On hereditary reducibility of 2-monomial matrices over commutative rings
In this paper we introduce the notion of hereditary reducibility for some matrices and indicate one general condition of the introduced reducibility.
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Цитувати: | On hereditary reducibility of 2-monomial matrices over commutative rings / V.M. Bondarenko, J. Gildea, A.A. Tylyshchak, N.V. Yurchenko // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 1–11. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1884172023-03-01T01:26:48Z On hereditary reducibility of 2-monomial matrices over commutative rings Bondarenko, V.M. Gildea, J. Tylyshchak, A.A. Yurchenko, N.V. In this paper we introduce the notion of hereditary reducibility for some matrices and indicate one general condition of the introduced reducibility. 2019 Article On hereditary reducibility of 2-monomial matrices over commutative rings / V.M. Bondarenko, J. Gildea, A.A. Tylyshchak, N.V. Yurchenko // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 1–11. — Бібліогр.: 6 назв. — англ. 1726-3255 2010 MSC: 15B33, 15A30 http://dspace.nbuv.gov.ua/handle/123456789/188417 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we introduce the notion of hereditary reducibility for some matrices and indicate one general condition of the introduced reducibility. |
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Bondarenko, V.M. Gildea, J. Tylyshchak, A.A. Yurchenko, N.V. |
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Bondarenko, V.M. Gildea, J. Tylyshchak, A.A. Yurchenko, N.V. On hereditary reducibility of 2-monomial matrices over commutative rings Algebra and Discrete Mathematics |
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Bondarenko, V.M. Gildea, J. Tylyshchak, A.A. Yurchenko, N.V. |
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Bondarenko, V.M. |
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On hereditary reducibility of 2-monomial matrices over commutative rings |
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On hereditary reducibility of 2-monomial matrices over commutative rings |
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On hereditary reducibility of 2-monomial matrices over commutative rings |
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On hereditary reducibility of 2-monomial matrices over commutative rings |
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On hereditary reducibility of 2-monomial matrices over commutative rings |
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on hereditary reducibility of 2-monomial matrices over commutative rings |
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Інститут прикладної математики і механіки НАН України |
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2019 |
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http://dspace.nbuv.gov.ua/handle/123456789/188417 |
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On hereditary reducibility of 2-monomial matrices over commutative rings / V.M. Bondarenko, J. Gildea, A.A. Tylyshchak, N.V. Yurchenko // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 1–11. — Бібліогр.: 6 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT bondarenkovm onhereditaryreducibilityof2monomialmatricesovercommutativerings AT gildeaj onhereditaryreducibilityof2monomialmatricesovercommutativerings AT tylyshchakaa onhereditaryreducibilityof2monomialmatricesovercommutativerings AT yurchenkonv onhereditaryreducibilityof2monomialmatricesovercommutativerings |
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2025-07-16T10:27:04Z |
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2025-07-16T10:27:04Z |
_version_ |
1837798916314103808 |
fulltext |
“adm-n1” — 2019/3/22 — 12:03 — page 1 — #9
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 27 (2019). Number 1, pp. 1–11
c© Journal “Algebra and Discrete Mathematics”
On hereditary reducibility of 2-monomial
matrices over commutative rings
Vitaliy M. Bondarenko, Joseph Gildea,
Alexander A. Tylyshchak∗, and Natalia V. Yurchenko
Communicated by V. V. Kirichenko
Abstract. A 2-monomial matrix over a commutative ring
R is by definition any matrix of the form M(t, k, n) = Φ
(
Ik 0
0 tIn−k
)
,
0 < k < n, where t is a non-invertible element of R, Φ the companion
matrix to λn − 1 and Ik the identity k× k-matrix. In this paper we
introduce the notion of hereditary reducibility (for these matrices)
and indicate one general condition of the introduced reducibility.
Introduction
This paper is devoted to one class of monomial matrices over commu-
tative rings which first arose in studying indecomposable representations
of finite p-groups over local rings ([1]). They were studied more extensively
(in a more generally) in [2]–[6].
Let R be a commutative ring with Jacobson radical J(R) 6= 0 and t a
non-zero element from J(R). An n×n matrix over R is called 2-monomial
∗The paper was written during the research stay of the third author at the University
of Presov under the National Scholarship Programme of the Slovak Republic.
2010 MSC: 15B33, 15A30.
Key words and phrases: commutative ring, Jacobson radical, 2-monomial mat-
rix, hereditary reducible matrix, similarity, linear operator, free module.
“adm-n1” — 2019/3/22 — 12:03 — page 2 — #10
2 On hereditary irreducibility of 2-monomial matrices
concerning t, if it is a permutation similar to a matrix of the following form:
M(t, k, n) := Φn
(
Ik 0
0 tIn−k
)
=
0 . . . 0 0 . . . 0 t
1 . . . 0 0 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 1 0 . . . 0 0
0 . . . 0 t . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 0 0 . . . t 0
,
where 0 < k < n, Φn is the companion matrix to the polynomial xn − 1
and Is is the identity s× s matrix. Such a matrix M = M(t, k, n) is said
to be hereditary reducible if it similar to a matrix
M ′ =
(
M(t, k′, n′) ∗
0 N
)
, n′ 6= n,
and hereditary irreducible if otherwise.
The aim of this paper is to prove the following result.
Theorem 1. A 2-monomal matrix M(t, k, n) is hereditary reducible if k
and n are not coprime.
In the next section, we indicate a more detailed interpretation of the
idea of this statement.
1. Generalization of Theorem 1: formulation and proof
In this section we prove a more general theorem (from which Theorem 1
follows). Instead of R we consider the ring Z[λ] (of integer polynomials).
Let (n, k) denote the greatest common divisor of the natural numbers n
and k.
Theorem 2. Let n > k be positive integers, such that (n, k) > 1. Then for
any positive divisors d > 1 of the number (n, k), the matrix M(λ, k, n) ∈
M(n,Z[λ]) similar to a matrix of the following form
(
M(λ, k′, n′) B
0 A
)
∈ M(n,Z[λ]),
where k′ = k
d
and n′ = n
d
.
“adm-n1” — 2019/3/22 — 12:03 — page 3 — #11
Bondarenko, Gildea, Tylyshchak, Yurchenko 3
Through this section 0 < k < n, 0 < k′ < n′, and 0 < n′ < n. Before
we prove Theorem 2, we provide four other important results which we
need for the proof.
Proposition 1. Let n′|n. Then there exists an n× n′-matrix
S =
λs1 0 . . . 0
0 λs2 . . . 0
...
...
. . .
...
0 0 . . . λs
n′
λs
n′+1 0 . . . 0
0 λs
n′+2 . . . 0
...
...
. . .
...
0 0 . . . λs2n′
. . . . . . . . . . . . . . . . . . . . . . . . . . .
λs
n−n′+1 0 . . . 0
0 λs
n−n′+2 . . . 0
...
...
. . .
...
0 0 . . . λsn
,
where si > 0, i = 1, . . . , n, such that M(λ, k, n)S = SM(λ, k′, n′) if and
only if n
n′ =
k
k′
.
Proof. Let l1 = · · · = lk = 0, lk+1 = · · · = ln = 1, r1 = · · · = rk′ = 0 and
rk′+1 = · · · = rn′ = 1 be such that
M(λ, k, n) =
0 . . . 0 0 . . . 0 λln
λl1 . . . 0 0 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . λlk 0 . . . 0 0
0 . . . 0 λlk+1 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 0 0 . . . λln−1 0
and
M(λ, k′, n′) =
0 . . . 0 0 . . . 0 λr
n′
λr1 . . . 0 0 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . λr
k′ 0 . . . 0 0
0 . . . 0 λr
k′+1 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 0 0 . . . λr
n′
−1 0
.
“adm-n1” — 2019/3/22 — 12:03 — page 4 — #12
4 On hereditary irreducibility of 2-monomial matrices
We denote by (i, j) the scalar equality
(M(λ, k, n)S)ij = (SM(λ, k′, n′))ij .
Obviously, in each of matrices M(λ, k, n)S and SM(λ, k′, n′) there are
exactly n non-zero element, which are in the i, j positions (i-th row, j-th
column), where i ≡ j + 1 (mod n′). Let δn′(i) = (i− 1) mod n′ + 1 or,
equivalently, δn′(i) ≡ i (mod n′), 1 6 δn′(i) 6 n′. Thus M(λ, k, n)S =
SM(λ, k′, n′) if and only if scalar equalities
{
(i+ 1, δn′(i)) : λliλsi = λsi+1λ
rδ
n′ (i) (i = 1, . . . , n− 1),
(1, n′) : λlnλsn = λs1λr
n′
hold. Obviously, these equalities are equivalent to the equalities
{
(i+ 1, δn′(i)) : li + si = si+1 + rδ
n′ (i) (i = 1, . . . , n− 1),
(1, n′) : ln + sn = s1 + rn′ .
(1)
Assume that for some si > 0, i = 1, . . . , n M(λ, k, n)S = SM(λ, k′, n′).
Then (1) holds. Summing the equations (1), we obtain
n−1
∑
i=1
li +
n−1
∑
i=1
si + ln + sn =
n−1
∑
i=1
si+1 +
n−1
∑
i=1
rδ
n′ (i) + s1 + rn′ .
But since δn′(n) = n′ we have that
n
∑
i=1
li +
n
∑
i=1
si =
n
∑
i=1
si +
n
∑
i=1
rδ
n′ (i),
or
∑n
i=1 li =
∑n
i=1 rδn′ (i). This is equivalent to
∑n
i=1 li =
n
n′
∑n′
i=1 ri or
k = n
n′k
′ and n
n′ =
k
k′
.
Now, assume that n
n′ =
k
k′
and we want to prove that for some si > 0,
i = 1, . . . , n,
M(λ, k, n)S = SM(λ, k′, n′).
It remains to prove that the equations in (1) hold for non negative inte-
gers si. We will prove it for arbitrary integers si since that addition of
any number to si will also be a solution. Let s1 = 0, si+1 = li + si − rδ
n′ (i)
(i = 1, . . . , n − 1). It follows immediately that all, accept last equa-
tion in (1) hold and sn =
∑n−1
i=1 li −
∑n−1
i=1 rδ
n′ (i). If we replace sn by
“adm-n1” — 2019/3/22 — 12:03 — page 5 — #13
Bondarenko, Gildea, Tylyshchak, Yurchenko 5
∑n−1
i=1 li −
∑n−1
i=1 rδ
n′ (i) and s1 by 0 in the last equation in (1), we obtain
the following equation:
ln +
n−1
∑
i=1
li −
n−1
∑
i=1
rδ
n′ (i) = rn′ or ln +
n−1
∑
i=1
li =
n−1
∑
i=1
rδ
n′ (i) + rn′ .
This equation is equivalent to
n
∑
i=1
li =
n
∑
i=1
rδ
n′ (i)
and
∑n
i=1 li =
n
n′
∑n′
i=1 ri which in turn is equivalent to k = n
n′k
′ or n
n′ =
k
k′
.
The proof is complete.
Using a similar argument applied in the previous proof, we can state
the following result:
Proposition 2. Let n′|n, li > 0 (i = 1, . . . , n),
∑n
i=1 li = k,
M =
0 . . . 0 0 . . . 0 λln
λl1 . . . 0 0 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . λlk 0 . . . 0 0
0 . . . 0 λlk+1 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 0 0 . . . λln−1 0
.
Then there exists an n× n′-matrix
S =
λs1 0 . . . 0
0 λs2 . . . 0
...
...
. . .
...
0 0 . . . λs
n′
λs
n′+1 0 . . . 0
0 λs
n′+2 . . . 0
...
...
. . .
...
0 0 . . . λs2n′
. . . . . . . . . . . . . . . . . . . . . . . . . . .
λs
n−n′+1 0 . . . 0
0 λs
n−n′+2 . . . 0
...
...
. . .
...
0 0 . . . λsn
,
“adm-n1” — 2019/3/22 — 12:03 — page 6 — #14
6 On hereditary irreducibility of 2-monomial matrices
where si > 0, i = 1, . . . , n, such that MS = SM(λ, k′, n′) if and only if
n
n′ =
k
k′
.
Next, we provide a result regarding the similarity of M(λ, k, n) and a cer-
tain matrix.
Proposition 3. Let n′|n, n
n′ =
k
k′
. Than k′ < k and M(λ, k, n) is similar
(over Z[λ]) to a matrix of the form
M =
0 . . . 0 0 . . . 0 λln
λl1 . . . 0 0 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . λl
k′ 0 . . . 0 0
0 . . . 0 λl
k′+1 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 0 0 . . . λln−1 0
,
where l1 = · · · = lk′ = 0, lk′+1 = · · · = lk′+n−k = 1, lk′+n−k+1 = · · · =
ln = 0.
Proof. Clearly k′ < k as k
k′
= n
n′ > 1. Now, rearrange the rows and columns
of the matrix M(λ, k, n) in the order k−k′+1, k−k′+2, . . . n, 1, 2, . . . , k−k′
and denote the new matrix by M :
M =
0 . . . 0 0 . . . 0 λln
λl1 . . . 0 0 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . λl
k′ 0 . . . 0 0
0 . . . 0 λl
k′+1 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 0 0 . . . λln−1 0
,
where l1 = · · · = lk′ = 0, lk′+1 = · · · = lk′+n−k = 1, lk′+n−k+1 = · · · =
ln = 0.
The next result connects the previous two results.
“adm-n1” — 2019/3/22 — 12:03 — page 7 — #15
Bondarenko, Gildea, Tylyshchak, Yurchenko 7
Proposition 4. Let n′|n, k′ < k.
M =
0 . . . 0 0 . . . 0 λln
λl1 . . . 0 0 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . λl
k′ 0 . . . 0 0
0 . . . 0 λl
k′+1 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 0 0 . . . λln−1 0
,
where l1 = · · · = lk′ = 0, lk′+1 = · · · = lk′+n−k = 1, lk′+n−k+1 = · · · =
ln = 0. Then there exists an n× n′-matrix
S =
1 0 . . . 0
0 1 . . . 0
...
...
. . .
...
0 0 . . . 1
1 0 . . . 0
0 λs
n′+2 . . . 0
...
...
. . .
...
0 0 . . . λs2n′
. . . . . . . . . . . . . . . . . . . . . . . . . . .
λs
n−n′+1 0 . . . 0
0 λs
n−n′+2 . . . 0
...
...
. . .
...
0 0 . . . λsn
,
where si > 0, (i = n′ + 2, . . . , n), such that MS = SM(λ, k′, n′) if and
only if n
n′ =
k
k′
.
Proof. Clearly if MS = SM(λ, k′, n′), then n
n′ = k
k′
by Proposition 2.
Assume that n
n′ =
k
k′
. Let r1 = · · · = rk′ = 0, rk′+1 = · · · = rn′ = 1 such
that
M(λ, k′, n′) =
0 . . . 0 0 . . . 0 λr
n′
λr1 . . . 0 0 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . λr
k′ 0 . . . 0 0
0 . . . 0 λr
k′+1 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 0 0 . . . λr
n′
−1 0
.
“adm-n1” — 2019/3/22 — 12:03 — page 8 — #16
8 On hereditary irreducibility of 2-monomial matrices
We will prove that M(λ, k, n)S = SM(λ, k′, n′) for some si > 0 (i =
n′ + 2, . . . , n).
It remains to prove that (1) holds, where s1 = · · · = sk′ = sk′+1 = 0.
Let
s1 = 0, si+1 = li + si − rδ
n′ (i) (i = 1, . . . , n− 1). (2)
Using a similar argument to the one used in (1) all equations hold.
Furthermore, if n−k
n′
−k′
= n
n′ =
k
k′
> 1 it follows n′ − k′ < n− k,
l1 = · · · = lk′ = 0 = r1 = · · · = rk′
and
lk′+1 = · · · = ln′ = 1 = rk′+1 = · · · = rn′ .
Therefore li = ri (i = 1, . . . , n′) and si+1 = si (i = 1, . . . , n′) by (2). We
can also see that s1 = · · · = sn′ = sn′+1 = 0.
It remains to prove that si > 0 (i = n′ + 2, . . . , n). Let sn+1 = s1. It
follows from (2) end last equation from (1), that sn+1 = ln + sn − rδ
n′ (n),
which is equivalent to s1 = sn+1 = 0 and
si+1 =
i
∑
j=1
lj −
i
∑
j=1
rδ
n′ (j) (i = 1, . . . , n). (3)
Let us consider s(i) = si, as a function of an integer i (1 6 i 6 n + 1).
Then s(i) = 0 if 1 6 i 6 n′ +1. Thus, s(i) is a constant for 1 6 i 6 n′ +1.
If n′ + 1 6 i < i + 1 6 k′ + n − k + 1, then it follows from (3) that
s(i + 1) − s(i) = li − rδ
n′ (i) = 1 − rδ
n′ (i) > 0. Therefore s(i) either
increases or remains constant for each step and s(i) > s(n′ + 1) = 0.
If k′ + n − k + 1 6 i < i + 1 6 n + 1, then it follows from (3) that
s(i + 1) − s(i) = li − rδ
n′ (i) = 0 − rδ
n′ (i) 6 0. Consequently s(i) either
decreases or remains constant for each step and s(i) > s(n + 1) = 0.
Therefore, si = s(i) > 0 (i = n′ + 2, . . . , n).
Finally, we are in a position to prove our main result.
Proof of Theorem 2. Recall that 0 < k < n, k
k′
= n
n′ = d > 1 and
0 < n′ < n, 0 < k′ < k. By Propositin 3, M(λ, k, n) is similar (over Z[λ])
“adm-n1” — 2019/3/22 — 12:03 — page 9 — #17
Bondarenko, Gildea, Tylyshchak, Yurchenko 9
to a matrix of the form
M =
0 . . . 0 0 . . . 0 λln
λl1 . . . 0 0 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . λl
k′ 0 . . . 0 0
0 . . . 0 λl
k′+1 . . . 0 0
...
. . .
...
...
. . .
...
...
0 . . . 0 0 . . . λln−1 0
,
where l1 = · · · = lk′ = 0, lk′+1 = · · · = lk′+n−k = 1, lk′+n−k+1 = · · · =
ln = 0.
By Proposition 4 there exists an n× n′-matrix
S =
1 0 . . . 0
0 1 . . . 0
...
...
. . .
...
0 0 . . . 1
1 0 . . . 0
0 λs
n′+2 . . . 0
...
...
. . .
...
0 0 . . . λs2n′
. . . . . . . . . . . . . . . . . . . . . . . . . . .
λs
n−n′+1 0 . . . 0
0 λs
n−n′+2 . . . 0
...
...
. . .
...
0 0 . . . λsn
=
(
In′
S′
)
,
where
S′ =
1 0 . . . 0
0 λs
n′+2 . . . 0
...
...
. . .
...
0 0 . . . λs2n′
. . . . . . . . . . . . . . . . . . . . . . . . . . .
λs
n−n′+1 0 . . . 0
0 λs
n−n′+2 . . . 0
...
...
. . .
...
0 0 . . . λsn
,
“adm-n1” — 2019/3/22 — 12:03 — page 10 — #18
10 On hereditary irreducibility of 2-monomial matrices
In′ is the identity n′ × n′ matrix, si > 0 (i = n′ + 2, . . . , n) such that
MS = SM(λ, k′, n′). Now,
(
In′ 0
S′ In−n′
)
−1
M
(
In′ 0
S′ In−n′
)
=
(
In′ 0
−S′ In−n′
)
M
(
In′ 0
S′ In−n′
)
.
If we omit the last n− n′ columns of the last matrix, we obtain
(
In′ 0
−S′ In−n′
)
M
(
In′
S′
)
=
(
In′ 0
−S′ In−n′
)(
In′
S′
)
M(λ, k′, n′)
=
(
In′
0
)
M(λ, k′, n′) =
(
M(λ, k′, n′)
0
)
.
In conclusion, we note that matrix M and the matrix M(λ, k, n) are
similar (over Z[λ]) to a matrix of the form
(
M(λ, k′, n′) B
0 A
)
∈ M(n,Z[λ]),
as claimed.
Note that Theorem 1 folows from the last theorem and the existence of
the homomorphism of rings f : Z[λ] → R where f(1) = 1 and f(λ) = t.
References
[1] P. M. Gudivok, O. A. Tylyshchak, On irreducible modular representations of finite
p-groups over commutative local rings, Nauk. Visn. Uzhgorod. Univ. Ser. Math.
(1998), no. 3, pp. 78–83 (in Ukrainian).
[2] V. M. Bondarenko, M. Yu. Bortos, R. F. Dinis, O. A. Tylyshchak, Reducibility and
irreducibility of monomial matrices over commutative rings, Algebra Discrete Math.
16 (2013), no. 2, pp. 171–187.
[3] V. M. Bondarenko, M. Yu. Bortos, On (∗, 2)-reducible monomial matrices over
commutative rings, Nauk. Visn. Uzhgorod Univ. Ser. Math. Inform. 29 (2016),
no. 2, pp. 22–30 (in Ukrainian).
[4] V. M. Bondarenko, M. Yu. Bortos, R. F. Dinis, O. A. Tylyshchak, Indecomposable
and irreducible t-monomial matrices over commutative rings, Algebra Discrete Math.
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[5] V. M. Bondarenko, M. Yu. Bortos, Sufficient conditions of reducibility in the category
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Ser. Math. Inform. 30 (2017), no. 1, pp. 11–24 (in Ukrainian).
[6] V. M. Bondarenko, M. Yu. Bortos, Indecomposable and isomorphic objects in the
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no. 7, pp. 889–904.
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Bondarenko, Gildea, Tylyshchak, Yurchenko 11
Contact information
V. M. Bondarenko Institute of Mathematics, Tereshchenkivska str.,
3, 01601 Kyiv, Ukraine
E-Mail(s): vitalij.bond@gmail.com
J. Gildea Faculty of Science and Engineering, University of
Chester, Thornton Science Park Pool Lane, Ince,
CH2 4NU, Chester, UK
E-Mail(s): j.gildea@chester.ac.uk
A. A. Tylyshchak,
N. V. Yurchenko
Faculty of Mathematics, Uzhgorod National
Univ., Universytetsyka str., 14, 88000 Uzhgorod,
Ukraine
E-Mail(s): alxtlk@gmail.com,
nataliia.yurchenko@uzhnu.edu.ua
Received by the editors: 10.02.2019.
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