On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents
Let I be a finite set (without 0) and J a subset of I × I without diagonal elements. Let S(I, J) denotes the semigroup generated by e₀ = 0 and ei, i ∈ I, with the following relations: e²i = ei for any i ∈ I, eiej = 0 for any (i, j) ∈ J. In this paper we prove that, for any finite semigroup S = S(I...
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Інститут прикладної математики і механіки НАН України
2012
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| Назва видання: | Algebra and Discrete Mathematics |
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| Цитувати: | On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents / V. Bondarenko, O. Tertychna // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 168–173. — Бібліогр.: 3 назв. — англ. |
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irk-123456789-1522362019-06-10T01:26:16Z On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents Bondarenko, V. Tertychna, O. Let I be a finite set (without 0) and J a subset of I × I without diagonal elements. Let S(I, J) denotes the semigroup generated by e₀ = 0 and ei, i ∈ I, with the following relations: e²i = ei for any i ∈ I, eiej = 0 for any (i, j) ∈ J. In this paper we prove that, for any finite semigroup S = S(I, J) and any its matrix representation M over a field k, each matrix of the form ∑i∈IαiM(ei) with αi ∈ k is similar to the direct sum of some invertible and zero matrices. We also formulate this fact in terms of elements of the semigroup algebra. 2012 Article On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents / V. Bondarenko, O. Tertychna // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 168–173. — Бібліогр.: 3 назв. — англ. 1726-3255 2010 MSC:16G, 20M30. http://dspace.nbuv.gov.ua/handle/123456789/152236 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let I be a finite set (without 0) and J a subset of I × I without diagonal elements. Let S(I, J) denotes the semigroup generated by e₀ = 0 and ei, i ∈ I, with the following relations: e²i = ei for any i ∈ I, eiej = 0 for any (i, j) ∈ J. In this paper we prove that, for any finite semigroup S = S(I, J) and any its matrix representation M over a field k, each matrix of the form ∑i∈IαiM(ei) with αi ∈ k is similar to the direct sum of some invertible and zero matrices. We also formulate this fact in terms of elements of the semigroup algebra. |
| format |
Article |
| author |
Bondarenko, V. Tertychna, O. |
| spellingShingle |
Bondarenko, V. Tertychna, O. On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents Algebra and Discrete Mathematics |
| author_facet |
Bondarenko, V. Tertychna, O. |
| author_sort |
Bondarenko, V. |
| title |
On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_short |
On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_full |
On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_fullStr |
On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_full_unstemmed |
On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_sort |
on 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/152236 |
| citation_txt |
On 0-semisimplicity of linear hulls of generators for semigroups generated by idempotents / V. Bondarenko, O. Tertychna // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 168–173. — Бібліогр.: 3 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT bondarenkov on0semisimplicityoflinearhullsofgeneratorsforsemigroupsgeneratedbyidempotents AT tertychnao on0semisimplicityoflinearhullsofgeneratorsforsemigroupsgeneratedbyidempotents |
| first_indexed |
2025-07-13T02:37:11Z |
| last_indexed |
2025-07-13T02:37:11Z |
| _version_ |
1837497570562146304 |
| fulltext |
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 14 (2012). Number 2. pp. 168 – 173
c© Journal “Algebra and Discrete Mathematics”
On 0-semisimplicity of linear hulls of generators
for semigroups generated by idempotents
Vitaliy M. Bondarenko, Olena M. Tertychna
Communicated by V. V. Kirichenko
Dedicated to 75th Birthday of Andrej V. Roiter
Abstract. Let I be a finite set (without 0) and J a subset of
I × I without diagonal elements. Let S(I, J) denotes the semigroup
generated by e0 = 0 and ei, i ∈ I, with the following relations:
e2
i = ei for any i ∈ I, eiej = 0 for any (i, j) ∈ J . In this paper
we prove that, for any finite semigroup S = S(I, J) and any its
matrix representation M over a field k, each matrix of the form
∑
i∈I αiM(ei) with αi ∈ k is similar to the direct sum of some
invertible and zero matrices. We also formulate this fact in terms of
elements of the semigroup algebra.
Introduction
We study matrix representations over a field k of semigroups generated
by idempotents.
Let I be a finite set without 0 and J a subset of I × I without the
diagonal elements (i, i), i ∈ I. Let S(I, J) denotes the semigroup with
zero generated by ei, i ∈ I ∪ 0, with the following defining relations:
1) e2
0 = e0, e0ei = eie0 = e0 for any i ∈ I ∪ 0, i. e. e0 = 0 is the zero
element;
2010 MSC: 16G, 20M30.
Key words and phrases: semigroup, matrix representations, defining relations,
0-semisimple matrix.
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V. Bondarenko, O. Tertychna 169
2) e2
i = ei for any i ∈ I;
3) eiej = 0 for any pair (i, j) ∈ J .
Every semigroup S(I, J) ∈ I is called a semigroup generated by idem-
potents with partial null multiplication (see, e.g., [2]). The set of all semi-
groups S(I, J) with |I|=n will be denoted by In. Put I = ∪∞
n=1In.
With each semigroup S = S(I, J) ∈ I we associate the directed
graph Λ(S) with set of vertices Λ0(S) = {ei | i ∈ I} and set of arrows
Λ1(S) = {ei → ej | (i, j) ∈ J}. Denote by Λ(S) the directed graph which
is the complement of the graph Λ(S) to the full directed graph without
loops, i.e. Λ0(S) = Λ0(S) and ei → ej belongs to Λ1(S) if and only if
i 6= j and ei → ej does not belong to Λ1(S). Obviously, the semigroup
S ∈ I is uniquely determined by each of these directed graphs.
In [1] the authors proved that a semigroup S = S(I, J) is finite if and
only if the graph Λ(S) is acyclic.
We call a quadratic matrix A over a field k α-semisimple, where α ∈ k,
if one of the following equivalent conditions holds:
a) rank(A − αE)2 = rank(A − αE) (E denotes the identity matrix);
b) A − αE is similar to the direct sum of some invertible and zero
matrices;
c) the minimal polynomial mA(x) of A is not devided by (x − α)2;
d) there is a polynomial f(x) = (x − α)g(x) such that g(α) 6= 0 and
f(A) = 0.
If A is α-semisimple for all α ∈ k, then it is obviously semisimple in
the classical sense.
In this paper we study 0-semisimple matrices associated with matrix
representations of a finite semigroup S from I (formulating also the
received results in terms of elements of the semigroup algebra).
1. Formulation of the main results
Let S be a semigroup and k be a field. Let Mm(k) denotes the algebra
of all m × m matrices with entries in k.
A matrix representation of S (of degree m) over k is a homomorphism
R from S to the multiplicative semigroup of Mm(k). If there is a zero
(resp. an identity) element a ∈ S, one can assume that the matrix R(a) is
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170 On 0-semisimplicity of linear hulls of generators
zero (resp. identity)1. Two representation R : S → Mm(k) and R′ : S →
Mm(k) are called equivalent if there is an invertible matrix C such that
C−1R(x)C = R′(x) for all x ∈ S.
In this paper we prove the following theorem2.
Theorem 1. Let S = S(I, J) be a finite semigroup from I and R a
matrix representation of S. Then, for any αi ∈ k, where i runs over I,
the matrix
∑
i∈I αiR(ei) is 0-semisimple.
Reformulate the theorem in terms of elements of the semigroup algebra
kS1, where S1 = S ∪ {1}. As usual, we identify the zero element of the
semigroup with the zero element of the semigroup algebra; then
kS1 = {
∑
s∈S\0
βss + β11 | βs, β1 ∈ k}.
We call an element g ∈ kS1 0-semisimple if the minimal polynomial mg(x)
of g is not devided by x2.
Set EI = {ei | i ∈ I} and let kEI denotes the k-linear hull of the
generators ei ∈ EI , i. e. kEI = {
∑
i∈I αiei | αi ∈ k}.
Theorem 1 is equivalent to the following one.
Theorem 2. Let S = S(I, J) be a finite semigroup from I. Then any
element g ∈ kEI is 0-semisimple.
Note that Theorem 1 follows from the results of [2, 3] on a normal
form of matrix representations of finite semigroups S(I, J), but here we
prove this fact directly.
2. Proof of Theorem 1
We apply induction on n = |I|. The case n = 1 is obvious since
any matrix representation of the semigroup S({1},∅) is given by an
idempotent matrix.
Suppose that Theorem 1 is proved for all matrix representations of
all finite semigroups S(I, J) ∈ In, and prove that the theorem holds for
S(I, J) ∈ In+1.
1It is easy to show that in this case we "lose" the only indecomposable representation
P of degree 1 with P (x) = 0 for all x ∈ S \ a and P (a) = 1 (resp. P (x) = 0 for all
x ∈ S).
2Notice that the theorem is also valid without the restrictions which has been
discussed in note 1.
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V. Bondarenko, O. Tertychna 171
Let S = S(I, J) be an arbitrary finite semigroup from In+1. One
may assume without loss of generality that I = {1, 2, . . . , n + 1}. We
show that for a fixed matrix representation R of S(I, J) and a vec-
tor α = (α1, . . . , αn+1) ∈ kn+1, the matrix P (α) = P (α1, . . . , αn+1) =
∑n+1
i=1
αiR(ei) is 0-semisimple.
Put A1 = R(e1), . . . , An+1 = R(en+1). Then A2
i = Ai for all i ∈ I,
AiAj = 0 for all (i, j) ∈ J and P (α) = α1A1 + . . . αn+1An+1.
Since the directed graph Λ(S) is acyclic (see Introduction), one can
fix a vertex el such that there are no arrows l → s, where s ∈ I. Consider
the subsemigroup S′ of S generated by e′
0 = e0, e′
1 = e1, . . . , e′
l−1
=
el−1, e′
l = el+1, . . . , e′
n = en+1. Obviously, the directed graph Λ(S′) coin-
cides with Λ(S) \ el. By the induction hypothesis for the restriction T of
the representation R on S′, the matrix
α1A1 + . . . + αl−1Al−1 + αl+1Al+1 + . . . + αn+1An+1
is 0-semisimple. Denote this matrix by P ′(α1, . . . , αl−1, αl+1, . . . , αn+1) =
P ′(α′), where α′ = (α1, . . . , αl−1, αl+1, . . . , αn+1). Then
P (α) = P ′(α′) + αlAl.
From the fact that there are no arrows l → s it follows that, for j 6= l,
elej = 0 and consequently AlAj = 0. Then AlP
′(α′) = 0 and it remains
only to apply the following statement: if A is an idempotent matrix, B is
a 0-semisimple matrix and AB = 0 then γA + δB is 0-semisimple for any
γ, δ ∈ k.
Instead we prove a more general statement.
Proposition 1. Let A and B be 0-semisimple matrices of size m×m such
that AB = 0. Then, for any γ, δ ∈ k, the matrix γA + δB is 0-semisimple.
Because λM with λ ∈ k is 0-semisimple provided that so is M , it is
sufficient to consider the case γ = δ = 1.
By condition b) of the definition of a 0-semisimple matrix there is an
invertible matrix X such that
X−1AX =
(
A0 0
0 0
)
(1)
where A0 is invertible. From AB = 0 it follows that
X−1BX =
(
0 0
P Q
)
(2)
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172 On 0-semisimplicity of linear hulls of generators
for some matrices P and Q (the matrices in the right parts of (1) and (2)
are partitioned conformally).
From condition a) for the matrix B (see the definition of a 0-semisimple
matrix) we have that
rank Q
(
P Q
)
= rank
(
P Q
)
(3)
But since rank Q
(
P Q
)
≤ rank Q (by the formula rank MN ≤
rank M) and rank
(
P Q
)
≥ rank Q, it follows from (3) that
rank
(
P Q
)
= rank Q (4)
and consequently there exists an invertible matrix Y such that P = QY .
Then
(
E1 0
−Y E2
)−1(
0 0
P Q
)(
E1 0
−Y E2
)
=
(
0 0
0 Q
)
(5)
where E1, E2 are the identical matrices.
From (2) and (5) it follows that the 0-semisimple matrix B is similar
to the matrix
(
0 0
0 Q
)
and hence the matrix Q is 0-semisimple. Then (by condition b) of the
definition of a 0-semisimple matrix) there is an invertible matrix Z such
that
Z−1QZ =
(
Q0 0
0 0
)
where Q0 is invertible, and consequently
(
E3 0
0 Z
)−1(
0 0
P Q
)(
E3 0
0 Z
)
=
0 0 0
P0 Q0 0
P1 0 0
(6)
where E3 is the identical matrix and
(
P0
P1
)
= Z−1P ; moreover by the
equality (4) we have that
P1 = 0. (7)
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V. Bondarenko, O. Tertychna 173
So, if one denotes the product of the matrices X and
(
E3 0
0 Z
)
by
T , then (see (1), (2), (6), (7))
T −1AT =
A0 0 0
0 0 0
0 0 0
, T −1BT =
0 0 0
P0 Q0 0
0 0 0
,
from which it follows that the matrix A + B is similar to the direct sum
of the invertible matrix
(
A0 0
P0 Q0
)
and some zero matrix.
Proposition 1, and therefore Theorem 1, are proved.
References
[1] 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, N3 (2006), pp. 23-44 (in Russian).
[2] V. M. Bondarenko, O. M. Tertychna, On tame semigroups generated by idempotents
with partial null multiplication // Algebra Discrete Math. – 2008. – N4. – P. 15–22.
[3] O. M. Tertychna, Matrix representations of semigroups generated by idempotents
with partial null multiplication. Thesis for a candidate’s degree by speciality 01.01.06
– algebra and number theory. – Kyiv National Taras Shevchenko University, 2009. –
167 p (In Ukrainian).
Contact information
V. M. Bondarenko Institute of Mathematics, NAS, Kyiv, Ukraine
E-Mail: vitalij.bond@gmail.com
O. M. Tertychna Vadim Hetman Kyiv National Economic Uni-
versity, Kiev, Ukraine
E-Mail: olena-tertychna@mail.ru
Received by the editors: 19.11.2012
and in final form 09.01.2013.
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