On classifying the non-Tits P-critical posets
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...
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irk-123456789-1887462023-03-15T01:27:14Z On classifying the non-Tits P-critical posets Bondarenko, V.M. Styopochkina, M.V. 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. 2021 Article On classifying the non-Tits P-critical posets / V.M. Bondarenko, M.V. Styopochkina // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 185-196. — Бібліогр.: 16 назв. — англ. 1726-3255 DOI:10.12958/adm1912 2020 MSC: 15B33, 15A30 http://dspace.nbuv.gov.ua/handle/123456789/188746 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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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. |
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Bondarenko, V.M. Styopochkina, M.V. |
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Bondarenko, V.M. Styopochkina, M.V. On classifying the non-Tits P-critical posets Algebra and Discrete Mathematics |
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Bondarenko, V.M. Styopochkina, M.V. |
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On classifying the non-Tits P-critical posets |
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On classifying the non-Tits P-critical posets |
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On classifying the non-Tits P-critical posets |
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On classifying the non-Tits P-critical posets |
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On classifying the non-Tits P-critical posets |
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on classifying the non-tits p-critical posets |
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Інститут прикладної математики і механіки НАН України |
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On classifying the non-Tits P-critical posets / V.M. Bondarenko, M.V. Styopochkina // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 185-196. — Бібліогр.: 16 назв. — англ. |
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Algebra and Discrete Mathematics |
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2025-07-16T10:56:53Z |
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© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 32 (2021). Number 2, pp. 185ś196
DOI:10.12958/adm1912
On classifying the non-Tits P -critical posets
V. M. Bondarenko and M. V. Styopochkina
Communicated by A. P. Petravchuk
Abstract. In 2005, the authors described all introduced by
them P -critical posets (minimal őnite 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 deőned 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.
1. Introduction
Quadratic Tits forms play an important role in modern representation
theory and its applications. They were őrst introduced by P. Gabriel for
quivers [1].
Let Q be a őnite quiver with the set of vertices Q0 and the set of
arrows Q1. By deőnition, the quadratic Tits form of the quiver Q is the
quadratic form qQ : Zn → Z, n = |Q0|, given by the following equality:
qQ(z) =
∑
i∈Q0
z2i −
∑
i→j
zizj ,
where i → j runs through the set Q1 (i.e., multiplied by -1, the difference
between the number of parameters of all representations of any őxed
2020 MSC: 15B33, 15A30.
Key words and phrases: Hasse diagram, Kleiner’s poset, minimax equivalence,
quadratic Tits form, 0-balanced subposet, P -critical poset, Tits P -critical poset.
186 On classifying the non-Tits P -critical posets
vector-dimension z = {zi | i ∈ Q0} and the number of parameters of the
matrix group acting on these representations). For an undirected graph G
the quadratic Tits form qG is, by deőnition, the quadratic Tits form of
a quiver Q = Q(G, ε) with some orientation ε on the edges of G (qG does
not depend on choice of ε).
In [1] P. Gabriel proved that, for a connected quiver Q, the following
conditions are equivalent:
(1q) Q is of őnite representation type over a őeld k;
(2q) the quadratic Tits form of Q is positive;
(3q) the underlying graph of Q is a (simply faced) Dynkin diagram.
Now let S be a őnite poset (without an element 0). By analogy with
the Tits quadratic form of a quiver (the counting the number of parameters
for matrix representations of posets), the Tits quadratic form qS(z) of S
has the following form:
qS(z) := z20 +
∑
i∈S
z2i +
∑
i<j,i,j∈S
zizj − z0
∑
i∈S
zi
(matrix representations were introduced by L. A. Nazarova and A. V. Roi-
ter [2]; see for more details [3]).
If one talks on őnite representational type of posets, the main role is
played by weakly positive forms i.e. positive on the set of vectors with non-
negative coordinates. Namely, Yu. A. Drozd [4] proved that, for a poset
S, the following conditions are equivalent:
(1p) S is of őnite representation type over a őeld k;
(2p) the quadratic Tits form of S is weakly positive.
On the other hand, M. M. Kleiner [5] proved that a poset S is of
őnite representation type if and only if it does not contain (full) subposets
K1śK5 with the Hasse diagrams of the form
s s s s
K1
s s
ss
s
s
K2
s
s
s
s
s
s
s
K3
V. M. Bondarenko and M. V. Styopochkina 187
s
s
s
s
s
s
s
s
�
�
K4
s
s
s
s
s
s
s
s
K5
So, for a poset S, the following conditions are equivalent:
(2p) the quadratic Tits form of S is weakly positive;
(3p) S does not contain as subposets the Kleiner’s posets K1śK5.
A poset S is said to be WP -critical if its quadratic Tits form is not
weakly positive, but the Tits form of any proper subposet of S is weakly
positive. The equivalence of conditions (2p) and (3p) implies that the
Kleiner’s posets K1śK5 form a complete system of WP -critical posets.
Since for posets, in contrast to quivers, the sets of those with weakly
positive and with positive Tits forms do not coincide, research related
to the positivity of the quadratic Tits form of posets are natural. In
particular, posets with positive quadratic Tits form (that are analogs of
the Dynkin diagrams).were studied by the authors in [6] ś [9] (all such
posets were classiőed in [7]).
In [7] the author also classiőed (introduced by them) the P -critical
posets as the minimal posets with non-positive quadratic Tits form (their
number is 132 up to isomorphism and 75 up to isomorphism and duality).
More precisely, a poset S is called P -critical if the following conditions
hold:
(a) the quadratic Tits form qS(z) of S is not positive;
(b) the quadratic Tits form of any proper subposet of S is positive.
Condition (b) means that if qS(z) is considered with zi = 0 for any
őxed 0 ̸= i ∈ S, then it is positive.
Later A. Polak and D. Simson [10] offered an alternative way of
describing the P -critical posets by using computer algebra tools (mainly
symbolic computation in Maple and numeric computation in C#). In
doing this, they deőned and described the Tits P -critical posets as a special
case of the P -critical ones. Namely, a P -critical poset S is said to be a Tits
P -critical if the quadratic form qS(z) with z0 = 0 is positive1.
1For an obvious reason, in [10] P -critical posets are called also as almost Tits
P -critical posets.
188 On classifying the non-Tits P -critical posets
Using a relationship between P -critical and WP -critical posets [7], in
this article we describe the non-Tits P -critical posets without using the
(previously obtained in [7]) list of all P -critical ones.
2. Main result
Throughout the paper, all posets are assumed to be őnite.
A poset T is called dual to a poset S and is denoted by Sop if T = S
as usual sets and x < y in T if and only if x > y in S. When T and T op
are isomorphic, the poset T is called self-dual (otherwise, non-self-dual).
The P -critical posets were described by the authors in [7]. Their
number is 132 up to isomorphism and 75 up to isomorphism and duality.
Later A. Polak and D. Simson [10] offered an alternative way of describing
the P -critical posets by using computer algebra tools; in doing this, they
also described the Tits P -critical posets.
In this paper we classify all the non-Tits P -critical posets without
using the list of all P -critical ones. Their number is 17 up to isomorphism
and 11 up to isomorphism and duality (this is brieŕy written in the őrst
cell of Table 1 below; self-dual posets are marked by sd).
Theorem 1. Up to duality, the non-Tits P -critical posets are given by
Table 1.
From this theorem (and what was said above on all P -critical posets)
it follows that the number of the Tits P -critical posets is 115 up to
isomorphism and 64 up to isomorphism and duality.
The table of the Tits P -critical posets (as indicated in [7] Table 1 of
all the P -critical posets without the posets of the above Table 1) is given
in the last section.
3. P -critical posets and minimax equivalence
3.1. Minimax equivalence
In this section we recall notation and results from [7].
By a subposet we always mean a full one, and singletons are identiőed
with the elements themselves. Although, by deőnition, posets are of order
n > 0, sometimes (in deőnitions and statements) we admit empty posets
which are or may be later subposets of some posets.
Let S be a poset. For a minimal (resp. maximal) element a of S,
denote by T = S↑
a (resp. T = S↓
a) the following poset: T = S as usual
V. M. Bondarenko and M. V. Styopochkina 189
Table 1.
Up to duality=11:
self-dual=5
non-self-dual=6
All=17 s s
ss
�
�
❅
❅
nTPC1 sd
s s
s
s
s
�
�
�
�
s
❍❍❍❍
nTPC2 sd
s
s s
�
�
ss
s
nTPC3 sd
s
s
s
s
✁
✁
✁
✁
s
s
s
nTPC4
s s
s
s
✂
✂
✂
✂
✂
✂✂
s
s
s
s
nTPC5
s s
s
s
s
�
�
s
�
�
s
nTPC6
s s
s
s
s
�
�
s
�
�
s
s
nTPC7
s s
s
s
s
s
�
�
✁
✁
✁
✁
s
s
nTPC8
s
s
s s
s
�
�
s�
�
ss
nTPC9 sd
s s
s
s
✁
✁
✁
✁
s
✁
✁
✁
✁
s
s
s
nTPC10 sd
s s
s
s s
s
�
�
✁
✁
✁
✁
�
�
s
s
nTPC11
sets, T \ a = S \ a as posets, the element a is maximal (resp. minimal) in
T , and a is comparable with x in T if and only if they are incomparable
in S. A poset T is called minimax equivalent or (min, max)-equivalent
to a poset S, if there are posets S1, . . . , Sp (p ⩾ 0) such that, if one puts
S = S0 and T = Sp+1, then, for every i = 0, 1, . . . , p, either Si+1 = (Si)
↑
xi
or Si+1 = (Si)
↓
yi(this notion was introduced in [11]).
The notion of minimax equivalence can be naturally continued to the
notion of minimax isomorphism: posets S and S′ are minimax isomorphic
if there exists a poset T , which is minimax equivalent to S and isomorphic
to S′.
The deőnition of posets of the form T = S↑
a (resp. T = S↓
a) can be
extended to subposets. Namely, let S be a poset and A its lower (resp.
190 On classifying the non-Tits P -critical posets
upper) subposet, i.e. x ∈ A whenever x < y (resp. x > y) and y ∈ A. By
T = S↑
A (resp. T = S↓
A) we denote the following poset: T = S as usual sets,
partial orders on A and S\A are the same as before, but comparability and
incomparability between elements of x ∈ A and y ∈ S \A are interchanged
and the new comparability can only be of the form x > y (resp. x < y).
Note that S and S↑
A (resp. S and S↓
A) are minimax equivalent.
From the deőnitions we have the following lemma.
Lemma 1. (a) S↓
A = S↑
S\A; (b) (S↓
A)
op = (Sop)↑Aop .
Corollary 1. If a poset S is self-dual, then, for any lower subposet B,
S↑
B = (S↑
S\Bop)
op.
Indeed A = Bop is an upper subposet of Sop = S and from equality
(b) one has (S↓
Bop)op = S↑
B, whence, by equality (a) S↑
B = (S↑
S\Bop)
op.
We write S↑↓
XY (resp. S↓↑
Y X) instead of (S↑
X)↓Y (resp. (S↓
Y )
↑
X). It is easy
to see that S↑↓
XY = S↓↑
Y X if X < Y i.e. x < y for any x ∈ X, y ∈ Y .
The main motivation for introducing the notion of minimax equivalen-
ce is the fact that the quadratic Tits forms of minimax equivalent posets
are Z-equivalent. This follows from the next proposition.
Proposition 1. Let S be a poset and let T = S↑
A or T = S↓
A. Then
qS(z) = qT (z
′), where z′0 = z0 −
∑
a∈A za, z′x = −zx for x ∈ A and
z′x = zx for x /∈ A.
In [7] we indicated a three-step algorithm for őnding all (up to isomor-
phism) posets minimax isomorphic to a given poset S:
Step I. Describe the set L = L(S) of lower subposets X ̸= S of S
(∅ ∈ L), and for all such X to build the posets S↑
X .
Step II. Describe the set LU< = LU<(S) of pairs (X,Y ) consisting
of proper lower and upper subposets X,Y of S such that X < Y , and for
them to build the posets S↑↓
XY (= S↓↑
Y X).
Step III. Among the posets constructed in I and II choose a complete
system of pairwise non-isomorphic ones.
This algorithm is denoted by MM-ALG.
We call subposets X and Y (of a poset S) having the form, indicated
in I, Aut-equivalent if φ(X) = Y for some automorphism φ of S. Similarly,
the indicated in II pairs (X,Y ) and (X ′, Y ′) are called Aut-equivalent if,
for some automorphism φ, φ(X) = X ′ and φ(Y ) = Y ′.
If elements of the sets L and LU< at the Steps I and II of Algorithm
MM-ALG are taken up to Aut-equivalent (we will denote them L◦ = L◦(S)
V. M. Bondarenko and M. V. Styopochkina 191
and LU◦
< = LU◦
<(S)), the corresponding algorithm is denoted by MM◦-
ALG; the modiőed steps of the new algorithm are denoted I◦, II◦ and
III◦.
3.2. Relationship between P -critical and WP -critical posets
P -critical posets were introduced and studied in detail for the őrst time
in [7]2. Recall also that a poset S is called WP -critical if its quadratic Tits
form is not weakly positive, but the Tits form of any proper subposet of S
is weakly positive. The Kleiner’s posets K1śK5 form a complete system of
WP -critical posets. In practice, it is natural to single out one representative
from each isomorphism class. Up to isomorphism, the Kleiner’s posets are
of the form
K1 = {1, 2, 3, 4} (without relations),
K2 = {1, 2, 3, 4, 5, 6 | 1 ≺ 2, 3 ≺ 4, 5 ≺ 6},
K3 = {1, 2, 3, 4, 5, 6, 7 | 2 ≺ 3 ≺ 4, 5 ≺ 6 ≺ 7},
K4 = {1, 2, 3, 4, 5, 6, 7, 8 | 1 ≺ 2 ≺ 3 ≺ 4, 5 ≺ 6, 7 ≺ 8, 5 ≺ 8},
K5 = {1, 2, 3, 4, 5, 6, 7, 8 | 2 ≺ 3, 4 ≺ 5 ≺ 6 ≺ 7 ≺ 8}3.
For a poset S of order n, deőne the kernel of qS(z) as follows:
Ker qS(z) := {u ∈ Z
n+1 | qS(u) = 0}. The following statement shows that,
for each Kleiner’s poset K, the kernel of qK(z) is an inőnite cyclic group.
Proposition 2. Put qi(z) := qKi
(z) (1 ⩽ i ⩽ 5). The quadratic forms
qi(z) are non-negative and
Ker q1(z) = (2, 1, 1, 1, 1)Z,
Ker q2(z) = (3, 1, 1, 1, 1, 1, 1)Z,
Ker q3(z) = (4, 2, 1, 1, 1, 1, 1, 1)Z,
Ker q4(z) = (5, 1, 1, 1, 1, 1, 2, 2, 1)Z,
Ker q5(z) = (6, 3, 2, 2, 1, 1, 1, 1, 1)Z.
Indeed, the relations ⊇ are veriőed by direct calculations, and then
the relations = follows from the results of [13]; non-negativity also follows
from [13]4 (see also [15]).
Corollary 2. Let v = (v0, v1, . . . , vm) ∈ Ker qK(z) with K to be a Klei-
ner’s poset. Then 2v0 = v1 + v2 + . . .+ vm.
2In fact, a little earlier under the name of critical posets they were őrst considered
in [12].
3
K4 and K5, as notation, often change places in other papers.
4See in both cases Theorem 2 of Sect. 1.0 [14], in which the main results of this
paper are summarized.
192 On classifying the non-Tits P -critical posets
Theorem 2 (Theorem 2 [7]). A poset S is P -critical if and only if it
minimax isomorphic to a Kleiner’s poset.
From Propositions 1, 2 and Theorem 2, we have the following corollary.
Corollary 3. Let S be a P -critical poset of order n. Then the quadratic
Tits form of S is non-negative and Ker qS(z) = vZ for some vector v =
(vi)i∈0∪S ∈ Z
n+1 with vi ≠ 0 for any i ̸= 0, which is determined by S
uniquely up to the sign,
4. Proof of Theorem 1
4.1. 0-balanced subposets
The vector v = (vi)i∈0∪S ∈ Z
n+1 speciőed in Corollary 3 is denoted by
vS = (vSi )i∈0∪S
5. A subposet X of a P -critical poset S is said to be small
if |X| does not exceed the integer part of (|S|+ 1)/2, and 0-balanced if
vS0 =
∑
i∈X vSi . From these deőnititons and Proposition 2 it follows the
next two lemmas.
Lemma 2. For any Kleiner posets Ki (1 ⩽ i ⩽ 5), there are no 0-balanced
subposets of the form X = A ∪B, A < B.
Lemma 3. For Kleiner posets Ki, small 0-balanced lower subposets Aij
are exhausted, up to automorphism (of Ki), by the following ones:
(1) A11 = {1, 2};
(2) A21 = {1, 2, 3}, A22 = {1, 3, 5};
(3) A31 = {1, 2, 3}, A32 = {1, 2, 5};
(4) A41 = {1, 2, 3, 7}, A42 = {1, 2, 5, 6}, A43 = {1, 2, 5, 7},
A44 = {1, 5, 7, 8}, A45 = {5, 6, 7};
(5) A51 = {1, 2, 4}, A52 = {1, 4, 5, 6}, A53 = {2, 3, 4, 5}.
The following lemma can be proved by complex enumeration of all
cases (using the previous lemma), but it follows directly from Corollaries 1
and 2 (by Corollaries 2, for any Kleiner’s poset K, subposets X and K \X
at the same time are or are not 0-balanced).
Lemma 4. Let X be a non-small 0-balanced lower subposet of a Kleiner’s
poset K = Ki. Then there is a small 0-balanced lower subposet Y of K
such that K↑
X = (K↑
Y )
op.
5A little ambiguity of the vector vS is not essential (one can, for example, őx vs ≠ 0
and choose v
S with v
S
s > 0).
V. M. Bondarenko and M. V. Styopochkina 193
4.2. The application of Algorithm MM◦◦◦-ALG
From Corollary 3 (taking into account the deőnitions of P -critical and
Tits P -critical posets) it follows the next statement.
Proposition 3. A P -critical poset S is not Tits P -critical if and only if
vS0 = 0.
By Theorem 2 and Proposition 1, 3, to prove Theorem 1 it is enough to
show that the set of all posets obtained as a result of applying Algorithm
MM◦◦◦-ALG to all 0-balanced lower subposets X (see Step I◦) and 0-
balanced pairs of subposets (X,Y ) (see Step II◦) of all the Kleiner’s posets
K = K1, . . . ,K5 coincides, up to isomorphism and duality, with the set
of all posets from Table 1.
It is follows from Lemma 2 that Step II◦ of the algorithm is in this case
empty, and by Lemma 4 on the Step I◦ one can take only small subposets.
So, it remains for us to calculate the posets KA with K = K1, . . . ,K5 and
A running the subposets indicated in Lemma 3, and then compare them
with the posets of Table 1. We have:
for K = K1, A = A11, the poset KA is isomorphic to nTPC1;
for K = K2, A = A21, the poset KA is isomorphic to nTPC3;
for K = K2, A = A22, the poset KA is isomorphic to nTPC2;
for K = K3, A = A31, the poset KA is isomorphic to nTPC4op;
for K = K3, A = A32, the poset KA is isomorphic to nTPC6op;
for K = K4, A = A41, the poset KA is isomorphic to nTPC11;
for K = K4, A = A42, the poset KA is isomorphic to nTPC10;
for K = K4, A = A43, the poset KA is isomorphic to nTPC9;
for K = K4, A = A44, the poset KA is isomorphic to nTPC11op;
for K = K4, A = A45, the poset KA is isomorphic to nTPC8op;
for K = K5, A = A51, the poset KA is isomorphic to nTPC7op.
for K = K5, A = A52, the poset KA is isomorphic to nTPC5op;
for K = K5, A = A53, the poset KA is isomorphic to nTPC5.
Thus, the set of all posets of the form KA coincides, up to isomorphism
and duality, with the set of posets from Table 1.
Theorem 1 is proved.
5. The table of the Tits P -critical posets
In this section we write out a table (Table 2) of all Tits P -critical
posets up to isomorphism and duality, i.e. Table 1 [7] (see also [16] ) of
the P -critical posets without the posets from Table 1 of Section 1.
194 On classifying the non-Tits P -critical posets
Table 2 contains 64 posets; self-dual posets are marked (in the upper
right corners) by sd. If we add all the posets dual to unmarked ones, we
obtain the Tits P -critical posets up to isomorphism; their number is 115.
Table 2.
q q q�
q
✟✟
TPC1
q q q q
TPC2 sd
q
q
q
q�
q
q
TPC3
q
q
q
�
�qq
q
TPC4 sd
q q
q
q
qq
TPC5 sd
q q q
q
q
�q
TPC6
q q
q
q
q
�
✁
✁
q
TPC7
q q
q
q
✂
✂
✂✂
q
q
q
TPC8
q
q
q q
q�
q
q
TPC9
q
q
q q
q
�q
q
TPC10
q q
q
q
✂
✂
✂✂
q
q
TPC11
q� q q
q
q
q
qq
TPC12 sd
q q q
q
q
✂
✂
✂✂
q
q
TPC13
q q
q
q
q
✂
✂
✂✂q
q
TPC14
q
q q
q
q
✂
✂
✂✂
q
�q
TPC15 sd
q q
q
q
q
✁
✁
q
�
q
q
TPC16
q qq
q
q
✁
✁q�
q
TPC17
q q
q
q
✁
✁
q
q
q
q
TPC18
q q
q
q
☎
☎
☎
☎
☎☎
q
q
q
q
TPC19
q q
q
q
�
q
q
q
q
TPC20
q q
q
q
✄
✄
✄
✄q
q
q
q
TPC21
q q
q
q
✄
✄
✄
✄
q
q
q
q
TPC22
q q
q
q
✂
✂
✂✂q
q
q
q
TPC23
q q
q
q
✁
✁
qq
q
q
TPC24
q q
q
q
✁
✁
q
q
q
q�
TPC25
q q
q
q
☎
☎
☎
☎
☎☎
q
q
q
q�
TPC26 sd
q q
q
q
�
q
�
q
q
q
TPC27
q
q
q
q
�
q
q
q
✄
✄
✄
✄
q
TPC28
q q
q
q
✄
✄
✄
✄q
✄
✄
✄
✄q
q
q
TPC29
q
q
q
q
�
q
�
q
q
q
TPC30
q q
q
q
✄
✄
✄
✄q
q
q
q�
TPC31
q q
q
q
✂
✂
✂✂
q
✂
✂
✂✂
q
q
q
TPC32
q q
q
q
✁
✁
q
q
q�
q
TPC33
q q
q
q
�
q
�
q
q
q�
TPC34
q q
q
q
�
q
�q
q�
q
TPC35 sd
q q
q
q
q
q
q
q
TPC36 sd
V. M. Bondarenko and M. V. Styopochkina 195
q q q
q
q
✁
✁
q
q
q
TPC37
q q q
q
q
☎
☎
☎
☎
☎☎
q
q
q
TPC38
q q
q
q
q
�
q
q
q
TPC39
q q
q
q
q
✄
✄
✄
✄
q
q
q
TPC40
q q
q
q
q
✁
✁q
q
q
TPC41
q q
q
q
q
✁
✁q
q
q
TPC42
q q
q
q
q
✁
✁q
q
q
TPC43
q q
q
q
q
�
q
q
q
TPC44 sd
q
q q
q
q
✁
✁
q
q
�q
TPC45
q q
q
q
q
�
q
�
q
q
TPC46
q
q q
q
q
✁
✁
q
�q
q
TPC47 sd
q
q q
q
q
q
✁
✁
q
q�
TPC48
q q
q
q
q
�
q
�
q
q
TPC49
q
q
q q q
q
✁
✁
q
q�
TPC50 sd
q q
q
q
q
�
q
�
q
q
TPC51
q q
q
q
q
✄
✄
✄
✄
q
�
q
q
TPC52
q q
q
q
q
✂
✂
✂✂q
�
q
q
TPC53
q q
q
q q
q
�✁
✁
q
q
TPC54
q q
q
q q
q✂
✂
✂✂
✁
✁
q
q
TPC55
q q
q
q q
q✁
✁
�
q
q
TPC56
q q
q
q q
q✁
✁
✁
✁
q
q
TPC57
q q
q
q
q
q�✁
✁ q
q
TPC58
q q
q
q
q
q
�☎
☎
☎
☎
☎☎q
q
TPC59 sd
q q
q
q
q
�
q
�
q
q
TPC60
q
q
q
q
q
�
q
�
q
q�
TPC61
q
q q
q
q
q
�
q
�
q�
TPC62 sd
q q
q
q q
q
�
��q
q
TPC63
q
q q
q
q
q�
�
�
q
q
TPC64
References
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[2] L. A. Nazarova, A. V. Roiter, Representations of partially ordered sets, Zap. Nauch.
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(1975), pp. 585ś606.
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Contact information
Vitaliy
M. Bondarenko
Institute of Mathematics, Tereshchenkivska str.,
3, 01024 Kyiv, Ukraine
E-Mail(s): vitalij.bond@gmail.com
Web-page(s): http://www.imath.kiev.ua
Maryna
V. Styopochkina
Polissia National University, Staryi Boulevard,
7, 10008 Zhytomyr, Ukraine
E-Mail(s): stmar@ukr.net
Received by the editors: 12.11.2021.
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