The classification of serial posets with the non-negative quadratic Tits form being principal
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 qS(z) : Zˢ⁺¹ → Z of S is non-negative; (2) KerqS(z) := {t | qS(t) = 0} is an infinite cyclic group...
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irk-123456789-1884332023-03-02T01:27:12Z The classification of serial posets with the non-negative quadratic Tits form being principal Bondarenko, V.M. Styopochkina, M.V. 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 qS(z) : Zˢ⁺¹ → Z of S is non-negative; (2) KerqS(z) := {t | qS(t) = 0} is an infinite cyclic group (equivalently, the corank of the symmetric matrix of qS(z) is equal to 1); (3) for any m ∈ N, there is a poset S(m) ⊃ S such that S(m) satisfies (1), (2) and |S(m) \ S| = m 2019 Article The classification of serial posets with the non-negative quadratic Tits form being principal / V.M. Bondarenko, M.V. Styopochkina // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 202–211. — Бібліогр.: 18 назв. — англ. 1726-3255 2010 MSC: 15B33, 15A30. http://dspace.nbuv.gov.ua/handle/123456789/188433 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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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 qS(z) : Zˢ⁺¹ → Z of S is non-negative; (2) KerqS(z) := {t | qS(t) = 0} is an infinite cyclic group (equivalently, the corank of the symmetric matrix of qS(z) is equal to 1); (3) for any m ∈ N, there is a poset S(m) ⊃ S such that S(m) satisfies (1), (2) and |S(m) \ S| = m |
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Bondarenko, V.M. Styopochkina, M.V. |
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Bondarenko, V.M. Styopochkina, M.V. The classification of serial posets with the non-negative quadratic Tits form being principal Algebra and Discrete Mathematics |
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Bondarenko, V.M. Styopochkina, M.V. |
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Bondarenko, V.M. |
| title |
The classification of serial posets with the non-negative quadratic Tits form being principal |
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The classification of serial posets with the non-negative quadratic Tits form being principal |
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The classification of serial posets with the non-negative quadratic Tits form being principal |
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The classification of serial posets with the non-negative quadratic Tits form being principal |
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The classification of serial posets with the non-negative quadratic Tits form being principal |
| title_sort |
classification of serial posets with the non-negative quadratic tits form being principal |
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Інститут прикладної математики і механіки НАН України |
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2019 |
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The classification of serial posets with the non-negative quadratic Tits form being principal / V.M. Bondarenko, M.V. Styopochkina // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 202–211. — Бібліогр.: 18 назв. — англ. |
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Algebra and Discrete Mathematics |
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“adm-n2” — 2019/7/14 — 21:27 — page 202 — #52
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 27 (2019). Number 2, pp. 202–211
c© Journal “Algebra and Discrete Mathematics”
The classification of serial posets with the
non-negative quadratic Tits form being principal
Vitalij M. Bondarenko, Marina V. Styopochkina
Communicated by A. P. Petravchuk
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 qS(z) : Z
|S|+1 → Z of S is non-negative; (2) Ker qS(z) :=
{t | qS(t) = 0} is an infinite cyclic group (equivalently, the corank
of the symmetric matrix of qS(z) is equal to 1); (3) for any m ∈ N,
there is a poset S(m) ⊃ S such that S(m) satisfies (1), (2) and
|S(m) \ S| = m.
1. Introduction
In [11], for a finite quiver 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) =
∑
i∈Q0
z2i −
∑
i→j
zizj ,
where i → j runs through the set Q1. He proved that a connected quiver
is of finite type over a field (i.e. has only finitely many isomorphism classes
of indecomposable representations) if and only if its underlying graph
is one of the (simply faced) Dynkin diagrams and that the quivers of
2010 MSC: 15B33, 15A30.
Key words and phrases: quiver, serial poset, principal poset, quadratic Tits
form, semichain, minimax equivalence, one-side and two-side sums, minimax sum.
“adm-n2” — 2019/7/14 — 21:27 — page 203 — #53
V. Bondarenko, M. Styopochkina 203
finite type coincide with the quivers whose quadratic Tits form is positive.
This Gabriel’s work laid the foundations of a new direction in the theory
of algebras dealing with the investigation of the relationships between
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 6∋ 0 [10]:
qS(z) = z20 +
∑
i∈S
z2i +
∑
i<j,i,j∈S
zizj − z0
∑
i∈S
zi.
In [10] Yu. A. Drozd showed that a poset S is of finite type if and only if
its quadratic Tits form is weakly positive, i.e. takes positive value on any
non-zero vector with non-negative coordinates (representations of posets
over a field were introduced by L. A. Nazarova and A. V. Roiter [16]).
In contrast to quivers, the sets of posets with weakly positive and with
positive Tits form do not coincide. Therefore the investigations of posets
with positive Tits form seems to be quite natural: they are analogs of the
Dynkin diagrams. Posets of this type were studied by the authors in many
papers (see e.g. [3] – [6]). In particular, the classification of posets with
positive Tits form was obtained (see Section 2).
One has a similar situation for quivers and posets of tame type. A
quiver Q is of tame type over a field if and only if its quadratic Tits
form is non-negative. This follows from the facts that a connected quiver
is of tame infinite type iff its underlying graph is an extended Dynkin
diagram [9, 15], and that the connected quivers with non-negative, but
not positive, Tits form coincide with the quivers, the underlying graphs
of which are extended Dynkin diagrams [8]. A poset S is of tame type if
and only if its quadratic Tits form is weakly non-negative (see [18] and
[17, Theorem 15.23]). Since (in contrast to quivers) the sets of posets with
weakly non-negative and with non-negative Tits forms do not coincide,
the investigations of posets with non-negative Tits form also are actual.
The present paper is devoted to the study of one class of posets with
non-negative quadratic Tits form.
2. Positive and principal posets
Throughout the paper, all posets are finite of order n > 0 without the
element 0. In the case, when the elements of a poset are numbered by
integer numbers, the relation of partial order is denoted by ≺ (and one
always assumes that i ≺ j implies i < j).
“adm-n2” — 2019/7/14 — 21:27 — page 204 — #54
204 The classification of serial posets
2.1. Definitions on posets. 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.
By a subposet we always mean a full one, and singletons are identified
with the elements themselves. Sometime (in definitions or statements) we
admit empty posets which are or may be later subposets of some posets.
A poset S is called a sum of subposets A1, A2, . . . , Am and write
S = A1 + A2 + · · ·+ Am, if S = ∪m
i=1Ai and Ai ∩ Aj = ∅ for any i and
j 6= i. If any two elements of different summands are incomparable, the
sum is called direct and one writes also
∐
instead of +.
A sum S = A + B with A,B 6= ∅ is said to be left (resp. right) if
a < b (resp. b < a) for some a ∈ A, b ∈ B and there is no a′ ∈ A, b′ ∈ B
satisfying a′ > b′ (resp. b′ > a′). Both left and right sums are called
one-sided. A sum S = A+ B is called two-sided if a < b and a′ > b′ for
some a, a′ ∈ A, b, b′ ∈ B. Finally, a one-sided (left or right) or two-sided
sum S = A + B is called minimax if x < y with x and y belonging to
different summands implies that x is minimal and y maximal in S.
2.2. The quadratic Tits form. Let S be a poset. The quadratic
form of S is by definition the following quadratic form q : Z|S|+1 → Z:
q = qS(z) = z20 +
∑
i∈S
z2i +
∑
i<j,i,j∈S
zizj − z0
∑
i∈S
zi
(see [10]). Here Z denotes as usual the ring of integer numbers. Obviously,
one can assume, without loss of generality, that S = {1, 2, . . . , n} (n > 1);
then Z
|S|+1 = Z
n+1 consists of the integer vectors (z0, z1, . . . , zn).
2.3. Positive posets. A poset with the quadratic Tits form being
positive (i.e. qS(z) > 0 for all non-zero vectors z ∈ Z
|S|+1) is called
positive.
In this subsection we adhere to the results of [4].
The positive posets are of two types: serial and non-serial.
A positive poset S is called serial if for any m ∈ N, there is a positive
poset S(m) ⊃ S such that |S(m) \ S| = m, and non-serial otherwise.
There are 108 non-serial posets up to isomorphism and duality, and 194
up to isomorphism (see Table 2 in [4]).
Now formulate two theorems on serial positive posets.
A linear ordered set with n > 0 elements is called a chain of length n.
A poset with one pair of incomparable elements a1 < . . . < ap < {b, c} <
d1 < . . . dq (p, q > 0) is called an almost chain of length n = p + q + 1
(ap < {b, c} < d1 means that ap < b < d1, ap < c < d1; b and c are
incomparable).
“adm-n2” — 2019/7/14 — 21:27 — page 205 — #55
V. Bondarenko, M. Styopochkina 205
Theorem 1. A poset T is serial positive if and only if it is isomorphic
to one of the following poset S:
(1) S is a direct sum of a chain of length k > 0 and a chain of length
s > 1, where k 6 s;
(2) S is a left minimax sum of two chains of lengths k > 1 and s > 1,
where k + s > 3;
(3) S is a direct sum of an almost chain of length k > 1 and a chain
of length s > 0, where k + s > 1.
Moreover, all these posets are pairwise non-isomorphic.
Theorem 2. Any positive poset of order n > 7 is serial.
2.4. Principal posets. A poset S and the quadratic Tits form qS(z)
are called principal (see [12]) if the following conditions hold:
(1) qS(z) is non-negative (i.e. which accepts only non-negative values);
(2) Ker qS(z) := {t | qS(t) = 0} is an infinite cyclic group, i.e. Ker qS(z)
= t′Z for some t′ 6= 0 (equivalently, the corank of the symmetric matrix
of qS(z) is equal to 1).
The principal posets form a natural class of the posets with non-
negative quadratic Tits form.
By analogy with the definition of a serial positive poset, we call a
principal poset S serial if for any m ∈ N, there is a principal poset
S(m) ⊃ S such that |S(m) \ S| = m.
Some class of principal posets of order n = 6, 7, 8 (which in our
terminology means the non-serial ones) were written by G. Marczak,
D. Simson and K. Zaja̧c with the help of programming in Maple and
Python (see the paper [13] for n = 6, 7 and the preprint [14] for n = 8).
In this paper, using the method of (min, max)-equivalence, we classify
all serial principal posets.
3. Main results
We adhere to the definitions of Section 2 and give some new definitions.
For subposets A,B of a poset S we write A < B if a < b for any
a ∈ A, b ∈ B. A poset P is called a semichain of length s if it has the
form P = ∪s
i=1Pi with P1 < P2 < · · · < Ps, where every Pi consists of one
or two incomparable elements. The number of two-element Pi is said to
be 2-length of the semichain P . Note that chains and almost chains (see
subsection 2.3) are semichains of 2-length 0 and 1, respectively.
We can now formulate the main theorems of this paper.
“adm-n2” — 2019/7/14 — 21:27 — page 206 — #56
206 The classification of serial posets
Theorem 3. A poset T is serial principal if and only if it is isomorphic
to one of the following poset S:
(I) S is a direct sum of a chain of length k > 0, and a semichain of
length s > 2 and 2-length 2;
(Il) S is a direct sum of a semichain of length k > 1 and 2-length 1,
and a semichain of length s > 1 and 2-length 1, where k 6 s;
(III) S is a left minimax sum of a chain of length k > 1, and a
semichain of length s > 2 and 2-length 1 with the only maximal element;
(IV) S is a left minimax sum of a semichain of length k > 2 and
2-length 1 with the only minimal element, and a chain of length s > 1;
(V) S is a two-sided minimax sum of a chain of length k > 2 and a
chain of length s > 3, where k 6 s.
Moreover, all these posets are pairwise non-isomorphic.
In the language of Hasse diagrams the posets indicated in the theorem
have the following form
r
r
(I)
t t
r
r
��
❅❅
❅❅
��
t t
r
r
��
❅❅
❅❅
��
r
r
(II)
t t
r
r
��
❅❅
❅❅
��
r
r
t t
r
r
��
❅❅
❅❅
��
r
r
(III)
t t
r
r
��
❅❅
❅❅
��
✡
✡
✡
✡
✡
✡✡ rt
tr
(IV)
t t
r
r
��
❅❅
❅❅
��
rt
tr
✡
✡
✡
✡
✡
✡✡
(V)
✡
✡
✡
✡
✡
✡✡
❏
❏
❏
❏
❏
❏❏tt
t
t
t
Here vertical lines are chains, and inclined segments do not contain
intermediate points. The large points indicated in the figures (unlike small
and intermediate ones) must always be present.
The second main theorem is the following.
Theorem 4. Any principal poset of order n > 8 is serial.
“adm-n2” — 2019/7/14 — 21:27 — page 207 — #57
V. Bondarenko, M. Styopochkina 207
4. Minimax equivalence of posets
In this section we recall notation and results from the papers [1], [4]
and formulate some corollaries which will be used in the proof of the main
theorems.
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
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, then, for every i = 0, 1, . . . , p, either Si+1 = (Si)
↑
xi
or
Si+1 = (Si)
↓
yi (the case p = 0 means that S is minimax equivalent to S).
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 definition 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.
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.
We write S↑↑
AB instead of (S↑
A)
↑
B , S↑↓
AB instead of (S↑
A)
↓
B , etc. Obviously,
S↑↓
AA = S, S↓↑
AA = S, S↑
A = S↓
S\A, S↓
A = S↑
S\A.
From the definitions we have the following corollary.
Corollary 1. (S↓
A)
op = (Sop)↑Aop .
The main motivation for introducing the notion of minimax equiva-
lence is the fact that the 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.
Corollary 2. Let S and T be the same as in Proposition 2, and let x /∈ A.
Then T \ x is positive if so is S \ x.
“adm-n2” — 2019/7/14 — 21:27 — page 208 — #58
208 The classification of serial posets
Corollary 3. A poset minimax equivalent to a principal one is also
principal.
5. Proofs of Theorems 3 and 4
We first prove that all posets of the form (I)–(V) are principal; then
their seriality is obvious.
It is easy to see that
(a) if S is of the form (I) and L denotes the chain of length k > 0,
then S↑
L is also a poset of the form (I) (with the empty chain);
(b) if S is of the form (II) and L denotes the first semichain, then S↑
L
is a poset of the form (I) (with the empty chain);
(c) if S is of the form (III) and p denotes the minimal element of the
chain of length k > 1, then S↑
p is a poset of the form (I);
(d) if S is of the form (IV) and p denotes the minimal element of the
semichain of length k > 2, then S↑
p is a poset of the form (II);
(e) if S is of the form (V) and p denotes the minimal element of the
second chain, then S↑
p is a poset of the form (IV).
So, by Corollary 3, it is sufficient to consider only the case of posets
of the form (I) with k = 0, i.e. the case of semichains of 2-length 2. By
formulas (3) and (18) of [2], the quadratic Tits form qP (z) of a semichain
P = {P1 < P2 < · · · < Ps} of 2-length 2 with two-element sets Pi =
{u1, u2}, Pj = {v1, v2} (i 6= j) and one-element sets Pk = {pk} (p 6= i, j)
satisfies the following equality:
2qP (z) = z20 + (z0 −
∑
k 6=i,j
zpk − zu1
− zu2
− zv1 − zv2)
2
+
∑
k 6=i,j
z2pk + (zu1
− zu2
)2 + (zv1 − zv2)
2.
From here it follows that qP (z) is principal, and so P is principal.
Thus the sufficiency of the Theorem 3 is proved.
Since all subposets of posets of the forms (I)–(V) also have such forms
(and all posets S(m) in the definition of serial principal posets are also the
same ones), for the proof of the necessity of Theorem 3 and Theorem 4 it
suffices to show that the next statement holds.
Proposition 2. Any principal poset of order n > 8 is one of the form
(I)–(V).
We prove first the following lemma.
“adm-n2” — 2019/7/14 — 21:27 — page 209 — #59
V. Bondarenko, M. Styopochkina 209
Lemma 1. Let A = {a}
∐
B be a principal poset of order n > 8 with B
to be a positive poset. Then B = {b}
∐
C, where C is an almost chain
(consequently, A is of the form (II) with k = 1).
Proof. For proof, we need the following facts: The posets
T1 = {1, 2, 3, 4, 5, 6, 7, 8 | 2 ≺ 3 ≺ 4, 5 ≺ 6 ≺ 7 ≺ 8},
T2 = {1, 2, 3, 4, 5, 6, 7, 8, 9 | 2 ≺ 3, 4 ≺ 5 ≺ 6 ≺ 7 ≺ 8 ≺ 9},
T3 = {1, 2, 3, 4, 5, 6, 7, 8, 9 | 2 ≺ 9, 3 ≺ 4 ≺ 5 ≺ 6 ≺ 7 ≺ 8 ≺ 9},
T4 = {1, 2, 3, 4, 5, 6, 7, 8, 9 | 2 ≺ 3, 2 ≺ 9, 4 ≺ 5 ≺ 6 ≺ 7 ≺ 8 ≺ 9},
T5 = {1, 2, 3, 4, 5 | 4 ≺ 5}.
are not non-negative1. They are follows from the equalities
qT1
(8, 4, 2, 2, 2, 2, 2, 1, 1) = qT2
(12, 6, 4, 4, 2, 2, 2, 2, 1, 1) =
= qT3
(8, 4, 6, 2, 2, 2, 2, 1, 1,−4) = qT4
(8, 4, 4, 2, 2, 2, 2, 1, 1,−2) =
= qT5
(4, 2, 2, 2, 1, 1) = −1.
Since the poset B is positive of order n′ = n− 1 > 7, it is of the form
(1) or (2) or (3) (see Theorems 1 and 2).
In case (1) {a}
∐
B is positive if k 6 1 (by (1) and (3) of Theorem 1),
and is not non-negative if k > 1 (because it contains a subposet isomorphic
to T2 when k = 2, and to T1 when k > 2). So A can not be principal.
In case (2) {a}
∐
B is not non-negative, because it contains a subposet
isomorphic to T3 when k = 1, to T4 when k = 2, to T1 when k = 3, 4, 5,
to T op
4 when k = 6, and to T op
3 when k > 7. So A can not be principal.
In case (3) {a}
∐
B is positive if s = 0 (by (3) of Theorem 1), and
is not non-negative if s > 1 (because it contains a subposet isomorphic
to T5). So s = 1 and B has the form indicated in the formulation of our
lemma (then A is principal as a poset of the form (II)).
Let now S be a principal poset of order n > 8 and let t′ 6= 0 be such
that qS(t
′) = 0. Fix d ∈ S such that t′d 6= 0. Then by the definition of
principal poset S0 = S \ d is a positive poset. Put A := {x ∈ S |x < d}
and B := {x ∈ S |x > d}. Then Sd := S↑↓
AB = {d}
∐
T for some subposet
T of S. By Corollaries 2 and 3 the poset T is positive and the poset Sd is
principal. Since |T | > 7, it follows from Theorems 1, 2 and Lemma 1 that
T is of the form (3) with s = 1.
We will mention the posets of the form (I), (II), . . . , (V) up to replace-
ment left sums by right sums.
1
T1–T5 are minimal posets, that are not non-negative (i.e. with the quadratic
Tits form, which are not non-negative). All such posets are classificated in [7]. This
classification provides a criterion for a poset S to be non-negative.
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210 The classification of serial posets
Let T = {x0}
∐
C with an almost chain C. Since S↑↓
AB = {d}
∐
T ,
implies S = ({d}
∐
T )↑↓AB, or equivalently S = {d}
∐
T ↑↓
AB (because d /∈
A,B), and the set of all posets of the form (I)–(V) is closed with respect
to duality, then by Lemma 1 (written as S↓
A = [(Sop)↑Aop ]op), to complete
the proof it suffices to show that {d}
∐
T ↑
A is one of the form (I)–(V) for
any lower subposet of T .
Put C := {x1 < x2 < · · ·xp < {u, v} < y1 < y2 < · · · < yq} and write
out all types of lower subposets of T = {x0}
∐
C:
A0 = ∅; A1 = {x1, . . . , xi}, 1 6 i 6 p; A2 = {x1, . . . , xp, u};
A3 = A2 ∪ v; A4 = A3 ∪ {y1, . . . , yj}, 1 6 j 6 q; A5 = x0;
A6 = x0 ∪A1; A7 = x0 ∪A2; A8 = x0 ∪A3; A9 = x0 ∪A4.
By the definition of S↑
A we have that {d}
∐
T ↑
A = {d}
∐
({x0}
∐
C)↑A
is of the form (II) for A = A0, A1, of the form (III) for A = A2, of the
form (I) for A = A3, A4, of the form (III) for A = A5, A6, of the form (V)
for A = A7, of the form (IV) for A = A8, A9.
Proposition 2 is proved.
The fact that all posets of the form (I)–(V) (see Theorem 3) are
pairwise non-isomorphic is obvious.
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Contact information
V. M. Bondarenko Institute of Mathematics, Tereshchenkivska str.,
3, 01024 Kyiv, Ukraine
E-Mail(s): vitalij.bond@gmail.com
M. V. Styopochkina Zhytomyr National Agroecological Univ.,
Staryi Boulevard, 7, 10008 Zhytomyr, Ukraine
E-Mail(s): stmar@ukr.net
Received by the editors: 14.03.2019.
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