Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range
UDC 512.64 Let $V$ be a vector space and let $T(V)$ denote the semigroup (under composition) of all linear transformations from $V$ into $V$. For a fixed subspace $W$ of $V$, let $T(V,W)$ be the semigroup consisting of all linear transformations from $V$ into $W$. It is known that \[ F(V,W) =\{\alph...
Збережено в:
| Дата: | 2021 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2021
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/1289 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507158995533824 |
|---|---|
| author | Sommanee, W. Sommanee, W. |
| author_facet | Sommanee, W. Sommanee, W. |
| author_sort | Sommanee, W. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:46:08Z |
| description | UDC 512.64
Let $V$ be a vector space and let $T(V)$ denote the semigroup (under composition) of all linear transformations from $V$ into $V$. For a fixed subspace $W$ of $V$, let $T(V,W)$ be the semigroup consisting of all linear transformations from $V$ into $W$. It is known that \[ F(V,W) =\{\alpha\in T(V,W): V\alpha\subseteq W\alpha\} \] is the largest regular subsemigroup of $T(V,W)$. In this paper, we prove that any regular semigroup $S$ can be embedded in $F(V,W)$ with $\dim(V) = |S^1|$ and $\dim(W) = |S|$, and determine all the maximal subsemigroups of $F(V,W)$ when $W$ is a finite dimensional subspace of $V$ over a finite field. |
| doi_str_mv | 10.37863/umzh.v73i12.1289 |
| first_indexed | 2026-03-24T02:04:52Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i12.1289
UDC 512.64
W. Sommanee (Chiang Mai Rajabhat Univ., Thailand)
EMBEDDING THEOREMS AND MAXIMAL SUBSEMIGROUPS OF SOME
LINEAR TRANSFORMATION SEMIGROUPS WITH RESTRICTED RANGE
ТЕОРЕМИ ПРО ВКЛАДЕННЯ ТА МАКСИМАЛЬНI ПIДНАПIВГРУПИ
ДЕЯКИХ НАПIВГРУП ЛIНIЙНИХ ПЕРЕТВОРЕНЬ
З ОБМЕЖЕНИМ ОБРАЗОМ
Let V be a vector space over a field and let T (V ) denote the semigroup of all linear transformations from V into V. For
a fixed subspace W of V, let F (V,W ) be the subsemigroup of T (V ) consisting of all linear transformations \alpha from
V into W such that V \alpha \subseteq W\alpha . In this paper, we prove that any regular semigroup S can be embedded in F (V,W )
with \mathrm{d}\mathrm{i}\mathrm{m}(V ) = | S1| and \mathrm{d}\mathrm{i}\mathrm{m}(W ) = | S| , and determine all the maximal subsemigroups of F (V,W ) when W is a finite
dimensional subspace of V over a finite field.
Нехай V — векторний простiр над деяким полем, а T (V ) — напiвгрупа всiх лiнiйних перетворень з V у V. Для
фiксованого пiдпростору W простору V нехай F (V,W ) — пiднапiвгрупа напiвгрупи T (V ), яка складається з усiх
лiнiйних перетворень \alpha з V у W таких, що V \alpha \subseteq W\alpha . Доведено, що будь-яку регулярну напiвгрупу S можна
вкласти у F (V,W ) з \mathrm{d}\mathrm{i}\mathrm{m}(V ) = | S1| i \mathrm{d}\mathrm{i}\mathrm{m}(W ) = | S| , та визначено всi максимальнi пiднапiвгрупи з F (V,W ),
якщо W — скiнченновимiрний пiдпростiр V над скiнченним полем.
1. Introduction. Let T (X) be the set of all full transformations from a nonempty set X into itself.
It is well-known that T (X) is a regular semigroup under composition of functions. The properties of
T (X) have been widely studied. In 1959, Hall (see [5], Theorem 1.10) showed that every semigroup
S can be embedded in a full transformation semigroup T (S1) by using the extended right regular
representation of S. In [3] (Theorem 8.5) showed that any right cancellative, right simple semigroup
S without idempotents can be embedded in a Bear – Levi semigroup of type (p, p) where p = | S| . In
[2] (Theorem 1.20) proved that any inverse semigroup S can be embedded in the symmetric inverse
semigroup I(S) of all injective partial transformations of S.
If X = \{ 1, 2, . . . , n\} with n \in \BbbZ +, we write Tn instead of T (X). In 1966 Bayramov [1]
characterized all the maximal subsemigroups of Tn, which is either the union of a maximal subgroup
of the symmetric group Sn and Tn\setminus Sn or it is the union of the set of all transformations \alpha \in Tn with
| X\alpha | \leq n - 2 and Sn. Later in 2002, You [20] determined all the maximal regular subsemigroups
of all ideals of Tn. In 2004, Yang and Yang [19] completely described the maximal subsemigroups
of ideals of Tn. And in 2015, East, Michell and Péresse [4] classified the maximal subsemigroups of
T (X) when X is an infinite set containing certain subgroups of the symmetric group on X.
For a fixed nonempty subset Y of a set X, let
T (X,Y ) = \{ \alpha \in T (X) : X\alpha \subseteq Y \} ,
where X\alpha denotes the image of \alpha . Then T (X,Y ) is a subsemigroup of T (X). In 1975, Symons
[18] described all the automorphisms of T (X,Y ). He also determined when T (X1, Y1) is isomorphic
to T (X2, Y2). In 2005, Nenthein, Youngkhong and Kemprasit [8] characterized the regular elements
of T (X,Y ). In 2008, Sanwong and Sommanee [12] defined
c\bigcirc W. SOMMANEE, 2021
1714 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
EMBEDDING THEOREMS AND MAXIMAL SUBSEMIGROUPS OF SOME LINEAR . . . 1715
F (X,Y ) = \{ \alpha \in T (X,Y ) : X\alpha \subseteq Y \alpha \}
and showed that F (X,Y ) is the largest regular subsemigroup of T (X,Y ). This semigroup plays a
crucial role in characterization of Green’s relations on T (X,Y ). Moreover, they determined a class
of maximal inverse subsemigroups of T (X,Y ). In 2011, Sanwong [11] described Green’s relations,
ideals and all the maximal regular subsemigroups of F (X,Y ). Also, the author proved that every
regular semigroup S can be embedded in F (S1, S). Later in 2013, Sommanee and Sanwong [15]
computed the rank of F (X,Y ) when X is a finite set. Furthermore, they obtained the rank and
idempotent rank of its ideals. Recently in 2018, Sommanee [13] described the maximal inverse
subsemigroups of F (X,Y ) and completely determined all the maximal regular subsemigroups of its
ideals.
For a vector space V over a field F, let T (V ) be the set of all linear transformations from V
into V. It is known that T (V ) is a regular semigroup under composition of functions (see [2, p. 57]).
In 2004, Mendes-Gonçalves and Sullivan [7] (Theorem 3.12) proved that any right simple, right
cancellative semigroup S without idempotents can be embedded in some GS(m,m), the linear
Baer – Levi semigroup on V. After that in 2012, Sullivan [16] (Theorem 3) proved that any semigroup
S can be embedded in T (V ) for some vector space V with dimension | S1| .
For a fixed subspace W of a vector space V, let
T (V,W ) = \{ \alpha \in T (V ) : V \alpha \subseteq W\} .
Then T (V,W ) is a subsemigroup of T (V ). In 2007, Nenthein and Kemprasit [9] proved that \alpha \in
\in T (V,W ) is a regular element of T (V,W ) if and only if V \alpha = W\alpha . As a consequence, they
showed that T (V,W ) is regular if and only if either V = W or W = \{ 0\} . Later in 2008, Sullivan
[17] proved that the set
F (V,W ) = \{ \alpha \in T (V,W ) : V \alpha \subseteq W\alpha \} ,
consisting of all regular elements in T (V,W ), is the largest regular subsemigroup of T (V,W ). He
characterized Green’s relations on T (V,W ) and showed that the semigroup F (V,W ) is always a
right ideal of T (V,W ). The author also described all the ideals of F (V,W ) and T (V,W ). Re-
cently in 2017, Sommanee and Sangkhanan [14] determined the maximal regular subsemigroups of
F (V,W ) when W is a finite dimensional subspace of V over a finite field F. Moreover, they com-
puted the rank and the idempotent rank of F (V,W ) when V is a finite dimensional vector space
over a finite field F.
Here, we prove that any regular semigroup S can be embedded in F (V,W ) where \mathrm{d}\mathrm{i}\mathrm{m}(V ) =
= | S1| and \mathrm{d}\mathrm{i}\mathrm{m}(W ) = | S| , and determine all the maximal subsemigroups of F (V,W ) when W is
a finite dimensional subspace of V over a finite field F.
2. Preliminaries and notations. Let S be a semigroup. We call a \in S a regular element if
a = axa for some x \in S, and S is said to be a regular semigroup if every element of S is regular.
An element e \in S is called an idempotent if e2 = e. A nonempty subset A of S is said to be an
ideal if SA \subseteq A and AS \subseteq A. A proper (regular) subsemigroup M of S is a maximal (regular)
subsemigroup of S if, whenever M \subseteq T \subseteq S for some a (regular) subsemigroup T of S, then
M = T or T = S.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1716 W. SOMMANEE
Let a and b be elements of a semigroup S. The Green’s relations \scrL , \scrR , \scrH and \scrJ on S are
defined as follows: a\scrL b if S1a = S1b, a\scrR b if aS1 = bS1, \scrH = \scrL \cap \scrR and a\scrJ b if S1aS1 = S1bS1,
where S1 is a semigroup S with the identity adjoined, if necessary. For each a \in S, we denote \scrL -
class, \scrR -class, \scrH -class and \scrJ -class containing a by La, Ra, Ha and Ja, respectively.
A semigroup S is said to be embedded in a semigroup T if there exists an injective function \varphi :
S \rightarrow T such that (xy)\varphi = (x\varphi )(y\varphi ) for all x, y \in S.
Let V be a vector spaces over a field F. A function \alpha : V \rightarrow V is a linear transformation on V
if
(u+ v)\alpha = u\alpha + v\alpha and (au)\alpha = a(u\alpha )
for all vectors u, v \in V and scalar a \in F. The set T (V ) of all linear transformations from V
into V is a semigroup with respect to the composition operation. This semigroup is called a linear
transformation semigroup. We denote by \Theta V the zero map in T (V ), that is, \Theta V : V \rightarrow \{ 0\} .
For a fixed subspace W of a vector space V, let
T (V,W ) = \{ \alpha \in T (V ) : V \alpha \subseteq W\} and F (V,W ) = \{ \alpha \in T (V,W ) : V \alpha \subseteq W\alpha \} .
Then T (V,W ) is a subsemigroup of T (V ) and F (V,W ) is the largest regular subsemigroup of
T (V,W ).
For any set A, | A| means the cardinality of the set A.
In this paper, a subspace of a vector space V over a field F generated by a linearly independent
subset \{ ei : i \in I\} of V is denoted by \langle ei\rangle . If we write U = \langle ei\rangle when U is a subspace of V, it
means the set \{ ei : i \in I\} is a basis of U with \mathrm{d}\mathrm{i}\mathrm{m}(U) = | I| . Let \{ ui : i \in I\} be a subset of V.
Then the notation
\sum
aiui means the linear combination
ai1ui1 + ai2ui2 + . . .+ ainuin
for some n \in \BbbZ +, ui1 , ui2 , . . . , uin \in \{ ui : i \in I\} and scalars ai1 , ai2 , . . . , ain \in F.
A construction of a map \alpha \in T (V ), we first choose a basis \{ ei : i \in I\} for a vector space V
and a subset \{ ui : i \in I\} of V, and then let ei\alpha = ui for each i \in I and extending this action
by linearity to the whole of V. To shorten this process, we simply say, given \{ ei : i \in I\} and \{ ai :
i \in I\} within the context. Then \alpha \in T (V ) is defined by letting
\alpha =
\biggl(
ei
ui
\biggr)
.
Let S1, S2, . . . , Sn be subspaces of a vector space V where n \geq 2. We call V the internal direct
sum of S1, S2, . . . , Sn, and we write
V = S1 \oplus S2 \oplus . . .\oplus Sn,
if V = S1 + S2 + . . .+ Sn = \{ s1 + s2 + . . .+ sn : si \in Si, 1 \leq i \leq n\} and Si \cap (S1 + . . .+ Si - 1 +
+ Si+1 + . . .+ Sn) = \{ 0\} for all 1 \leq i \leq n. We note that if U is a subspace of V, then there exists
a subspace T of V such that V = U \oplus T (see [10], Theorem 1.4).
The external direct sum of a family of rings \{ Ri : i \in I\} , denoted
\sum
i\in I
Ri, is the set of all
sequences (ri) where ri \in Ri and at most finitely many ri are non-zero.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
EMBEDDING THEOREMS AND MAXIMAL SUBSEMIGROUPS OF SOME LINEAR . . . 1717
3. Embedding theorems. In 2011, Sanwong proved that every regular semigroup S can be
embedded in F (S1, S) (see [11], Theorem 3). Here, we prove a linear version of that result.
Remark 3.1 ([6, p. 182], Remark (c)). Let X be any nonempty set and R a ring with identity.
Let V be the external direct sum
\sum
Ri with the copies of R indexed by the set X. Then V is a free
R-module on the set X such that X is a basis of V. In particular, if R is a field, then V =
\sum
Ri is
a vector space over R with dimension | X| .
Lemma 3.1 ([16], Theorem 3). Any semigroup S can be embedded in T (U) where U =
\sum
Fi
is the external direct sum of the copies of a field F indexed by the semigroup S1 (U is a vector
space with dimension | S1| ).
Theorem 3.1. Let W be a subspace of a vector space V. Then T (W ) can be embedded in
F (V,W ).
Proof. It is clear that if V = W, then T (W ) = F (V, V ) = F (V,W ) and so T (W ) \sim = F (V,W ).
But, when W = \{ 0\} , we see that T (W ) = \{ \Theta W \} and F (V,W ) = \{ \Theta V \} . Thus, they are isomorphic.
Now, suppose that \{ 0\} \not = W \varsubsetneq V. Let W = \langle wi\rangle and V = \langle wi\rangle \oplus \langle vj\rangle for some subspace \langle vj\rangle
of V. Then we have \{ wi : i \in I\} \not = \varnothing \not = \{ vj : j \in J\} . Let \alpha \in T (W ) and write
\alpha =
\biggl(
wi
wi\alpha
\biggr)
.
Define \alpha \prime \in T (V,W ) as follows:
\alpha \prime =
\biggl(
wi vj
wi\alpha 0
\biggr)
.
We obtain V \alpha \prime \subseteq W\alpha \prime , which implies that \alpha \prime \in F (V,W ). For any element w \in W, we can write
w =
\sum
aiwi and so w\alpha \prime = (
\sum
aiwi)\alpha
\prime =
\sum
ai(wi\alpha
\prime ) =
\sum
ai(wi\alpha ) = (
\sum
aiwi)\alpha = w\alpha . Also, if
\alpha , \beta \in T (W ) and w \in W, then w\alpha \in W and thus (w\alpha )\beta \prime = (w\alpha )\beta . We define
\Phi : T (W ) \rightarrow F (V,W ) by \alpha \Phi = \alpha \prime for all \alpha \in T (W ).
We prove that \Phi is a monomorphism. Let \alpha , \beta \in T (W ). If \alpha \Phi = \beta \Phi , then \alpha \prime = \beta \prime . For w \in W,
w =
\sum
aiwi and w\alpha =
\bigl( \sum
aiwi
\bigr)
\alpha =
\sum
ai(wi\alpha ) =
\sum
ai(wi\alpha
\prime ) =
\sum
ai(wi\beta
\prime ) =
\sum
ai(wi\beta ) =
=
\bigl( \sum
aiwi
\bigr)
\beta = w\beta . So, \alpha = \beta and hence \Phi is injective. Let v \in V. Then we can write v =
=
\sum
biwi+
\sum
cjvj and v(\alpha \prime \beta \prime ) =
\bigl( \sum
biwi+
\sum
cjvj
\bigr)
(\alpha \prime \beta \prime ) =
\sum
bi
\bigl(
wi(\alpha
\prime \beta \prime )
\bigr)
+
\sum
cj
\bigl(
vj(\alpha
\prime \beta \prime )
\bigr)
=
=
\sum
bi
\bigl(
(wi\alpha
\prime )\beta \prime \bigr) +
\sum
cj
\bigl(
(vj\alpha
\prime )\beta \prime \bigr) =
\sum
bi(wi\alpha )\beta
\prime +
\sum
cj(0\beta
\prime ) =
\sum
bi(wi\alpha )\beta +
\sum
cj(0) =
=
\sum
bi
\bigl(
wi(\alpha \beta )
\bigr)
+
\sum
cj(0) =
\sum
bi
\bigl(
wi(\alpha \beta )
\prime \bigr) + \sum
cj
\bigl(
vj(\alpha \beta )
\prime \bigr) =
\bigl( \sum
biwi +
\sum
cjvj
\bigr)
(\alpha \beta )\prime =
= v(\alpha \beta )\prime . Whence, (\alpha \beta )\prime = \alpha \prime \beta \prime , it follows that (\alpha \beta )\Phi = (\alpha \Phi )(\beta \Phi ). Thus, \Phi is a monomorphism
and therefore T (W ) can be embedded in F (V,W ).
Theorem 3.1 is proved.
By Lemma 3.1, any semigroup S can be embedded in T (W ) for some vector space W with
dimension | S1| . And by Theorem 3.1, T (W ) can be embedded in F (V,W ) when V is any vector
space which contains W. So, we have the following corollary.
Corollary 3.1. Any semigroup S can be embedded in F (V,W ) for some subspace W of V with
\mathrm{d}\mathrm{i}\mathrm{m}(W ) = | S1| , where V is any vector space which contains W.
Lemma 3.2. Let S be any semigroup and x \in S. We write S1 = \{ ai : i \in I\} and define \rho x :
S1 \rightarrow S1 by ai\rho x = aix for all i \in I. Let F be any field and V the external direct sum
\sum
Fi with
the copies of F indexed by S1. Then:
(1) \rho x can be extended by linearity to an element of T (V ),
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1718 W. SOMMANEE
(2) the mapping \rho : S \rightarrow T (V ) is given by x\rho = \rho x for all x \in S, is a monomorphism.
Proof. See the proof as given in [16] (Theorem 3).
Lemma 3.3. Let
\sum
Fi be the external direct sum of the copies of a field F indexed by some set
I with | I| \geq 2. We fix k \in I and let J = I \setminus \{ k\} . Let G be the external direct sum of \{ 0\} \cup \{ Fj :
j \in J\} , where 0 \in Fk = F, and
\sum
Fj is the external direct sum of the copies of a field F indexed
by the set J. Then:
(1) G is a subspace of
\sum
Fi,
(2)
\sum
Fj is isomorphic to G.
Proof. (1) It is easy to verify that G is a subspace of
\sum
Fi.
(2) For each (rj) \in
\sum
Fj , we construct an element (r\prime i) in G by
(r\prime i) =
\Biggl\{
0, if i = k,
rj , if i \in I \setminus \{ k\} = J.
Define \varphi :
\sum
Fj \rightarrow G by (rj)\varphi = (r\prime i) for all (rj) \in
\sum
Fj . Then \varphi is bijective. Let (rj), (sj) \in
\in
\sum
Fj and c \in F. It is routine to show [(rj) + (sj)]\varphi = (rj)\varphi + (sj)\varphi and [c(rj)]\varphi = c[(rj)\varphi ].
Thus, \varphi is an isomorphism and so
\sum
Fj
\sim = G.
Lemma 3.3 is proved.
Theorem 3.2. Any regular semigroup S can be embedded in F (V,W ) for some subspace W
of a vector space V, where \mathrm{d}\mathrm{i}\mathrm{m}(V ) = | S1| and \mathrm{d}\mathrm{i}\mathrm{m}(W ) = | S| .
Proof. Assume that S is a regular semigroup and let V be the external direct sum
\sum
Fi with
the copies of a field F indexed by S1. We note that V =
\bigl\langle
S1
\bigr\rangle
and \mathrm{d}\mathrm{i}\mathrm{m}(V ) = | S1| by Remark 3.1.
There are two cases to consider.
Case 1: 1 \in S. Then we have S1 = S. Let W = V. It follows from Lemma 3.1 that S can be
embedded in T (V ) = F (V,W ) such that \mathrm{d}\mathrm{i}\mathrm{m}(W ) = \mathrm{d}\mathrm{i}\mathrm{m}(V ) = | S1| = | S| .
Case 2: 1 /\in S. This implies that | S1| \geq 2 and S = S1 \setminus \{ 1\} . Let G be the external direct sum
of \{ 0\} \cup \{ Fj : j \in S\} , where 0 \in F1 = F = Fj for all j \in S. It follows from Lemma 3.3 that\sum
Fj
\sim = G \subseteq V, where
\sum
Fj is the external direct sum of \{ Fj : j \in S\} with the copies of the field
F indexed by S. Here, we let W =
\sum
Fj . Thus, we have W =
\sum
Fj = \langle S\rangle , \mathrm{d}\mathrm{i}\mathrm{m}(W ) = | S| and
W \subseteq V in the sense of embedding. Now, we write S1 = \{ ai : i \in I\} . For each x \in S, define \rho x :
S1 \rightarrow S1 by ai\rho x = aix for all i \in I. Then by Lemma 3.2 (1), we obtain \rho x \in T (V ) and it is clear
that ai\rho x = aix \in S for all i \in I. Notice that there exists t \in S such that x = xtx since S is regular.
We prove \rho x \in F (V,W ). Let v\rho x \in V \rho x for some v \in V =
\bigl\langle
S1
\bigr\rangle
. So, we can write v =
\sum
diai and
v\rho x =
\sum
di(ai\rho x) \in \langle S\rangle = W. Whence, V \rho x \subseteq W. Next, we prove V \rho x \subseteq W\rho x. If v =
\sum
diai
for some ai \in S, then v\rho x =
\bigl( \sum
diai
\bigr)
\rho x \in \langle S\rangle \rho x = W\rho x. If v = d \cdot 1 for some scalar d \in F, then
v\rho x = (d \cdot 1)\rho x = d(1\rho x) = dx = d(xtx) = d((xt)x) = d((xt)\rho x) = (d(xt))\rho x \in \langle S\rangle \rho x = W\rho x.
Hence, V \rho x \subseteq W\rho x and so \rho x \in F (V,W ). We define \rho : S \rightarrow F (V,W ) by x\rho = \rho x for all x \in S.
Then by Lemma 3.2 (2), we have \rho is a monomorphism. Therefore, we conclude that S can be
embedded in F (V,W ).
Theorem 3.2 is proved.
4. Maximal subsemigroups. In 2017, Sommanee and Sangkhanan determined the maximal
regular subsemigroups of F (V,W ), when W is a finite dimensional subspace of a vector space V
over a finite field F (see [14], Theorem 4.9).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
EMBEDDING THEOREMS AND MAXIMAL SUBSEMIGROUPS OF SOME LINEAR . . . 1719
In general, if S is a regular semigroup and T is a maximal regular subsemigroup of S, then
T may not be a maximal subsemigroup of S (see [19, 20], Theorem 2). Here, we prove that the
maximal subsemigroups and the maximal regular subsemigroups of F (V,W ) coincide.
We begin by recalling some notations and results from [14] that will be useful in this section.
Lemma 4.1 ([14], Lemma 2.3). Let W be a subspace of a vector space V and \alpha , \beta \in F (V,W ).
Then:
(1) \alpha \scrJ \beta if and only if \mathrm{d}\mathrm{i}\mathrm{m}(V \alpha ) = \mathrm{d}\mathrm{i}\mathrm{m}(V \beta ),
(2) \alpha \scrH \beta if and only if V \alpha = V \beta and \mathrm{k}\mathrm{e}\mathrm{r}\alpha = \mathrm{k}\mathrm{e}\mathrm{r}\beta ,
where \mathrm{k}\mathrm{e}\mathrm{r}\alpha = \{ v \in V : v\alpha = 0\} .
Lemma 4.2 ([14], Theorem 2.4). Let W be a subspace of a vector space V. Then the ideals of
F (V,W ) are precisely the sets Qk = \{ \alpha \in F (V,W ) : \mathrm{d}\mathrm{i}\mathrm{m}(V \alpha ) \leq k\} , where 0 \leq k \leq \mathrm{d}\mathrm{i}\mathrm{m}(W ).
We note that Qk is a regular subsemigroup of F (V,W ) (see [14], Lemma 2.5).
Let n \geq 0 be an integer and W an n-dimensional subspace of a vector space V over a finite
field F.
For 0 \leq k \leq n = \mathrm{d}\mathrm{i}\mathrm{m}(W ), define J(k) = \{ \alpha \in F (V,W ) : \mathrm{d}\mathrm{i}\mathrm{m}(V \alpha ) = k\} . Then J(k) is a
\scrJ -class of F (V,W ). Let Qk be defined as in Lemma 4.2. We have Qk = J(0) \cup J(1) \cup . . . \cup J(k)
and Qn = F (V,W ).
Remark 4.1. The following facts are directly obtained from the definitions of J(k) and Qk :
(1) Q0 = J(0) contains exactly one element \Theta V , the zero map;
(2) for each \alpha \in J(n), V \alpha = W since V \alpha \subseteq W and \mathrm{d}\mathrm{i}\mathrm{m}(V \alpha ) = n = \mathrm{d}\mathrm{i}\mathrm{m}(W ) is finite.
We will use the notation GL(U) as a set of all automorphisms of a vector space U over a field
F. It is well-known that GL(U) is a group under the composition of functions.
Lemma 4.3 ([14], Lemma 3.2). Let \varepsilon \in F (V,W ) be an idempotent. Then H\varepsilon
\sim = GL(V \varepsilon ).
From now on, we suppose that n \geq 1 and let E
\bigl(
J(n)
\bigr)
= \{ \varepsilon p : p \in P\} be the set of all
idempotents in J(n). Then we have
J(n) =
\bigcup
p\in P
H\varepsilon p
is a disjoint union of groups all of which are isomorphic (see [14], Lemma 3.3). Moreover, J(n) is
a regular subsemigroup of F (V,W ) (see [14], Lemma 3.6).
Lemma 4.4 ([14], Lemma 4.1). J(n - 1) \subseteq J(n)\alpha J(n) for all \alpha \in J(n - 1).
Lemma 4.5 ([14], Theorem 4.2). For n \geq 2, the set Qn - 2 \cup J(n) is a maximal regular sub-
semigroup of F (V,W ).
For each \varepsilon p \in E
\bigl(
J(n)
\bigr)
, H\varepsilon p
\sim = GL(V \varepsilon p) = GL(W ) by Lemma 4.3 and Remark 4.1 (2). We
let \Phi p : H\varepsilon p \rightarrow GL(W ) be an isomorphism and U a fixed maximal subgroup of GL(W ). For each
p \in P, we define
Mp = U\Phi - 1
p .
Then Mp is a maximal subgroup of H\varepsilon p for all p \in P (for details, see [14, p. 409]).
Lemma 4.6 ([14], Lemma 4.3). Let Mp be defined as above and M =
\bigcup
p\in P
Mp. Then M is
a maximal regular subsemigroup of J(n).
Lemma 4.7 ([14], Theorem 4.4). Let M be as in Lemma 4.6. Then Qn - 1 \cup M is a maximal
regular subsemigroup of F (V,W ).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1720 W. SOMMANEE
Lemma 4.8 ([14], Lemma 4.6). T is a maximal regular subsemigroup of J(n) if and only if
there is a maximal subgroup U of GL(W ) such that T =
\bigcup
p\in P
Mp with Mp = U\Phi - 1
p , where \Phi p
is defined as previous Lemma 4.6 (p \in P ).
Recall that if A is a subset of a semigroup S, then \langle A\rangle denotes the subsemigroup of S generated
by A.
Lemma 4.9 ([14], Lemma 4.7). For 0 \leq k \leq n - 1, Qk = \langle J(k)\rangle .
To prove the main results, we prepare the following two lemmas.
Lemma 4.10. Every subsemigroup of J(n) is a regular subsemigroup of J(n).
Proof. Assume that T is a subsemigroup of J(n) =
\bigcup
p\in P
H\varepsilon p . Let R = \{ r \in P : T\cap H\varepsilon r \not = \varnothing \}
and Tr = T \cap H\varepsilon r for all r \in R. It is clear that T =
\bigcup
r\in R
Tr. Since Tr = T \cap H\varepsilon r \not = \varnothing , we obtain
Tr is a finite subsemigroup of the group H\varepsilon r . Thus, Tr is a subgroup of H\varepsilon r and so Tr is a regular
subsemigroup of H\varepsilon r for all r \in R. Therefore, T is a regular subsemigroup of J(n).
Lemma 4.10 is proved.
From Lemma 4.10, we easily verify the following lemma.
Corollary 4.1. The maximal subsemigroups and the maximal regular subsemigroups of J(n)
coincide.
The following lemma is directly obtained from Lemma 4.8 and Corollary 4.1.
Lemma 4.11. T is a maximal subsemigroup of J(n) if and only if there is a maximal subgroup
U of GL(W ) such that T =
\bigcup
p\in P
Mp with Mp = U\Phi - 1
p where \Phi p is defined as previous Lemma
4.6 (p \in P ).
Lemma 4.12. For n \geq 2, Qn - 2 \cup J(n) is a maximal subsemigroup of F (V,W ).
Proof. Let n \geq 2. Then we have Qn - 2 \cup J(n) is a regular subsemigroup of F (V,W ) by
Lemma 4.5. Thus, it is a subsemigroup of F (V,W ). To prove that Qn - 2 \cup J(n) is a maximal
subsemigroup of F (V,W ), suppose that there is a subsemigroup S of F (V,W ) such that Qn - 2 \cup
\cup J(n) \varsubsetneq S \subseteq F (V,W ). We prove that S is a regular subsemigroup of F (V,W ). Let \alpha be any
element in S. Then there exists \alpha \prime \in F (V,W ) such that \alpha = \alpha \alpha \prime \alpha , since F (V,W ) is regular. We
note that if \alpha \in Qn - 2 \cup J(n), then \alpha is a regular element in S, since Qn - 2 \cup J(n) is regular
and Qn - 2 \cup J(n) \subseteq S. Suppose that \alpha /\in Qn - 2 \cup J(n), that is, \alpha \in J(n - 1). We assume that
\alpha \prime /\in S. Thus, \alpha \prime \in J(n - 1) \setminus S and we can write \alpha \prime = \beta \alpha \gamma for some \beta , \gamma \in J(n) \subseteq S by
Lemma 4.4. This implies that \alpha \prime \in S, a contradiction. Whence, \alpha \prime \in S and so S is a regular
subsemigroup of F (V,W ). Since Qn - 2 \cup J(n) is a maximal regular subsemigroup of F (V,W ), we
get S = F (V,W ). Therefore, Qn - 2 \cup J(n) is a maximal subsemigroup of F (V,W ).
Lemma 4.12 is proved.
Lemma 4.13. Let M be as in Lemma 4.6. Then Qn - 1 \cup M is a maximal subsemigroup of
F (V,W ).
Proof. Since Qn - 1 \cup M is a regular subsemigroup of F (V,W ) by Lemma 4.7, it is a subsemi-
group of F (V,W ). We prove that Qn - 1 \cup M is a maximal subsemigroup of F (V,W ). Let S be a
subsemigroup of F (V,W ) such that Qn - 1 \cup M \varsubsetneq S \subseteq F (V,W ). We see that S \cap J(n) \not = \varnothing . It fol-
lows that S\cap J(n) is a subsemigroup of J(n). Then by Lemma 4.10, we get that S\cap J(n) is a regular
subsemigroup of J(n). Thus, S = Qn - 1 \cup (S \cap J(n)) is a regular subsemigroup of F (V,W ). Since
Qn - 1 \cup M is a maximal regular subsemigroup of F (V,W ), we obtain S = F (V,W ). Therefore,
Qn - 1 \cup M is a maximal subsemigroup of F (V,W ).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
EMBEDDING THEOREMS AND MAXIMAL SUBSEMIGROUPS OF SOME LINEAR . . . 1721
Lemma 4.14. Let S be any maximal subsemigroup of F (V,W ). Then the following statements
hold:
(1) S \cap J(n) \not = \varnothing ,
(2) S \cap J(n) is a maximal subsemigroup of J(n).
Proof. (1) If S \cap J(n) = \varnothing , we get S \subseteq Qn - 1 \varsubsetneq Qn - 1 \cup M \varsubsetneq F (V,W ), where M is defined
in Lemma 4.6. But Qn - 1 \cup M is a maximal subsemigroup of F (V,W ) by Lemma 4.13, which
contradicts the maximality of S. Therefore, S \cap J(n) \not = \varnothing .
(2) It follows from (1) that S\cap J(n) is a subsemigroup of J(n). If S\cap J(n) is not maximal, then
there exists a maximal subsemigroup T of J(n) such that S \cap J(n) \varsubsetneq T \varsubsetneq J(n). It is easy to see
that Qn - 1 \cup T is a subsemigroup of F (V,W ) with S \varsubsetneq Qn - 1 \cup T \varsubsetneq F (V,W ), which contradicts
the maximality of S. Therefore, S \cap J(n) is a maximal subsemigroup of J(n).
Lemma 4.14 is proved.
Theorem 4.1. Let n \geq 2 and S a maximal subsemigroup of F (V,W ). Then S is either of the
form:
(1) Qn - 2 \cup J(n)
or
(2) Qn - 1 \cup M, where M is defined in Lemma 4.6.
Proof. By Lemmas 4.12 and 4.13, we have Qn - 2 \cup J(n) and Qn - 1 \cup M are maximal subsemi-
groups of F (V,W ). On the other hand, since S \cap J(n) \not = \varnothing by Lemma 4.14 (1). So, we consider
the following two cases.
Case 1: S \cap J(n) = J(n). Hence, J(n) \subseteq S. We suppose that S \nsubseteq Qn - 2 \cup J(n). Then there
exists \alpha \in S and \alpha /\in Qn - 2 \cup J(n), that is, \alpha \in J(n - 1). It follows from Lemma 4.4 that
J(n - 1) \subseteq J(n)\alpha J(n) \subseteq S\alpha S \subseteq S, and so Qn - 1 = \langle J(n - 1)\rangle \subseteq S by Lemma 4.9. Whence,
F (V,W ) = Qn - 1 \cup J(n) \subseteq S \subseteq F (V,W ). Thus, S = F (V,W ), which contradicts the maximality
of S. Therefore, S \subseteq Qn - 2 \cup J(n). But, Qn - 2 \cup J(n) is a maximal subsemigroup of F (V,W ) by
Lemma 4.12. This implies that S = Qn - 2 \cup J(n).
Case 2: S\cap J(n) \varsubsetneq J(n). By Lemma 4.14 (2), we have S\cap J(n) is a maximal subsemigroup of
J(n). Then by Lemma 4.11, we get that S \cap J(n) =
\bigcup
p\in P
Mp, where Mp = U\Phi - 1
p for all p \in P
with a fixed maximal subgroup U of GL(W ). We let M =
\bigcup
p\in P
Mp. Then M = S \cap J(n). Since
S \subseteq Qn - 1 \cup (S \cap J(n)) = Qn - 1 \cup M and Qn - 1 \cup M is a maximal subsemigroup of F (V,W ) by
Lemma 4.13, we obtain S = Qn - 1 \cup M.
Theorem 4.1 is proved.
Corollary 4.2. For n = 1, each maximal subsemigroup of F (V,W ) must be one of the forms:
J(1) or \{ \Theta V \} \cup M, where M is defined in Lemma 4.6.
Proof. Assume that n = 1. By Lemma 4.13, we obtain that Q0 \cup M = \{ \Theta V \} \cup M is a
maximal subsemigroup of F (V,W ) where M is defined in Lemma 4.6. Furthermore, if n = 1, then
F (V,W ) = J(0) \cup J(1) = \{ \Theta V \} \cup J(1), that is, J(1) = F (V,W ) \setminus \{ \Theta V \} . And, since J(1) is a
subsemigroup of F (V,W ), it is clear that J(1) is a maximal subsemigroup of F (V,W ).
Let S be any maximal subsemigroup of F (V,W ). Then we consider two cases.
Case 1: \Theta V /\in S. Then S \subseteq J(1). Since S \subseteq J(1) \varsubsetneq F (V,W ) and J(1) is a subsemigroup of
F (V,W ), whence S = J(1).
Case 2: \Theta V \in S. By Lemma 4.14 (1), we have S \cap J(1) \not = \varnothing . If S \cap J(1) = J(1), we get that
S = F (V,W ), a contradiction. Hence, S \cap J(1) \varsubsetneq J(1). Then by the same argument as in the proof
of Theorem 4.1 (Case 2), we obtain S = Q0 \cup M = \{ \Theta V \} \cup M, where M is defined in Lemma 4.6.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1722 W. SOMMANEE
Corollary 4.2 is proved.
Next, we consider the case when V = W and V is a finite dimensional vector space with
\mathrm{d}\mathrm{i}\mathrm{m}(V ) = n. Then we have F (V,W ) = T (V ), and it is easy to verify that J(n) = GL(V ). So, we
establish the following corollary.
Corollary 4.3. Let V be an n-dimensional vector space over a finite field F (n \geq 2) and S a
maximal subsemigroup of T (V ). Then S is either of the form:
(1) Qn - 2 \cup GL(V )
or
(2) Qn - 1 \cup M, where M is a maximal subgroup of GL(V ).
References
1. R. A. Bairamov, On the problem of completeness in a symmetric semigroup of finite degree, Diskret Analiz., 8, 3 – 26
(1966) (in Russian).
2. A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, vol. 1, Math. Surveys and Monogr., 7 (1961).
3. A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, vol. 3, Math. Surveys and Monogr., 7 (1967).
4. J. East, J. D. Michell, Y. Péresse, Maximal subsemigroups of the semigroup of all mappings on an infinite set, Trans.
Amer. Math. Soc., 367, № 3, 1911 – 1944 (2015).
5. J. M. Howie, An introduction to semigroup theory, Acad. Press, London (1976).
6. T. W. Hungerford, Algebra, Springer-Verlag, New York (1974).
7. S. Mendes-Gonçalves, R. P. Sullivan, Baer – Levi semigroups of linear transformations, Proc. Roy. Soc. Edinburgh
Sect. A, 134, № 3, 477 – 499 (2004).
8. S. Nenthein, P. Youngkhong, Y. Kemprasit, Regular elements of some transformation semigroups, Pure Math. and
Appl. (PU.M.A.), 16, № 3, 307 – 314 (2005).
9. S. Nenthein, Y. Kemprasit, Regular elements of some semigroups of linear transformations and matrices, Int. Math.
Forum, 2, № 4, 155 – 166 (2007).
10. S. Roman, Advanced linear algebra, 3rd ed., Grad. Texts in Math., Springer (2008).
11. J. Sanwong, The regular part of a semigroup of transformations with restricted range, Semigroup Forum, 83, № 1,
134 – 146 (2011).
12. J. Sanwong, W. Sommanee, Regularity and Green’s relations on a semigroup of transformations with restricted range,
Int. J. Math. and Math. Sci., Article ID 794013 (2008), 11 p.
13. W. Sommanee, The regular part of a semigroup of full transformations with restricted range: maximal inverse
subsemigroups and maximal regular subsemigroups of its ideals, Int. J. Math. and Math. Sci., Article ID 2154745
(2018), 9 p.
14. W. Sommanee, K. Sangkhanan, The regular part of a semigroup of linear transformations with restricted range, J.
Aust. Math. Soc., 103, № 3, 402 – 419 (2017).
15. W. Sommanee, J. Sanwong, Rank and idempotent rank of finite full transformation semigroups with restricted range,
Semigroup Forum, 87, № 1, 230 – 242 (2013).
16. R. P. Sullivan, Embedding theorems for semigroups of generalised linear transformations, Southeast Asian Bull.
Math., 36, № 4, 547 – 552 (2012).
17. R. P. Sullivan, Semigroups of linear transformations with restricted range, Bull. Aust. Math. Soc., 77, № 3, 441 – 453
(2008).
18. J. S. V. Symons, Some results concerning a transformation semigroup, J. Aust. Math. Soc., 19, № 4, 413 – 425 (1975).
19. H. Yang, X. Yang, Maximal subsemigroups of finite transformation semigroups K(n, r), Acta Math. Sin. (Engl.
Ser.), 20, № 3, 475 – 482 (2004).
20. T. You, Maximal regular subsemigroups of certain semigroups of transformations, Semigroup Forum, 64, № 3,
391 – 396 (2002).
Received 02.12.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
|
| id | umjimathkievua-article-1289 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:52Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/54/d816adbd58d727bcf0918895531c5154.pdf |
| spelling | umjimathkievua-article-12892025-03-31T08:46:08Z Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range Sommanee, W. Sommanee, W. linear transformation; restricted range; embedding; maximal subsemigroup UDC 512.64 Let $V$ be a vector space and let $T(V)$ denote the semigroup (under composition) of all linear transformations from $V$ into $V$. For a fixed subspace $W$ of $V$, let $T(V,W)$ be the semigroup consisting of all linear transformations from $V$ into $W$. It is known that \[ F(V,W) =\{\alpha\in T(V,W): V\alpha\subseteq W\alpha\} \] is the largest regular subsemigroup of $T(V,W)$. In this paper, we prove that any regular semigroup $S$ can be embedded in $F(V,W)$ with $\dim(V) = |S^1|$ and $\dim(W) = |S|$, and determine all the maximal subsemigroups of $F(V,W)$ when $W$ is a finite dimensional subspace of $V$ over a finite field. УДК 512.64 Tеореми про вкладення та максимальнi пiднапiвгрупи деяких напiвгруп лiнiйних перетвореньз обмеженим образом Нехай $V$ – векторний простір над деяким полем, а $T(V)$ – напівгрупа всіх лінійних перетворень з $V$ у $V.$ Для фіксованого підпростору $W$ простору $V$ нехай $F(V,W)$ – піднапівгрупа напівгрупи $T(V),$ яка складається з усіх лінійних перетворень $\alpha$ з $V$ у $W$ таких, що $V\alpha\subseteq W\alpha.$ Доведено, що будь-яку регулярну напівгрупу $S$ можна вкласти у $F(V,W)$ з $\dim(V) = |S^1|$ і $\dim(W) = |S|,$ та визначено всі максимальні піднапівгрупи з $F(V,W),$ якщо $W$ – скінченновимірний підпростір $V$ над скінченним полем.   Institute of Mathematics, NAS of Ukraine 2021-12-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1289 10.37863/umzh.v73i12.1289 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 12 (2021); 1714 - 1722 Український математичний журнал; Том 73 № 12 (2021); 1714 - 1722 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1289/9166 Copyright (c) 2021 Worachead Sommanee |
| spellingShingle | Sommanee, W. Sommanee, W. Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range |
| title | Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range |
| title_alt | Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range |
| title_full | Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range |
| title_fullStr | Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range |
| title_full_unstemmed | Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range |
| title_short | Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range |
| title_sort | embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range |
| topic_facet | linear transformation restricted range embedding maximal subsemigroup |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1289 |
| work_keys_str_mv | AT sommaneew embeddingtheoremsandmaximalsubsemigroupsofsomelineartransformationsemigroupswithrestrictedrange AT sommaneew embeddingtheoremsandmaximalsubsemigroupsofsomelineartransformationsemigroupswithrestrictedrange |