On tame semigroups generated by idempotents with partial null multiplication
Let I be a finite set without 0 and J a subset in I×I without diagonal elements (i,i). We define S(I,J) to be the semigroup with generators ei, where i∈I∪0, and the following relations: e0=0; e2i=ei for any i∈I; eiej=0 for any (i,j)∈J. In this paper we study finite-dimensional representations o...
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Інститут прикладної математики і механіки НАН України
2008
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| Цитувати: | On tame semigroups generated by idempotents with partial null multiplication / V.M. Bondarenko, O.M. Tertychn // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — С. 15–22. — Бібліогр.: 6 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1533682025-02-09T14:34:40Z On tame semigroups generated by idempotents with partial null multiplication Bondarenko, V.M. Tertychna, O.M. Let I be a finite set without 0 and J a subset in I×I without diagonal elements (i,i). We define S(I,J) to be the semigroup with generators ei, where i∈I∪0, and the following relations: e0=0; e2i=ei for any i∈I; eiej=0 for any (i,j)∈J. In this paper we study finite-dimensional representations of such semigroups over a field k. In particular, we describe all finite semigroups S(I,J) of tame representation type. 2008 Article On tame semigroups generated by idempotents with partial null multiplication / V.M. Bondarenko, O.M. Tertychn // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — С. 15–22. — Бібліогр.: 6 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 15A, 16G. https://nasplib.isofts.kiev.ua/handle/123456789/153368 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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Let I be a finite set without 0 and J a subset in I×I without diagonal elements (i,i). We define S(I,J) to be the semigroup with generators ei, where i∈I∪0, and the following relations: e0=0; e2i=ei for any i∈I; eiej=0 for any (i,j)∈J. In this paper we study finite-dimensional representations of such semigroups over a field k. In particular, we describe all finite semigroups S(I,J) of tame representation type. |
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Bondarenko, V.M. Tertychna, O.M. |
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Bondarenko, V.M. Tertychna, O.M. On tame semigroups generated by idempotents with partial null multiplication Algebra and Discrete Mathematics |
| author_facet |
Bondarenko, V.M. Tertychna, O.M. |
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Bondarenko, V.M. |
| title |
On tame semigroups generated by idempotents with partial null multiplication |
| title_short |
On tame semigroups generated by idempotents with partial null multiplication |
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On tame semigroups generated by idempotents with partial null multiplication |
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On tame semigroups generated by idempotents with partial null multiplication |
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On tame semigroups generated by idempotents with partial null multiplication |
| title_sort |
on tame semigroups generated by idempotents with partial null multiplication |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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https://nasplib.isofts.kiev.ua/handle/123456789/153368 |
| citation_txt |
On tame semigroups generated by idempotents with partial null multiplication / V.M. Bondarenko, O.M. Tertychn // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — С. 15–22. — Бібліогр.: 6 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT bondarenkovm ontamesemigroupsgeneratedbyidempotentswithpartialnullmultiplication AT tertychnaom ontamesemigroupsgeneratedbyidempotentswithpartialnullmultiplication |
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2025-11-26T21:36:15Z |
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2025-11-26T21:36:15Z |
| _version_ |
1849890415981887488 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2008). pp. 15 – 22
c© Journal “Algebra and Discrete Mathematics”
On tame semigroups generated by idempotents
with partial null multiplication
Vitaliy M. Bondarenko, Olena M. Tertychna
Communicated by V. V. Kirichenko
Abstract. Let I be a finite set without 0 and J a subset
in I × I without diagonal elements (i, i). We define S(I, J) to
be the semigroup with generators ei, where i ∈ I ∪ 0, and the
following relations: e0 = 0; e2i = ei for any i ∈ I; eiej = 0 for any
(i, j) ∈ J . In this paper we study finite-dimensional representations
of such semigroups over a field k. In particular, we describe all finite
semigroups S(I, J) of tame representation type.
Introduction
We study finite-dimensional representations over a field k of semigroups
generated by idempotents.
Let I be a finite set without 0 and J a subset in I×I without diagonal
elements (i, i). We define S(I, J) to be the semigroup with generators ei,
where i ∈ I ∪ 0, and the following relations:
1) e0 = 0;
2) e2i = ei for any i ∈ I;
3) eiej = 0 for any pair (i, j) ∈ J .
The set of all semigroups of the form S(I, J) is denoted by I. We
call S(I, J) ∈ I a semigroup generated by idempotents with partial null
multiplication.
2000 Mathematics Subject Classification: 15A, 16G.
Key words and phrases: semigroup, representation, tame type, the Tits form.
16 On tame semigroups generated by idempotents
In this paper we give a criterion for S(I, J) to be of finite represen-
tation type and a criterion for a finite S(I, J) to be tame (note that any
semigroup S(I, J) of finite type is finite).
1. Formulation of the main results
Throughout the paper, k denotes a field. All vector space are finite-
dimensional vector space over k. Under consideration maps, morphisms,
etc., we keep the right-side notation.
Let S be a semigroup and let Mn(k) denotes the algebra of all n× n
matrices with entries in k. A matrix representation of S (of degree n)
over k is a homomorphism T from S to the multiplicative semigroup of
Mn(k). If there is an identity (resp. zero) element a ∈ S, we assume that
the matrix T (a) is identity (resp. zero). Since Mn(k) can be considered
as the algebra of all linear transformations of any fixed n-dimensional
vector space, we can consider representations of the semigroup S in terms
of vector spaces and linear transformations. Thus, a representation of S
over k is a homomorphism ϕ from S to the multiplicative semigroup
of the algebra EndkU with U being a finite-dimensional vector space.
Two representation ϕ : S → EndkU and ϕ′ : S → EndkU
′ are called
equivalent if there is a linear map σ : U → U ′ such that ϕσ = ϕ′.
A representation ϕ : S → EndkU of S is also denoted by (U,ϕ). By
the dimension of (U,ϕ) one means the dimension of U . The represen-
tations of S form a category which will be denoted by repk S (it has as
morphisms from (U,ϕ) to (U ′, ϕ) the maps σ such that ϕσ = ϕ′). Since
representations X,Y ∈ S are equivalent iff they are isomorphic as objects
of repk S, we will use both the terms.
In an analogous way we can define representations of the semigroup
S over a (not necessarily finite-dimensional) k-algebra Λ; in this case
we must take free Λ-modules of finite rank instead of finite-dimensional
vector spaces.
We say that a semigroup is of finite representation type over k if it
has only finitely many equivalent classes of indecomposable representa-
tions (over k), and of infinite type if otherwise. Further, we say that a
semigroup is of tame (respectively, wild) type, or simply tame (respec-
tively, wild), if so is the problem of classifying its representations (precise
definitions are given below).
Let S = S(I, J) ∈ I and J = {(i, j) ∈ (I × I) \ J | i 6= j}. We
may assume, without loss of generality, that I = {1, 2, . . . ,m}. With the
semigroup S = S(I, J) we associate the quadratic form fS(z) : Zm → Z
V. Bondarenko, O. Tertychna 17
in the following way:
fS(z) =
∑
i∈I
z2
i −
∑
(i,j)∈J
zizj .
We call fS(z) the quadratic form of the semigroup S.
In this paper we prove the following theorems.
Theorem 1. A semigroup S(I, J) is of finite representation type over k
if and only if its quadratic form is positive (then S(I, J) is finite).
Theorem 2. Let S(I, J) be a finite semigroup. Then S(I, J) is tame
over k if its quadratic form is nonnegative, and wild if otherwise.
2. Connections between representations of S(I, J)S(I, J)S(I, J) and
representations of quivers
We first recall the notion of representations of a quiver [1].
Let Q = (Q0, Q1) be a (finite) quiver, where Q0 is the set of its
vertices and Q1 is the set of its arrows α : x→ y.
A representation of the quiver Q = (Q0, Q1) over a field k is a pair
R = (V, γ) formed by a collection V = {Vx |x ∈ Q0} of vector spaces
Vx and a collection γ = {γα |α : x → y runs throughQ1} of linear maps
γα : Vx → Vy. A morphism from R = (V, γ) to R′ = (V ′, γ ′) is given by
a collection λ = {λx |x ∈ Q0} of linear maps λx : Vx → V ′
x, such that
γαλy = λxγ
′
α for any arrow α : x→ y.
The category of representations of Q = (Q0, Q1) will be denoted by
repk Q.
In an analogous way we can define representations of the quiver Q over
a (not necessarily finite-dimensional) k-algebra Λ; in this case we must
take free Λ-modules of finite rank instead of finite-dimensional vector
spaces.
A quiver Q is said to be of finite representation type over k if repk Q
has only finitely many isomorphism classes of indecomposable represen-
tations (over k), and of infinite representation type if otherwise. Further,
Q is said to be of tame (respectively, wild) representation type, or simply
tame (respectively, wild), if so is the problem of classifying its represen-
tations (precise definitions are given below).
Now we proceed to investigate connections between representations
of S(I, J) and representations of quivers.
We identify a linear map α of U = U1 ⊕ . . . Up into V = V1 ⊕ . . . Vq
with the matrix (αij), i = 1, . . . , p, j = 1, . . . , q, where αij : Ui → Vj are
18 On tame semigroups generated by idempotents
the linear maps induced by α (then the sum and the composition of maps
are given by the matrix rules).
For a finite set X and Y ⊆ X ×X, we denote by Q(X,Y ) the quiver
with vertex set X and arrows a→ b, (a, b) ∈ Y .
Let S = S(I, J), where, as before, I = {1, 2, . . . ,m}. Define the func-
tor F from repk Q(I, J) to repk S(I, J) as follows. F = F (I, J) assigns
to each object (V, γ) ∈ repk Q(I, J) the object (V ′, γ ′) ∈ repk S(I, J),
where V ′ = ⊕i∈IVi, (γ ′(ei))jj = 1Vj
if i = j, (γ ′(ei))ij = γij if (i, j) ∈ J ,
and (γ ′(ei))js = 0 in all other cases. F assigns to each morphism λ of
repk Q(I, J) the morphism ⊕i∈Iλi of repk S(I, J).
Proposition 1. The functor F = F (I, J) : repk Q(I, J) → repk S(I, J)
is full and faithful.
Proof. It is obvious that the functor F is faithful. It remains to prove
that it is full. Let δ be a morphism from (V, γ)F = (V ′, γ ′) to (W,σ)F =
(W ′, σ′). In other words, δ is a linear map of V ′ into W ′ such that
γ ′(es)δ = δσ′(es) for s = 1, . . . ,m. We will consider these equalities as
matrix ones (taking into account that V ′ = ⊕i∈IVi and W ′ = ⊕i∈IWi)
and denote by [s, i, j] the scalar equality (γ ′(es)δ)ij = (δσ′(es))ij , induced
by the (matrix) equality γ ′(es)δ = δσ′(es).
From an equation [j, i, j] with j 6= i it follows that δij = 0, and
consequently δ is a diagonal matrix: δ = δ11⊕ δ22⊕· · ·⊕ δmm. Further, if
α : i→ j is an arrow of the quiver Q(I, J), then from the equation [i, i, j]
we have that γαδjj = δiiσα. Consequently, a collection δ = {δss | s =
1, . . . ,m} is a morphism from (V, γ) to (W,σ). Since δ = δ11⊕δ22⊕· · ·⊕
δmm, we have that δ = λF , where λ = δ, as claimed.
Proposition 2. If the quiver Q(I, J) has no oriented cycles, then each
object of repk S(I, J) is isomorphic to an object of the form XF (I, J) ⊕
(W, 0), where X is an object of repk Q(I, J) (W is a vector space of
dimension d ≥ 0 and 0 : W →W is the zero map).
Proof. For simplicity, the quiver Q(I, J) is denoted by Q = (Q0, Q1).
The proof will be by induction on m, the case m = 0, 1 being trivial.
Now let m > 1 and let R = (U,ϕ) be a representation of S(I, J). Fix
s ∈ Q0 such that there is no arrow i → s; obviously, one can assume
that s = m. We consider the subsemigroup S′ of S generated by ei,
i ∈ I ′ ∪ 0, where I ′ = {1, . . . ,m − 1}. Then S′ = S(I ′, J ′) with J ′ =
{(i, j) ∈ I × I | i, j ∈ I ′}, and Q′ = Q(I ′, J ′) is the full subquiver of Q
with vertex set Q′
0 = I ′.
Denote by R′ = (U,ϕ′) the restriction of R to S′ (ϕ′(x) = ϕ(x) for any
x ∈ S′). It follows by induction that R′ ∼= R′ = X ′F (I ′, J ′) ⊕ (W ′, 0),
V. Bondarenko, O. Tertychna 19
where X ′ is a representation of the quiver Q(I ′, J ′). Let R′ = (U,ϕ′)
and X ′ = (V ′, γ′) with V ′ = {V ′
i | i ∈ Q′
0} and γ′ = {γ′α |α : i →
j runs throughQ′
1}. Since R′ ∼= R′, there exists a linear map σ : U →
U = V ′
1⊕V
′
2⊕. . .⊕V
′
m−1⊕W
′ such that ϕ′σ = ϕ′. Then the representation
R = (U,ϕ) is equivalent to the representation R = (U,ϕ), where ϕ(ei) =
ϕ′(ei) for any i = 1, . . . ,m− 1 and ϕ(em) = ϕ(em)σ (because, for i 6= m,
ϕ′(ei) = ϕ′(ei)σ = ϕ(ei)σ, and so ϕ(x) = ϕ(x)σ for any x ∈ S).
We consider the representation R = (U,ϕ) in more detail. We set
Vm = W ′ and consider ϕ as a matrix (taking into account that U =
V1 ⊕ V2 ⊕ . . . ⊕ Vm−1 ⊕ Vm). For (p, q) ∈ J , we denote by [p, q, i, j]
the scalar equality [ϕ(ep)ϕ(eq)]ij = 0, induced by the (matrix) equality
ϕ(ep)ϕ(eq) = 0 (the last equation holds since epeq = 0 in S(I, J)). It
follows from [m, q, i, q] (for any fixed q 6= m) that (ϕ(em))iq = 0, and
consequently (ϕ(em))ij = 0 for any (i, j) ∈ I × I ′.
We first consider two special cases: a) ϕmm = 0; b) ϕmm = 1 = 1Vm
.
In case a) (ϕ)2 = ϕ implies ϕ = 0 and so R = XF (I, J)⊕ (W, 0) with
X = (V, γ), where V = {V ′
1 . . . , V
′
m−1, 0}, γα = γ′α for α ∈ Q′
1, γα = 0 for
α /∈ Q′
1 and W = W ′.
In case b) an equality [p,m, p,m] for (p,m) /∈ J implies (ϕ)pm = 0 and
so R = XF (I, J)⊕ (W, 0) with X = (V, γ), where V = {V ′
1 . . . , V
′
m−1, 0},
γα = γ′α for α ∈ Q′
1, γα = 0 for α /∈ Q′
1 and W = W ′.
Now we consider the general case. Since (ϕmm)2 = ϕmm, there is an
invertible map ν = (ν1, ν2) : Vm →W1 ⊕W2 such that
ϕmm(ν1, ν2) = (ν1, ν2)
(
1 0
0 0
)
,
where 1 = 1W1
. Then the representation R′ = (U,ϕ′) is isomorphic to
the the representation R̂′ = (Û , ϕ̂′), where Û = Û1⊕Û2⊕ . . .⊕Ûm+1 with
Ûi = Vi for i = 1, . . .m−1, Ûm = W1, Ûm+1 = W2, and ϕ̂′(ei) = ϕ′(ei) for
i = 1, . . .m− 1, (ϕ̂′(em))ij = (ϕ′(em))ij for (i, j) ∈ I ′× I ′, (ϕ̂′(em))ij = 0
for i = m,m + 1, j ∈ I ′, (ϕ̂′(em))m,mj = 1 = 1W1
, (ϕ̂′(em))m,m+1 = 0,
(ϕ̂′(em))m+1,m = 0, (ϕ̂′(em))m+1,m+1 = 0 (for instance, one can take the
isomorphism β : R̂′ → R′ with ϕ̂′(ei) = µ−1R′µ, where µ = 1U1
⊕ . . . ⊕
1Um−1
⊕ ν.
From (ϕ̂′(ei))
2 = ϕ̂′(ei) it follows that ϕ̂′(em))i,m+1 = 0 for any i ∈ I ′
(see the partial case a)); (then ϕ̂′(em))i,m+1 = 0 for any i = 1, . . . ,m+1).
From the scalar equalities [p,m, p,m] for (p,m) /∈ J implies (ϕ̂)pm = 0
(see the partial case b)). Thus, R = (Û , ϕ̂) ∼= R = (U,ϕ) has the form
XF (I, J) ⊕ (W, 0), where X = (V, γ) with V = {Ûi | i ∈ Q0}, γ =
{γα |α : i → j runs throughQ1} with γα = γ′α for α ∈ Q′
1, γα = ϕ̂(em)ij
for α /∈ Q′
1 (then j = m), and W = Ŵm+1, as claimed.
20 On tame semigroups generated by idempotents
Denote by rep◦
k S(I, J) the full subcategory of repk S(I, J) consist-
ing of all objects that have no objects (W, 0), with W 6= 0, as direct
summands.
We have as an immediate consequence of Propositions 1 and 2 the
following statement.
Corollary 1. If the quiver Q(I, J) has no oriented cycles, then the func-
tor F = F (I, J), viewed as a functor from repk Q(I, J) to rep◦
k S(I, J),
is an equivalence of categories.
3. Proof of Theorems 1 and 2
In [1] P. Gabriel introduced the quadratic Tits form qQ : ZQ0 → Z of a
quiver Q = (Q0, Q1):
qQ(z) =
∑
i∈Q0
z2
i −
∑
i→j
zizj ,
where i→ j runs throughQ1, and proved thatQ is of finite representation
type if and only if its Tits form is positive.
The definitions of the quadratic Tits form of a quivers and the quad-
ratic form of a semigroup S(I, J) ∈ I immediately imply the following
lemma.
Lemma 1. Let S = S(I, J) ∈ I and Q = Q(I, J). Then the quadratic
forms fS(z) and qQ(z) coincide.
Now we prove Theorem 1. In [3] one proves that a semigroup S(I, J)
is finite if and only if the quiver Q(I, J) has no oriented cycles (the
Tits form of which are not positive [6]). Then Theorem 1 follows from
Corollary 1, Lemma 1 and the above-mentioned Gabriel’s results.
Before we begin to prove Theorem 2, we recall precise definitions of
tame and wild semigroups (see general definitions in [2]).
For a semigroup S and a k-algebra Λ, we denote by RΛ(S) the set of
all representations of S over Λ. By L(Λ) we denote the category of left
finite-dimensional (over k) Λ-modules.
Let S be a semigroup and Λ = K1 = k[x]. We say that a representa-
tion N = (U,ϕ) from repk S is generated by a representation M = (V, ψ)
from RΛ(S) if, for some X ∈ L(Λ), N ∼= M ⊗X = (V ⊗X,ψ⊗ 1X) (the
tensor products are considered over Λ).
We assume first that the field k is separable closed. The semigroup
S is called tame if, for any fixed dimension d, there exist finitely many
elements Mi of RΛ(S) such that, up to isomorphism, each indecompo-
sable object of repk S (of the dimension d) is generated by Mi for some
V. Bondarenko, O. Tertychna 21
i. Such a set {Mi} is called a parametrizing family of representations of
S of dimension d.
When the field k is not separable closed, the semigroup S is called
tame, if it is tame over the separable closure k of k (in the case of infinite
k one can take k itself in place of k).
Now we give a definition of wild semigroups.
Let S be a semigroup and Λ = K2 = k < x, y > the free associative
k-algebra in two noncommuting variables x and y. A representation
M = (V, ψ) of S over Λ is said to be perfect if it satisfies the following
conditions:
1) the representation M ⊗X = (V ⊗X,ψ ⊗ 1X) (of S over k) with
X ∈ L(Λ) is indecomposable if so is X;
2) the representations M ⊗ X and M ⊗ X ′ are nonisomorphic if so
are X and X ′.
The semigroup S is called wild over k if it has a perfect representation
over Λ.
In an analogous way one can define tame and wild quivers; the set
of all representations of a quiver Q over an algebra Λ will be denote by
RΛ(Q).
Now we prove Theorem 2.
Let S = S(I, J) be a finite semigroup. Then the quiver Q(I, J) has
no oriented cycles (see above). From the papers [4, 5] on tame quivers
and the paper [6] on integral quadratic forms it follows that a quiver Q is
tame if its Tits form is nonnegative, and wild if otherwise. Then the first
part of Theorem 2 follows from Lemma 1, Corollary 1 and the obvious
fact that, for Λ = k[x], the map FΛ = FΛ(I, J) from RΛ(Q) to RΛ(S),
which is defined analogously to the functor F = F (I, J) on objects, “pre-
serves” (from left to right) parametrizing families of any fixed dimen-
sion. Analogously, the second part of Theorem 2 follows from Lemma 1,
Corollary 1 and the obvious fact that, for Λ = k < x, y >], the map
FΛ = FΛ(I, J) from RΛ(Q) to RΛ(S), which is defined analogously to the
functor F = F (I, J) on objects, “preserves” (from left to right) perfect
representations over Λ.
Theorems 1 and 2 are proved.
References
[1] P. Gabriel, Unzerlegbare Darstellungen, Manuscripts Math., V.6, 1972, pp. 71-
103,309.
[2] Yu. A. Drozd, Tame and wild matrix problems, Lecture Notes in Math., N.832,
1980, pp. 242-258.
[3] V. M. Bondarenko, O. M. Tertychna, On infiniteness of type of infinite semigroups
generated by idempotents with partial null multiplication, Trans. Inst. of Math. of
NAS of Ukraine, N.3, 2006, pp. 23-44 (Russian).
22 On tame semigroups generated by idempotents
[4] L. A. Nazarova, Representations of quivers of infinite type, Izv. Akad. Nauk. SSSR,
V.37, 1973, pp. 752-791 (Russian).
[5] P. Donovan, M. R. Freislich, The representation theory of finite graphs and asso-
ciated algebras, Carleton Lecture Notes, N.5, 1973, pp. 3-86.
[6] A. V. Roiter, Roots of integral quadratic forms, Trudy Mat. Inst. Steklov, V.148,
1978, pp. 201-210,277 (Russian).
Contact information
V. M. Bondarenko Institute of Mathematics, NAS, Kyiv,
Ukraine
E-Mail: vit-bond@imath.kiev.ua
O. M. Tertychna Kyiv National Taras Shevchenko University,
Kiev, Ukraine
E-Mail: tertychna@mail.ru
Received by the editors: 13.05.2008
and in final form 14.10.2008.
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