Coefficients of transitiveness of P-critical posets
We introduce an invariant of a finite poset, called the coefficient of transitiveness, and calculate it for all P-critical posets, which are an analog of the extended Dynkin diagrams.
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| author | Bondarenko, V. M. Styopochkina, M. V. Bondarenko, V. M. Styopochkina, M. V. |
| author_facet | Bondarenko, V. M. Styopochkina, M. V. Bondarenko, V. M. Styopochkina, M. V. |
| author_institution_txt_mv | [
{
"author": "V. M. Bondarenko",
"institution": "Institute of Mathematics of NAS of Ukraine"
},
{
"author": "M. V. Styopochkina",
"institution": "Zhytomyr National University of Agriculture and Ecology"
}
] |
| author_sort | Bondarenko, V. M. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-13T13:58:16Z |
| description | We introduce an invariant of a finite poset, called the coefficient of transitiveness, and calculate it for all P-critical posets, which are an analog of the extended Dynkin diagrams. |
| first_indexed | 2026-08-04T01:00:35Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2017, том 14, № 1, 46–51
УДК 512.64 + 512.56
V.M. Bondarenko 1, M.V. Styopochkina 2
1(Institute of Mathematics of NAS of Ukraine, Kyiv)
2(Zhytomyr National University of Agriculture and Ecology, Zhytomyr)
1 vit-bond@imath.kiev.ua, 2 StMar@ukr.net
Coefficients of transitiveness of
PPP -critical posets
Dedicated to Prof. Yu. B. Zelinskii on the occasion of his 70th birthday
Ми вводимо iнварiант для будь-якої скiнченної частково впорядкованої
множини, який називаємо коефiцiєнтом транзитивностi, i обчислюємо
його для всiх P -критичних частково впорядкованих множин, якi є ана-
логом розширених дiаграм Динкiна.
We introduce an invariant of a finite poset, called the coefficient of transi-
tiveness, and calculate it for all P -critical posets, which are an analog of
the extended Dynkin diagrams.
1. Introduction. In [1], for a finite quiver (directed graph) Q with
the set of vertices Q0 and the set of arrows Q1, P. Gabriel introduced a
quadratic form qQ : Zn −→ Z, n = |Q0|, called by him the quadratic Tits
form of the quiver Q:
qQ(z) = qQ(z1, . . . , zn) :=
∑
i∈Q0
z2i −
∑
i→j
zizj ,
where i → j runs through the set Q1. He proved that the quiver Q has
finite representation type over a field k (i.e., finitely many indecomposable
representations, up to isomorphism) if and only if its Tits form is positive.
This Gabriel’s work laid the foundations of a new direction in the theory
of algebra dealing with the investigation of the relationships between
c© V.M. Bondarenko, M.V. Styopochkina, 2017
Coefficients of transitiveness of P -critical posets 47
the properties of representations of various objects and the properties of
quadratic forms associated with these objects.
The above quadratic form is naturally generalized to a (finite) poset
S 63 0:
qS(z) = z20 +
∑
i∈S
z2i +
∑
i<j,i,j∈S
zizj − z0
∑
i∈S
zi.
In [2], Yu.A. Drozd showed that a poset S has finite representation type
if and only if its Tits form is weakly positive, i. e., takes positive value
on any nonzero vector with nonnegative coordinates (representations of
posets were introduced by L. A. Nazarova and A. V. Roiter in [3]).
For posets, in contrast to quivers, the sets of those with weakly positive
and with positive Tits forms do not coincide. Therefore the investigations
of posets with positive Tits form seems to be quite natural; notice that they
are analogs of the Dynkin diagrams. Posets of this type were studied by
the authors (from different points of view) in many papers (cf., e.g., [4–7]).
In particular, in [5] it is introduced the notion of P -critical poset: a
poset S is called P -critical if its Tits quadratic form is not positive, but
that of any proper subset of S is positive. If one gives the similar definition
for a quiver. then the set of P -critical quivers coincides with the set of
extended Dynkin diagrams. So the P -critical posets are analogs of the
extended Dynkin diagrams. All such posets are classified in [5] (see the
next section).
The present paper is devoted to the investigation of combinatorial
properties of P -critical posets.
Let S be a finite poset and S2
< := {(x, y) |x, y ∈ S, x < y}. If (x, y) ∈ S2
<
and there is no z satisfying x < z < y, then one says that x and y are
neighboring. We put nw = nw(S) := |S2
<| and denote by ne = ne(S) the
number of pairs of neighboring elements. On the language of the Hasse
diagram H(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 paths, up to parallelity,
going bottom-up (two path is called parallel if they start and terminate at
the same vertices). The ratio kt = kt(S) of the numbers nw − ne and nw
we call the coefficient of transitiveness of S. If nw = 0 (then ne = 0), we
assume kt = 0.
The aim of this paper is to calculate kt for all P -critical posets.
2. Preliminary. Indicate the table from [5], in which are written
all P -critical posets.
48 V.M. Bondarenko, M.V. Styopochkina
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Coefficients of transitiveness of P -critical posets 49
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q q q q
75
Note that the P -critical posets are written up to isomorphism and
anti-isomorphism.
50 V.M. Bondarenko, M.V. Styopochkina
3. Main result. We write all the coefficients of transitiveness kt up
to the second decimal place.
Theorem 1. The following holds for P -critical posets 1 – 75:
N ne nw kt N ne nw kt N ne nw kt
1 4 4 0,00 26 8 19 0,58 51 6 11 0,45
2 5 10 0,50 27 8 19 0,58 52 7 11 0,36
3 5 11 0,55 28 8 16 0,50 53 6 10 0,40
4 6 11 0,45 29 8 15 0,47 54 6 9 0,33
5 6 18 0,67 30 3 3 0,00 55 7 16 0,56
6 7 18 0,61 31 3 3 0,00 56 7 13 0,46
7 6 14 0,57 32 4 6 0,33 57 7 12 0,42
8 6 12 0,50 33 5 7 0,29 58 7 14 0,50
9 6 12 0,50 34 6 6 0,00 59 7 11 0,36
10 7 26 0,73 35 4 6 0,33 60 7 13 0,46
11 8 26 0,69 36 5 12 0,58 61 7 12 0,42
12 8 26 0,69 37 6 12 0,50 62 7 15 0,53
13 7 23 0,70 38 5 8 0,38 63 7 20 0,65
14 8 23 0,65 39 6 10 0,40 64 7 17 0,59
15 7 21 0,67 40 6 10 0,40 65 7 12 0,42
16 7 18 0,61 41 6 8 0,25 66 7 11 0,36
17 7 18 0,61 42 5 11 0,55 67 7 18 0,61
18 7 15 0,53 43 6 19 0,68 68 8 18 0,56
19 7 15 0,53 44 7 19 0,63 69 7 13 0,46
20 7 14 0,50 45 7 19 0,63 70 8 13 0,38
21 8 25 0,68 46 6 16 0,63 71 7 10 0,30
22 9 25 0,64 47 6 15 0,60 72 8 17 0,53
23 9 25 0,64 48 6 12 0,50 73 8 14 0,43
24 8 22 0,64 49 6 14 0,57 74 8 13 0,38
25 8 22 0,64 50 7 14 0,50 75 0 0 0,00
The proof is carried out by direct calculations.
Here we do not analyze fully this result, and formulate only one
corollary from the theorem.
Recall that an element of a poset T is called nodal, if it is comparable
with all elements of T . Obviously, each element of T is nodal iff T is a chain.
It follows from the table of Section 2 that any P -critical poset S is uniquely
represented in the form S = S−0 0∪S1∪S+
0 where S−0 , S+
0 are chains (maybe
Coefficients of transitiveness of P -critical posets 51
empty), S1 does not contain nodal elements and S−0 < S1 < S+
0 (X < Y
means that x < y for any x ∈ X, y ∈ Y ). Then S0 = S−0 ∪S
+
0 is the set of
all nodal elements of S.
Corollary 1. Let S be a P -critical poset. Then the following conditions
are equivalent :
a) kt(S) ≥ kt(T ) for any P -critical poset T ;
b) |S0| ≥ |T0| for any P -critical poset T , and S−0 or S+
0 is empty.
References
[1] Gabriel P. Unzerlegbare Darstellungen // Manuscripta Math. — 1972. —
6. — P. 71 – 103.
[2] Drozd Yu.A. Coxeter transformations and representations of partially
ordered sets // Funkc. Anal. i Priložen. — 1974. — 8, No. 3. — P. 34 –
42 (in Russian).
[3] Nazarova L.A., Roiter A.V. Representations of posets // Zap. Nauch.
Semin. LOMI. — 1972. — 28. — P. 5 – 31 (in Russian).
[4] Bondarenko V.M., Styopochkina M. V. On posets of width two with posi-
tive Tits form // Algebra and Discr. Math. — 2005. — No. 2. — P. 11 – 22.
[5] Bondarenko V.M., Stypochkina M.V. (Min, max)-equivalence of partially
ordered sets and the Tits quadratic form // Zb. Pr. Inst. Mat. NAN Ukr.
/Problems of Analysis and Algebra/. — K.: Institute of Mathematics of
NAS of Ukraine, 2005. — 2, No. 3. — P. 18 – 58 (in Russian).
[6] Bondarenko V.M., Stypochkina M.V. On finite posets of inj-finite type and
their Tits forms // Algebra and Discr. Math. — 2006. — No. 2. — P. 17 – 21.
[7] Bondarenko V.M., Stypochkina M.V. On the serial posets with positive-
definite quadratic Tits form // Nelin. Kolyv. — 2006. — 9, No. 3. — P. 320 –
325 (Ukrainian).
|
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| last_indexed | 2026-08-04T01:00:35Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
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| spelling | oai:trim.imath.kiev.ua:article-1012018-02-13T13:58:16Z Coefficients of transitiveness of P-critical posets Bondarenko, V. M. Styopochkina, M. V. Bondarenko, V. M. Styopochkina, M. V. We introduce an invariant of a finite poset, called the coefficient of transitiveness, and calculate it for all P-critical posets, which are an analog of the extended Dynkin diagrams. Інститут математики НАН України 2017-04-25 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/101 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 14 No. 1 (2017): Vol. 14 No. 1 (2017): Analysis and Applications; 46-51 Сборник Трудов Института математики НАН Украины; Том 14 № 1 (2017): Том 14 № 1 (2017): Анализ и приложения; 46-51 Збірник Праць Інституту математики НАН України; Том 14 № 1 (2017): Аналіз та застосування; 46-51 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/101/89 Авторське право (c) 2017 Праці Інституту математики НАН України |
| spellingShingle | Bondarenko, V. M. Styopochkina, M. V. Bondarenko, V. M. Styopochkina, M. V. Coefficients of transitiveness of P-critical posets |
| title | Coefficients of transitiveness of P-critical posets |
| title_full | Coefficients of transitiveness of P-critical posets |
| title_fullStr | Coefficients of transitiveness of P-critical posets |
| title_full_unstemmed | Coefficients of transitiveness of P-critical posets |
| title_short | Coefficients of transitiveness of P-critical posets |
| title_sort | coefficients of transitiveness of p-critical posets |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/101 |
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