the posets with weakly positive and
with positive Tits quadratic forms do not coincide. Therefore investiga-
tions related to posets with positive Tits form are natural. In [11], the
author together with V. M. Bondarenko classified all posets having posi-
tive Tits quadratic form and the minimal posets with nonpositive Tits
quadratic form (they are called respectively positive and P -critical).
We have a similar situation for quivers and posets of tame type. Ac-
cording the papers [12,13], a connected quiver is of tame infinite type if
and only if the corresponding non-oriented graph is an extended Dynkin
diagram. On the other hand, the connected quivers with nonnegative,
but not positive, Tits form coincide with the quivers, the corresponding
graphs of which are extended Dynkin diagrams [14]. A poset S is of tame
type if and only if its quadratic Tits form is weakly nonnegative [15].
Since (in contrast to quivers) the classes of posets with weakly nonnega-
tive and with nonnegative Tits forms do not coincide, the investigations
related to posets with nonnegative Tits form are also natural. In [16], the
author together with V. M. Bondarenko classified minimal posets with
nonnegative Tits quadratic form, which were called NP -critical.
The present paper, which is a natural continuation of the papers
[17, 18] on positive and P -critical posets, is devoted to the investigation
of combinatorial properties of NP -critical posets.
2. Main result
Let S be a poset and S2
≺ := {(x, y) |x, y ∈ S, x ≺ y}. Elements x and y
are called neighboring if (x, y) ∈ S2
< and there no z satisfying x ≺ z ≺ y.
Denote by nw = nw(S) the order of the set S2
≺ and by ne = ne(S) the
number of pairs (x, y) of neighboring elements of S. On the language of
262 On combinatorial properties of minimal posets
the Hasse diagram H(S) of S (that represents S in the plane), ne is equal
to the number of all its edges and nw to the number of all its ways going
bottom-up, up to parallelism (i.e. with the same start and terminate
vertices). The ratio kt = kt(S) of the numbers nw − ne and nw is called
the coefficient of transitivity (if nw = 0, one assumes that kt = 0); see,
e.g., [18].
The main result of this paper is the following theorem (h denotes the
height of a poset, i.e. the maximum length of its subchain).
Theorem 1. Let S and T be NP -critical posets. Then
(1) kt(T ) > kt(S) if h(T ) > h(S) + 2;
(2) kt(T ) > kt(S)− 1
5 if h(T ) = h(S) + 1;
(3) kt(T ) > kt(S)− 1
50 if h(T ) = h(S) + 2.
Since dual posets have the same coefficient of transitivity, under the
proving of the theorem we can use the classification of NP -critical poset
not only up to isomorphism, but simultaneously also up to duality.
3. Classification of NP -critical posets
For subsets X,Y of a poset S, we denote by X ⊔ Y their direct sum (i.e.
X ∪ Y , where x ∈ X and y ∈ Y are always incomparable). From Dil-
worth’s theorem it follows that any poset can be represented in the form
⊔m
i=1Xi, where all Xi are chains and there is allowed additional relations
y < z for the elements belonging to different components (which it is na-
tural to write up to transitivity). By As, Bs, Cs we denote, respectively,
the chains a1 < . . . < as, b1 < . . . < bs, c1 < . . . < cs.
The NP -critical posets were classified in [16]. We formulate the cor-
responding theorem with another numbering of the posets.
Theorem 2. The NP -critical posets are exhausted, up to isomorphism
and duality, by the posets
of order 5of order 5of order 5
NP1.1 = A1 ⊔B2 ⊔ C2, a1 < b2, a1 < c2, b1 < c2, c1 < b2;
NP1.2 = A1 ⊔B1 ⊔ C1 ⊔D2, a1 < d2, b1 < d2, c1 < d2;
NP1.3 = A1 ⊔B1 ⊔ C1 ⊔D1 ⊔E1; NP2.1 = A2 ⊔B3, a1 < b2, b1 < a2;
NP2.2 = A1 ⊔B2 ⊔ C2, a1 < b2, b1 < c2, c1 < b2;
NP2.3 = A1 ⊔B3 ⊔C1, a1, c1 < b2; NP2.4 = A1 ⊔B3 ⊔C1, a1, c1 < b3;
NP2.5 = A1 ⊔B3 ⊔ C1, a1 < b3, b1 < a1, b1 < c1, c1 < b3;
NP2.6 = A1⊔B1⊔C1⊔D2; NP2.7 = A1⊔B1⊔C2⊔D1, b1 < c2, d1 < c2;
of order 7of order 7of order 7
M. V. Styopochkina 263
NP3.1 = A3 ⊔B4, a2 < b3; NP3.2 = A2 ⊔B5, a2 < b3;
NP3.3 = A2 ⊔B5, a2 < b4; NP3.4 = A3 ⊔B4, a2 < b2;
NP3.5 = A3⊔B4, a1 < b1, a3 < b3; NP3.6 = A3⊔B4, a1 < b1, a3 < b4;
NP3.7 = A2 ⊔B2 ⊔ C3; NP3.8 = A2 ⊔B2 ⊔ C3, b2 < c3;
NP3.9 = A1 ⊔B3 ⊔ C3, a1 < b3, b2 < c3;
NP3.10 = A2 ⊔B2 ⊔ C3, a1 < b2, b1 < c2, c1 < a2;
NP3.11 = A1 ⊔B2 ⊔ C4, b2 < c3; NP3.12 = A1 ⊔B2 ⊔ C4, b2 < c4;
NP3.13 = A1 ⊔B3 ⊔ C3, b1 < c1, b3 < c3;
NP3.14 = A1 ⊔B3 ⊔ C3, a1 < b3, b2 < c2;
NP3.15 = A1 ⊔B4 ⊔ C2, a1 < b4, b3 < c2;
of order 8of order 8of order 8
NP4.1 = A4 ⊔B4, a1 < b2; NP4.2 = A3 ⊔B5, a1 < b2;
NP4.3 = A3 ⊔B5, a1 < b3; NP4.4 = A3 ⊔B5, a3 < b5;
NP4.5 = A2 ⊔B6, a1 < b3; NP4.6 = A2 ⊔B6, a1 < b4;
NP4.7 = A1 ⊔B7, a1 < b4; NP4.8 = A1 ⊔B7, a1 < b5;
NP4.9 = A2⊔B6, a1 < b1, a2 < b4; NP4.10 = A2⊔B6, a1 < b1, a2 < b5;
NP4.11 = A3 ⊔B5, a2 < b1, a3 < b4; NP4.12 = A1 ⊔B3 ⊔ C4;
NP4.13 = A3⊔B1⊔C4, b1 < c4; NP4.14 = A1⊔B3⊔C4, a1 < b2, b1 < c2;
NP4.15 = A1 ⊔B3 ⊔ C4, a1 < b2, b1 < c3;
NP4.16 = A2 ⊔B2 ⊔ C4, a2 < b2, b1 < c3;
NP4.17 = A2⊔B2⊔C4, a2 < b2, b1 < c4; NP4.18 = A2⊔B1⊔C5, b1 < c4;
NP4.19 = A2⊔B1⊔C5, b1 < c5; NP4.20 = A2⊔B2⊔C4, b1 < c1, b2 < c4;
NP4.21 = A1 ⊔B2 ⊔ C5, a1 < b2, b1 < c3;
NP4.22 = A1⊔B2⊔C5, a1 < b2, b1 < c4; NP4.23 = A1⊔B1⊔C6, b1 < c4;
NP4.24 = A1⊔B1⊔C6, b1 < c5; NP4.25 = A1⊔B2⊔C5, b1 < c1, b2 < c4;
NP4.26 = A1 ⊔B2 ⊔ C5, b1 < c1, b2 < c5;
of order 9 (part 1)of order 9 (part 1)of order 9 (part 1)
NP5.1 = A4 ⊔B5, a1 < b4; NP5.2 = A4 ⊔B5, a2 < b5;
NP5.3 = A3 ⊔B6, a1 < b5; NP5.4 = A3 ⊔B6, a2 < b6;
NP5.5 = A2 ⊔B7, a1 < b2; NP5.6 = A2 ⊔B7, a1 < b6;
NP5.7 = A2 ⊔B7, a2 < b7; NP5.8 = A1 ⊔B8, a1 < b3;
NP5.9 = A1 ⊔B8, a1 < b7; NP5.10 = A2 ⊔B7, a1 < b1, a2 < b3;
NP5.11 = A2 ⊔B7, a1 < b1, a2 < b7;
NP5.12 = A3 ⊔B6, a2 < b1, a3 < b3;
NP5.13 = A4 ⊔B5, a3 < b1, a4 < b3; NP5.14 = A4 ⊔B1 ⊔ C4, b1 < c3;
NP5.15 = A4 ⊔B2 ⊔ C3, b1 < c1, b2 < c3;
NP5.16 = A1 ⊔B4 ⊔ C4, a1 < b2, b1 < c4;
NP5.17 = A3 ⊔B1 ⊔ C5, b1 < c3; NP5.18 = A5 ⊔B1 ⊔ C3, b1 < c3;
NP5.19 = A3 ⊔B2 ⊔ C4, b1 < c1, b2 < c3;
264 On combinatorial properties of minimal posets
NP5.20 = A1 ⊔B3 ⊔ C5, a1 < b2, b1 < c5;
NP5.21 = A1 ⊔B5 ⊔ C3, a1 < b2, b1 < c3; NP5.22 = A1 ⊔B2 ⊔ C6;
NP5.23 = A2⊔B1⊔C6, b1 < c3; NP5.24 = A2⊔B2⊔C5, b1 < c1, b2 < c3;
NP5.25 = A2 ⊔B3 ⊔ C4, b2 < c1, b2 < c3;
NP5.26 = A1 ⊔B2 ⊔ C6, a1 < b2, b1 < c2;
NP5.27 = A1⊔B2⊔C6, a1 < b2, b1 < c6; NP5.28 = A1⊔B1⊔C7, b1 < c3;
NP5.29 = A1⊔B1⊔C7, b1 < c7; NP5.30 = A1⊔B2⊔C6, b1 < c1, b2 < c3;
NP5.31 = A1 ⊔B3 ⊔ C5, b2 < c1, b2 < c3;
of order 9 (part 2)of order 9 (part 2)of order 9 (part 2)
NP6.1 = A4⊔B5, a1 < b4, a2 < b5; NP6.2 = A3⊔B6, a1 < b5, a2 < b6;
NP6.3 = A2⊔B7, a1 < b2, a2 < b3; NP6.4 = A2⊔B7, a1 < b2, a2 < b7;
NP6.5 = A2⊔B7, a1 < b6, a2 < b7; NP6.6 = A6⊔B3, a1 < b2, a6 < b3;
NP6.7 = A6 ⊔B3, a5 < b2, a6 < b3;
NP6.8 = A3 ⊔B6, a1 < b1, a2 < b2, a3 < b3;
NP6.9 = A4 ⊔B5, a2 < b1, a3 < b2, a4 < b3;
NP6.10 = A3 ⊔B2 ⊔ C4, b1 < c2, b2 < c3;
NP6.11 = A4 ⊔B2 ⊔ C3, b1 < c2, b2 < c3;
NP6.12 = A1 ⊔B4 ⊔ C4, a1 < b3, b1 < c4;
NP6.13 = A3 ⊔B2 ⊔ C4, a2 < b2, b1 < c2;
NP6.14 = A3 ⊔B3 ⊔ C3, b1 < c1, b2 < c2, b3 < c3;
NP6.15 = A3 ⊔B3 ⊔ C3, a3 < b3, b1 < c1, b2 < c2;
NP6.16 = A5⊔B2⊔C2, a1 < b2; NP6.17 = A2⊔B2⊔C5, b1 < c2, b2 < c3;
NP6.18 = A1 ⊔B3 ⊔ C5, a1 < b3, b1 < c5;
NP6.19 = A1 ⊔B5 ⊔ C3, a1 < b3, b1 < c3;
NP6.20 = A2 ⊔B2 ⊔ C5, a1 < b2, b1 < c2;
NP6.21 = A2 ⊔B3 ⊔ C4, b1 < c1, b2 < c2, b3 < c3;
NP6.22 = A2 ⊔B3 ⊔ C4, a2 < b3, b1 < c1, b2 < c2;
NP6.23 = A4 ⊔B3 ⊔ C2, a4 < b3, b1 < c1, b2 < c2;
NP6.24 = A1 ⊔B2 ⊔ C6, b1 < c2; NP6.25 = A1 ⊔B2 ⊔ C6, b1 < c6;
NP6.26 = A1 ⊔B2 ⊔ C6, b1 < c2, b2 < c3;
NP6.27 = A1 ⊔B3 ⊔ C5, a1 < b3, b1 < c1;
NP6.28 = A1 ⊔B6 ⊔ C2, a1 < b3, b1 < c2;
NP6.29 = A1 ⊔B3 ⊔ C5, b1 < c1, b2 < c2, b3 < c3;
NP6.30 = A1 ⊔B4 ⊔ C4, b2 < c1, b3 < c2, b4 < c3;
NP6.31 = A1 ⊔B3 ⊔ C5, a1 < b3, b1 < c1, b2 < c2;
NP6.32 = A1 ⊔B7 ⊔ C1, a1 < b3, b1 < c1;
NP6.33 = A1 ⊔B7 ⊔ C1, a1 < b7, b1 < c1.
M. V. Styopochkina 265
4. Calculation of the transitivity coefficients.
Proof of Theorem 1
We first calculate the coefficients of transitivity kt of the NP -critical
posets, which are indicated in Theorem 2. The coefficients kt are calcu-
lated up to the fifth decimal place. If the number of decimal places is
less than five, then the decimal fraction is finite, and if it is five, then
infinite. When two decimal fractions are equal up to five digits, then
they are generally equal.
The following holds for the posets from Theorem 2:
N h ne nw kt N h ne nw kt
NP1.3 1 0 0 0 NP1.2 2 4 4 0
NP2.2 2 5 5 0 NP2.4 3 4 5 0,2
NP1.1 2 6 6 0 NP2.1 3 5 7 0,28571
NP2.6 2 1 1 0 NP2.5 3 5 7 0,28571
NP2.7 2 3 3 0 NP2.3 3 4 7 0,42857
N h ne nw kt N h ne nw kt
NP3.7 3 4 5 0,2 NP3.1 4 6 13 0,53846
NP3.10 3 7 9 0,22222 NP3.11 4 5 11 0,54545
NP3.8 3 5 7 0,28571 NP3.6 5 7 15 0,53333
NP3.9 3 6 9 0,33333 NP3.5 5 7 17 0,58824
NP3.12 4 5 9 0,44444 NP3.3 5 6 15 0,6
NP3.13 4 6 11 0,45455 NP3.4 5 6 15 0,6
NP3.14 4 6 11 0,45455 NP3.2 5 6 17 0,64706
NP3.15 4 6 11 0,45455
N h ne nw kt N h ne nw kt
NP4.17 4 7 11 0,36364 NP4.18 5 6 13 0,53846
NP4.13 4 6 10 0,4 NP4.3 5 7 16 0,5625
NP4.16 4 7 12 0,41667 NP4.4 5 7 16 0,5625
NP4.12 4 5 9 0,44444 NP4.2 5 7 17 0,58824
NP4.15 4 7 13 0,46154 NP4.26 6 7 17 0,58824
NP4.14 4 7 14 0,5 NP4.25 6 7 18 0,61111
NP4.1 4 7 15 0,53333 NP4.6 6 7 19 0,63158
NP4.20 5 7 13 0,46154 NP4.24 6 6 17 0,64706
NP4.19 5 6 12 0,5 NP4.5 6 7 20 0,65
NP4.22 5 7 14 0,5 NP4.23 6 6 18 0,66667
NP4.21 5 7 15 0,53333 NP4.10 7 8 24 0,66667
NP4.9 7 8 25 0,68 NP4.8 7 7 24 0,70833
NP4.11 7 8 25 0,68 NP4.7 7 7 25 0,72
266 On combinatorial properties of minimal posets
N h ne nw kt N h ne nw kt
NP5.15 4 8 14 0,42857 NP5.26 6 8 22 0,63636
NP5.14 4 7 14 0,5 NP5.23 6 7 20 0,65
NP5.16 4 8 16 0,5 NP5.6 7 8 24 0,66667
NP5.18 5 7 14 0,5 NP5.7 7 8 24 0,66667
NP5.19 5 8 16 0,5 NP5.29 7 7 22 0,68182
NP5.20 5 8 16 0,5 NP5.30 7 8 26 0,69231
NP5.1 5 8 18 0,55556 NP5.31 7 8 26 0,69231
NP5.2 5 8 18 0,55556 NP5.5 7 8 28 0,71429
NP5.21 5 8 18 0,55556 NP5.28 7 7 26 0,73077
NP5.17 5 7 16 0,5625 NP5.11 8 9 30 0,7
NP5.27 6 8 18 0,55556 NP5.9 8 8 30 0,73333
NP5.3 6 8 20 0,6 NP5.10 8 9 34 0,73529
NP5.4 6 8 20 0,6 NP5.12 8 9 34 0,73529
NP5.24 6 8 20 0,6 NP5.13 8 9 34 0,73529
NP5.25 6 8 20 0,6 NP5.8 8 8 34 0,76471
NP5.22 6 6 16 0,625
N h ne nw kt N h ne nw kt
NP6.11 4 8 13 0,38462 NP6.17 5 8 19 0,57895
NP6.14 4 9 15 0,4 NP6.2 6 9 21 0,57143
NP6.10 4 8 15 0,46667 NP6.27 6 8 19 0,57895
NP6.12 4 8 15 0,46667 NP6.25 6 7 17 0,58824
NP6.13 4 8 15 0,46667 NP6.31 6 9 23 0,60870
NP6.15 4 9 17 0,47059 NP6.28 6 8 21 0,61905
NP6.20 5 8 14 0,42857 NP6.29 6 9 25 0,64
NP6.16 5 7 13 0,46154 NP6.30 6 9 25 0,64
NP6.18 5 8 15 0,46667 NP6.24 6 7 21 0,66667
NP6.23 5 9 17 0,47059 NP6.26 6 8 25 0,68
NP6.1 5 9 19 0,52632 NP6.5 7 9 25 0,64
NP6.21 5 9 19 0,52632 NP6.6 7 9 25 0,64
NP6.22 5 9 19 0,52632 NP6.33 7 8 23 0,65217
NP6.19 5 8 17 0,52941 NP6.4 7 9 29 0,68966
NP6.7 7 9 29 0,68966 NP6.32 7 8 27 0,70370
NP6.8 7 10 33 0,69697 NP6.3 7 9 33 0,72727
NP6.9 7 10 33 0,69697
For calculation the coefficients from the tables we need the following
lemmas (see [17] or [18]).
Lemma 1. Let S = S1 ⊔ S2. Then
ne(S) = ne(S1) + ne(S2), nw(S) = nw(S1) + nw(S2).
M. V. Styopochkina 267
Lemma 2. Let S = Am. Then
ne(S) = m− 1, nw(S) =
(m−1)m
2 .
Lemma 3. Let S = {Am ⊔Bn, ai < bj}. Then
(a) ne(S) = m+ n− 1;
(b) nw(S) =
(m−1)m+(n−1)n
2 + i(n− j + 1).
Lemma 4. Let S = {Am ⊔ Bn, ai < bj , ai′ < bj′}, where i < i′, j < j′.
Then
(a) ne(S) = m+ n;
(b) nw(S) =
(m−1)m+(n−1)n
2 + i′(n− j′ + 1) + i(j′ − j).
Lemma 5. Let S = {Am⊔Bn⊔Cs, ai < bj , bj′ < ck, where j > j′. Then
(a) ne(S) = m+ n+ s− 1;
(b) nw(S) =
(m−1)m+(n−1)n+s(s−1)
2 + i(n− j + 1) + j′(s− k + 1).
Lemma 6. Let S = {Am ⊔Bn, ai < bj , ai+1 < bj+1, ai+2 < bj+2. Then
(a) ne(S) = m+ n+ 1;
(b) nw(S) =
(m−1)m+(n−1)n
2 + (i+ 2)n− i(j − 1)− (2j + 1).
Lemma 7. Let S = {Am⊔Bn⊔Cs, ai < bj , bj′ < ck, bj′+1 < ck+1, where
j > j′ + 1. Then
(a) ne(S) = m+ n+ s;
(b) nw(S) =
(m−1)m+(n−1)n+s(s−1)
2 + i(n− j+1)+(j′+1)(s−k)+ j′.
Namely, the transitivity coefficients indicated in the tables follows
from the following Lemmas:
Lemmas 1, 2 for NP3.7, Lemma 3 for NP3.1 − NP3.4, Lemmas 1–3
for NP3.8, NP3.11, NP3.12, Lemma 4 for NP3.5 −NP3.6, Lemmas 1, 2, 4
for NP3.13, Lemma 5 for NP3.9, NP3.14, NP3.15;
Lemmas 1, 2 for NP4.12, Lemma 3 for NP4.1−NP4.8, Lemmas 1–3
for NP4.13, NP4.18 − NP4.19, NP4.23 − NP4.24, Lemma 4 for NP4.9 −
NP4.11, Lemmas 1, 2, 4 forNP4.20, NP4.25, NP4.26, Lemma 5 forNP4.14−
NP4.17, NP4.21 −NP4.22;
Lemmas 1, 2 for NP5.22, Lemma 3 for NP5.1−NP5.9, Lemmas 1–3 for
NP5.14, NP5.17−NP5.18, NP5.23, NP5.28−NP5.29, Lemma 4 for NP5.10−
NP5.13, Lemmas 1, 2, 4 for NP5.15, NP5.19, NP5.24 − NP5.25, NP5.30 −
NP5.31, Lemma 5 for NP5.16, NP5.20 −NP5.21, NP5.26 −NP5.27;
Lemmas 1–3 for NP6.16, NP6.24 − NP6.25; Lemma 4 for NP6.1 −
NP6.7; Lemmas 1, 2, 4 for NP6.10 − NP6.11, NP6.17, NP6.26; Lemma 5
for NP6.12 −NP6.13, NP6.18 −NP6.20, NP6.27 −NP6.28, NP6.32 −NP6.33;
268 On combinatorial properties of minimal posets
Lemma 6 for NP6.8 −NP6.9; Lemmas 1, 6 for NP6.14, NP6.21, NP6.29 −
NP6.30; Lemma 7 for NP6.15, NP6.22 −NP6.23, NP6.31.
The coefficients in the cases NP1i, NP2j and NP3.10 are proved by
direct calculations.
Given the lexicographic notation in the tables, it is easy to check that
Theorem 1 follows from them.
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Contact information
M. V. Styopochkina Polissia National University, Staryi Boulevard,
7, 10008 Zhytomyr, Ukraine
E-Mail: stmar@ukr.net
Received by the editors: 25.04.2026.
https://doi.org/10.1070/IM1973v007n04ABEH001975
https://doi.org/10.1007/s11253-009-0245-6
https://doi.org/10.1007/s10958-023-06624-6
https://doi.org/10.12958/adm2151
Maryna V. Styopochkina
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| id | admjournalluguniveduua-article-2490 |
| institution | Algebra and Discrete Mathematics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-09T01:00:16Z |
| publishDate | 2026 |
| publisher | Lugansk National Taras Shevchenko University |
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| spelling | admjournalluguniveduua-article-24902026-07-08T07:55:33Z On combinatorial properties of minimal posets with nonnegative Tits quadratic form Styopochkina, Maryna V. height, neighboring elements, Hasse diagram, Dynkin diagram, Tits quadratic form, \(NP\)-critical poset, coefficient of transitivity 06A07, 11E04 In this paper, we study combinatorial properties of finite posets connected with the negativity of their Tits quadratic form. We calculate the coefficients of transitivity for all minimal posets with nonnegative Tits quadratic form (such posets are called \(NP\)-critical and their number is 115 up to isomorphism and duality). Some relationships between these coefficients and the heights of posets are established. Lugansk National Taras Shevchenko University 2026-07-08 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2490 10.12958/adm2490 Algebra and Discrete Mathematics; Vol 41, No 2 (2026) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2490/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2490/1380 Copyright (c) 2026 Algebra and Discrete Mathematics |
| spellingShingle | height neighboring elements Hasse diagram Dynkin diagram Tits quadratic form \(NP\)-critical poset coefficient of transitivity 06A07 11E04 Styopochkina, Maryna V. On combinatorial properties of minimal posets with nonnegative Tits quadratic form |
| title | On combinatorial properties of minimal posets with nonnegative Tits quadratic form |
| title_full | On combinatorial properties of minimal posets with nonnegative Tits quadratic form |
| title_fullStr | On combinatorial properties of minimal posets with nonnegative Tits quadratic form |
| title_full_unstemmed | On combinatorial properties of minimal posets with nonnegative Tits quadratic form |
| title_short | On combinatorial properties of minimal posets with nonnegative Tits quadratic form |
| title_sort | on combinatorial properties of minimal posets with nonnegative tits quadratic form |
| topic | height neighboring elements Hasse diagram Dynkin diagram Tits quadratic form \(NP\)-critical poset coefficient of transitivity 06A07 11E04 |
| topic_facet | height neighboring elements Hasse diagram Dynkin diagram Tits quadratic form \(NP\)-critical poset coefficient of transitivity 06A07 11E04 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2490 |
| work_keys_str_mv | AT styopochkinamarynav oncombinatorialpropertiesofminimalposetswithnonnegativetitsquadraticform |