Basic semigroups: theory and applications
A concept of basic matrix semigroups over fields (with some variations) is introduced and throughly investigated. Sections 1 and 2 contain main definitions, Section 3 treats some properties of basic semigroups, Section 4 is devoted to some application of basic semigroups: matrix representations (...
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Інститут прикладної математики і механіки НАН України
2004
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| Назва видання: | Algebra and Discrete Mathematics |
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| Цитувати: | Basic semigroups: theory and applications / J.S. Ponizovskii // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 60–65. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1565832019-06-19T01:26:11Z Basic semigroups: theory and applications Ponizovskii, J.S. A concept of basic matrix semigroups over fields (with some variations) is introduced and throughly investigated. Sections 1 and 2 contain main definitions, Section 3 treats some properties of basic semigroups, Section 4 is devoted to some application of basic semigroups: matrix representations (including faithful representations), finiteness theorems, the problem of Korjakov (when a matrix semigroup over field K is conjugate to a matrix semigroup over a proper subfield of K). The paper is a survey and contains no proofs (which may be found in papers from References). 2004 Article Basic semigroups: theory and applications / J.S. Ponizovskii // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 60–65. — Бібліогр.: 7 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/156583 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
A concept of basic matrix semigroups over fields
(with some variations) is introduced and throughly investigated.
Sections 1 and 2 contain main definitions, Section 3 treats some
properties of basic semigroups, Section 4 is devoted to some application of basic semigroups: matrix representations (including
faithful representations), finiteness theorems, the problem of Korjakov (when a matrix semigroup over field K is conjugate to a
matrix semigroup over a proper subfield of K). The paper is a
survey and contains no proofs (which may be found in papers from
References). |
| format |
Article |
| author |
Ponizovskii, J.S. |
| spellingShingle |
Ponizovskii, J.S. Basic semigroups: theory and applications Algebra and Discrete Mathematics |
| author_facet |
Ponizovskii, J.S. |
| author_sort |
Ponizovskii, J.S. |
| title |
Basic semigroups: theory and applications |
| title_short |
Basic semigroups: theory and applications |
| title_full |
Basic semigroups: theory and applications |
| title_fullStr |
Basic semigroups: theory and applications |
| title_full_unstemmed |
Basic semigroups: theory and applications |
| title_sort |
basic semigroups: theory and applications |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156583 |
| citation_txt |
Basic semigroups: theory and applications / J.S. Ponizovskii // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 60–65. — Бібліогр.: 7 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT ponizovskiijs basicsemigroupstheoryandapplications |
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2025-07-14T08:58:38Z |
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2025-07-14T08:58:38Z |
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1837612161820524544 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2004). pp. 60 – 65
c© Journal “Algebra and Discrete Mathematics”
Basic semigroups: theory and applications
J. S. Ponizovskii
Abstract. A concept of basic matrix semigroups over fields
(with some variations) is introduced and throughly investigated.
Sections 1 and 2 contain main definitions, Section 3 treats some
properties of basic semigroups, Section 4 is devoted to some ap-
plication of basic semigroups: matrix representations (including
faithful representations), finiteness theorems, the problem of Ko-
rjakov (when a matrix semigroup over field K is conjugate to a
matrix semigroup over a proper subfield of K). The paper is a
survey and contains no proofs (which may be found in papers from
References).
1. Notions
In what follows n > 1, K is a fixed field; Mn(K) denotes the multiplica-
tive semigroup of all n × n-matrices over K. If S is a semigroup then
T ≤ S (T E S) means that T is a subsemigroup (an ideal) of S. If
S ≤ Mn(K) then J(S) = {x ∈ Mn(K) | xS ∪ Sx ≤ S} is the idealizer of
S in Mn(K).
Let S ≤ Mn(K). Define H(S) as follows. If S = {0} (0 is the zero
matrix) then H(S) = {0}. If S 6= {0} and r is the least natural such that
r = rank(x) for some x ∈ S, then H(S) = {x ∈ S | rank(x) ≤ r}.
Clearly H(S) E S. H(S) is called the homogeneous ideal of S. A
semigroup T with zero 0 is called 0-prime if and only if the following
holds:
x, y ∈ T, xTy = 0 =⇒ [x = 0 ∨ y = 0].
Let W denote an n-dimensional linear space over K consisting of all
row-vectors of dimension n. Elements from Mn(K) act on W as right
operators. If V is a subspace of W then (V : K) stands for the dimension
of V . If A ⊆ Mn(K) then L(A) is the K-linear envelope of A.
J. S. Ponizovskii 61
2. Basic, strongly basic, weakly basic semigroups
Let S ≤ Mn(K). Denote by R(S) (C(S)) the row-space (the column-
space) of S: R(S) is a subspace of W spanned by all the rows of all
matrices from S (C(S) is defined similarly).
Theorem 1. The followings conditions are equivalent for any S ≤ Mn(K):
(i) (R(S) : K) = (C(S) : K) = n;
(ii) W · L(S) = W and if w ∈ W is such that w · S = 0 then w = 0;
(iii) if x ∈ Mn(K) is such that either xS = 0 or Sx = 0 then x = 0.
Definition. Let S ≤ Mn(K). Then:
S is basic ⇐⇒ any of (i), (ii), (iii) from Theorem 1 holds;
S is strongly basic ⇐⇒ H(S) is basic 0−prime;
S is weakly basic ⇐⇒ for any x ∈ Mn(K), xS = Sx = 0 implies x = 0.
The class of all basic (strongly basic, weakly basic) subsemigroups of
Mn(K) is denoted by B(K) (SB(K), WB(K)). The following holds:
SB(K) ⊂ B(K) ⊂ WB(K) (SB(K) 6= B(K) 6= WB(K)).
Examples.
(i) Any irreducible S ≤ Mn(K) is strongly basic.
(ii) Any indecomposable inverse S ≤ Mn(K) is strongly basic.
(iii) Any nonzero indecomposable commutative S = S2 ≤ Mn(K) is
basic but not necessarily strongly basic.
(iv) Let S be a set of all matrices from Mn(K) with nonzero entries
in the last row only. Then S is a weakly basic subsemigroup of Mn(K),
but S is not basic.
(v) Any S ≤ Mn(K) containing the identity matrix is basic but not
necessarily strongly basic.
3. Properties of basic (strongly basic, weakly basic) semi-
groups
3.1. Embedding theorem
For any abstract semigroup T , Ω(T ) denotes the translational hull of T .
Recall that a semigroup T is left weakly reductive if and only if following
holds:
let a, b ∈ T ; if xa = xb for all x ∈ T then a = b.
62 Basic semigroups: theory and applications
Right weakly reductive semigroups are defined similarly. A semigroup
T is weakly reductive if the following holds:
let a, b ∈ T ; if xa = xb and ax = ab for all x ∈ T then a = b.
Clearly left (right) weakly reductive semigroup is weakly reductive
but not vice versa.
It is well known that any weakly reductive semigroup has a standartd
embedding into Ω(T ) as an ideal. So if T is a weakly reductive semigroup
we put T ⊳ Ω(T ).
Theorem 2. Any weakly basic S ≤ Mn(K) is weakly reductive (hence
we may take S E Ω(S)).
Let S ≤ Mn(K) be weakly basic, S E Ω(S) as abstract semigroups.
Define a mapping ω : J(S) → Ω(S) as follows:
if a ∈ J(S) then ω(a) is defined by the rule: ω(a) · x = ax,
x · ω(a) = xa for all x ∈ S.
It is easy to show that ω is a homomorphism of semigroups. The
following fact is very important:
Theorem 3 (Embedding Theorem). If S ≤ Mn(K) is weakly basic then
ω is a monomorphism. If S ≤ Mn(K) is basic then ω is an isomorphism.
Remark. Theorem 3 shows that, for S basic, the pair S ⊂ Ω(S) may be
included into Mn(K). More exactly: there exists a commutative diagram
S
f
−→ Ω(S)
ε ↑ ↑ ω
S
g
−→ J(S)
where ε is an identity mapping, f and g are inclusions.
3.2. Closure theorem
Let S ≤ Mn(K) be homogeneous (i.e. S = H(S)). A semigroup S ≤
Mn(K) is called the closure of S if the following holds:
(i) S is completely 0-simple,
(ii) S ⊆ S,
(iii) if U ≤ Mn(K) is completely 0-simple such that S ⊆ U then
S ≤ U .
J. S. Ponizovskii 63
Theorem 4 (Closure Theorem). For any strongly basic S ≤ Mn(K)
there exists a closure S; moreover S is unigue and S meets all H-classes
of S.
Examples show that the condition "S is basic" cannot be omitted.
The meaning of closure is rather evident: it is a sort of completely 0-
simple approximation of a homogeneous semigroup.
3.3. Heritability properties
Theorem 5 (Heritability Theorem). Let S ≤ Mn(K) be strongly basic,
and let T, U be such that T E S E U ≤ Mn(K). Then T, U are strongly
basic.
The following theorem shows that an extension of a field K does not
change the idealizer of a basic semigroup.
Theorem 6. Let K ⊆ F be fields, and let S ≤ Mn(K) be basic. Then
the idealizer of S in Mn(K) is equal to the idealizer of S in Mn(F ).
4. Applications
4.1. Matrix representations of semigroups
See [1].
4.2. Finiteness theorems
Let S ≤ Mn(K) be strongly basic. Since H(S) E S, then H(S) is
strongly basic by Theorem 5. Now we formulate
Theorem 7. Let S ≤ Mn(K) be strongly basic. If a maximal nonzero
subgroup of H(S) is finite then S is finite (note that H(S) exists by The-
orem 3).
Theorem 8. Let S ≤ Mn(K) be periodic of bounded period. Assume
that there exists a set of strongly basic representations of S (over some
field F ) which separates points of S. Then S is finite (a representation
f : S → M(F ) is strongly basic if f(S) is strongly basic).
This is a generalization of theorem of Y. Zalcstein [7].
Theorem 9. Let S ≤ Mn(K) be regular irreducible with finite subgroups.
Then S is finite.
Applying the well known theorem by Shur we get
64 Basic semigroups: theory and applications
Theorem 10. Let S ≤ Mn(K) be irreducible, periodic and regular. Then
S is finite.
Theorem 8 is in [3]. Theorem 9 is published in [2].
4.3. Reduction to smaller fields
Results concern the problem:
Let F ⊂ K be a field extension, and let S ≤ Mn(K); when S is
conjugate to a subsemigroup of Mn(F )?
Some sufficient conditions are given in the following
Theorem 11. Let F ⊂ K be fields. Let S ≤ Mn(K) be strongly basic
and G be a maximal nonzero subgroup of H(S). Then S is conjugate to
a subsemigroup of Mn(F ) provided G has this property.
This theorem is a generalization of a result from [4].
Theorem 11 gives a positive answer to the question 3.39 of Kor-
jakov [6].
5. Faithful matrix representations of semigroups
Let S be a semigroup having a faithful matrix representation f : S →
Mn(K). Assume that T is a semigroup such that S E T ≤ Ω(S). When
f may be extended to a faithful representation F : S → Mn(K)?
It is always possible if f(S) is basic since then one can take F = ω (see
Theorem 3). But it is not so in general if f(S) is only weakly basic because
in this case ω maps J(S) into Mn(K) (more exactly into J(f(S))), a part
of Ω(S) only. The following theorem shows that sometimes such F may
be constructed in parts.
Theorem 12 ([5]). Let S be a weakly reductive semigroup (so that we
put S ⊳Ω(S)). Let {Si | i ∈ I} be a family of subsemigroups of Ω(S) such
that the following holds:
(i) S is an ideal of Si for all i ∈ I;
(ii) there exists a faithful representation f : S → Mn(K) such that
f(S) is weakly basic;
(iii) for any i ∈ I, there exists a faithful representation fi : Si →
Mn(K) such that
x ∈ S =⇒ fi(x) = f(x),
x ∈ S, y ∈ Si =⇒ f(xy) = f(x)fi(y), f(yx) = fi(y)f(x).
Let T be a subsemigroup of Ω(S) generated by all Si (i ∈ I). Then there
exists a faithful representation F : T → Mn(K) such that F extends f
and all fi (i ∈ I), i.e.
J. S. Ponizovskii 65
F (x) = f(x) for all x ∈ S,
F (y) = fi(y) for arbitrary i ∈ I and for all y ∈ Si.
References
[1] J. Okninski and J. S. Ponisovskii, A New Matrix representation Theorem for Semi-
groups. Semigroup Forum, 52(1996), no.3, 293-305.
[2] J. S. Ponisovskii, On matrix semigroups, Sovremennaja Algebra, Semigroups with
additional structures. Leningrad, 1986, 64-71 (in Russian).
[3] J. S. Ponisovskii, On a theorem by Y. Zalcstein, Semigroups, Automata and Lan-
guages, Proc. Conf. University of Porto 1994, World Sci. Singapore – New Jersey
– London – Hong Kong, 225-231.
[4] J. S. Ponisovskii. On Matrix Semigroups over a Field K conjugate to Matrix semi-
groups over a proper subfield of K. Proc. Conf. "Semigroups with Applications",
Oberwolfach, 1991, 1-5.
[5] J. S. Ponisovskii, On faithful representations of semigroups, Sovremennaja Algebra
3(23). Leningrad, 1998, 103-106 (in Russian).
[6] Sverdlovskaja tetrad III.Sverdlovsk, 1989, Problem 3.39 (in Russian).
[7] Y. Zalcstein. A finiteness theorem for linear semigroups, Proc. Amer. Math. Soc.
103(1988), no.2, 399-400.
Contact information
J. S. Ponizovskii Russian State Hydrometeororlogical Uni-
versity, Department of Mathematics, Mal-
ookhtinsky pr. 98, 195196 St-Peterburg,
Russia
E-Mail: JP@JP4518.spb.edu
Received by the editors: 21.05.2004
and final form in 06.12.2004.
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