Codecomposition of a Transformation Semigroup
The present paper deals with the concept of “codecomposition” of a transformation semigroup interacting with the phase semigroup. In this way, we distinguish new classes of transformation semigroups with meaningful relations, e.g., we show the class of all distal transformation semigroups ⊂, the cla...
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irk-123456789-1657132020-02-17T01:26:36Z Codecomposition of a Transformation Semigroup Sabbaghan, M. Ayatollah Zadeh Shirazi, F. Hosseini, A. Статті The present paper deals with the concept of “codecomposition” of a transformation semigroup interacting with the phase semigroup. In this way, we distinguish new classes of transformation semigroups with meaningful relations, e.g., we show the class of all distal transformation semigroups ⊂, the class of all transformation semigroups decomposable into distal semigroups ⊂, and the class of all transformation semigroups (here, ⊂ is strict inclusion). Розглянуто концепцію „корозкладу" трансформаційної напівгрупи, що взаємодiє з фазовою напівгрупою. Таким чином, ми вирізняємо новий клас трансформаційних напівгруп зі змістовними співвідношеннями. Так, показано, що „клас всіх дистальних трансформаційних напівгруп ⊂, клас всіх трансформаційних напівгруп, що розкладаються в дистальні напівгрупи ⊂, клас всіх трансформаційних напівгруп" (де ⊂, позначає строге включення). 2013 Article Codecomposition of a Transformation Semigroup / M. Sabbaghan, F. Ayatollah Zadeh Shirazi, A. Hosseini // Український математичний журнал. — 2013. — Т. 65, № 11. — С. 1506–1514. — Бібліогр.: 9 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165713 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Sabbaghan, M. Ayatollah Zadeh Shirazi, F. Hosseini, A. Codecomposition of a Transformation Semigroup Український математичний журнал |
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The present paper deals with the concept of “codecomposition” of a transformation semigroup interacting with the phase semigroup. In this way, we distinguish new classes of transformation semigroups with meaningful relations, e.g., we show the class of all distal transformation semigroups ⊂, the class of all transformation semigroups decomposable into distal semigroups ⊂, and the class of all transformation semigroups (here, ⊂ is strict inclusion). |
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Sabbaghan, M. Ayatollah Zadeh Shirazi, F. Hosseini, A. |
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Sabbaghan, M. Ayatollah Zadeh Shirazi, F. Hosseini, A. |
| author_sort |
Sabbaghan, M. |
| title |
Codecomposition of a Transformation Semigroup |
| title_short |
Codecomposition of a Transformation Semigroup |
| title_full |
Codecomposition of a Transformation Semigroup |
| title_fullStr |
Codecomposition of a Transformation Semigroup |
| title_full_unstemmed |
Codecomposition of a Transformation Semigroup |
| title_sort |
codecomposition of a transformation semigroup |
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Інститут математики НАН України |
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2013 |
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| citation_txt |
Codecomposition of a Transformation Semigroup / M. Sabbaghan, F. Ayatollah Zadeh Shirazi, A. Hosseini // Український математичний журнал. — 2013. — Т. 65, № 11. — С. 1506–1514. — Бібліогр.: 9 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT sabbaghanm codecompositionofatransformationsemigroup AT ayatollahzadehshirazif codecompositionofatransformationsemigroup AT hosseinia codecompositionofatransformationsemigroup |
| first_indexed |
2025-07-14T19:39:45Z |
| last_indexed |
2025-07-14T19:39:45Z |
| _version_ |
1837652493907001344 |
| fulltext |
UDC 517.9
M. Sabbaghan, F. Ayatollah Zadeh Shirazi (College Sci., Univ. Tehran, Iran),
A. Hosseini (Univ. Guilan, Rasht, Iran)
CODECOMPOSITION OF A TRANSFORMATION SEMIGROUP
КОРОЗКЛАД ТРАНСФОРМАЦIЙНОЇ НАПIВГРУПИ
The following text deals with the concept of “codecomposition” of a transformation semigroup which interact with the
phase semigroup. In this way we distinguish new classes of transformation semigroups with meaningful relations, e.g., we
will show “the class of all distal transformation semigroups ⊂ the class of all transformation semigroups decomposable to
distal ones ⊂ the class of all transformation semigroups” (where ⊂ is strict inclusion).
Розглянуто концепцiю „корозкладу” трансформацiйної напiвгрупи, що взаємодiє з фазовою напiвгрупою. Таким
чином, ми вирiзняємо новий клас трансформацiйних напiвгруп зi змiстовними спiввiдношеннями. Так, показано,
що „клас всiх дистальних трансформацiйних напiвгруп⊂ клас всiх трансформацiйних напiвгруп, що розкладаються
в дистальнi напiвгрупи⊂ клас всiх трансформацiйних напiвгруп” (де ⊂ позначає строге включення).
1. Introduction. The approach of decomposition of a transformation group has been studied in
several manuscripts, e.g., as it has been mentioned in [3] (Proposition 2.6) in transformation group
(X,T ) with X locally compact T2, every point of X is an almost periodic point if and only if
{xT : x ∈ X} is a partition (decomposition) of X consisting of compact sets; however about twenty
five years before that text, i.e., in 1944, one may find this theorem “In order that the homeomorphism
h give an orbit-closure decomposition it is sufficient that h be pointwise almost periodic; and in
case X is compact, this condition is also necessary” [8] (Theorem 2) (for metric space X and
homeomorphism h : X → X), also the author has mentioned that a direct proof of this theorem can
be found in [7]; or in [4] (Section 2) in a minimal transformation group (X,T ) a relatively dense
subgroup of T namely G has been considered and the elements of decomposition
{
xG : x ∈ X
}
has
been studied (for abelian T and compact metric X . . . one may find discussions on decomposition of
a transformation semigroup in several other papers like [2].
In most of the above mentioned cases the emphasis is on simplifying the matter by dividing
the phase space to some smaller subspaces, then study them and their interactions; in this text
by codecomposition of a transformation semigroup we mean certain collection of transformation
semigroups with smaller phase semigroup, and then study the interaction of these codecompositions
with the original one.
2. First steps towards codecomposition approach: multitransformation semigroups. By a
right transformation semigroup we mean a triple (X,S, π) or simply (X,S) where X is a topological
space, S is a topological semigroup with identity e and π : X×S → X (π(x, s) = xs) is a continuous
map such that for each x ∈ X and for each s, t ∈ S we have xe = x and x(st) = (xs)t. Left
transformation semigroup (S,X) is defined in a similar way. By a bitransformation semigroup we
mean (S2, X, S1), where (S2, X) is a left transformation semigroup, (X,S1) is a right transformation
semigroup, and for each s1 ∈ S1, s2 ∈ S2, x ∈ X we have (s2x)s1 = s2(xs1) (which is denoted
by s2xs1) (note to the fact that left and right transformation groups coincide, e.g., if (S,X) is a left
transformation group, then (X,S) is a right transformation group, where xs := s−1x (x ∈ X, s ∈
∈ S)). But how much it will be interesting if instead of two sides (left and right) one consider several
sides for a point x ∈ X? This will be our primary motivation to introduce multitransformation
semigroups.
c© M. SABBAGHAN, F. AYATOLLAH ZADEH SHIRAZI, A. HOSSEINI, 2013
1506 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
CODECOMPOSITION OF A TRANSFORMATION SEMIGROUP 1507
However in the following subsection we do our best to cause more interest for readers through
some examples.
2.1. A short motivation through examples. Consider a mill supplied with the following powers:
S1: water S2: wind,
S3: sun S4: manpower.
Regardless of the power source, all of these powers act on the grain in the mill. Furthermore, it
does not matter which force is applied first. In this example one may have general approach or
partial approach. Whenever we study (X,S), where X is the collection of our grain and S is the
power supplied for the mill (hence S is generated by S1 ∪ S2 ∪ S3 ∪ S4), we have general approach.
Whenever we study (X,S1), (X,S2), (X,S3) and (X,S4) we have partial approach. As a matter of
fact regarding our future topics, ((X,Si) : i ∈ {1, 2, 3, 4}) is a multitransformation semigroup and a
codecomposition of (X,S).
We continue this subsection with some more specialized examples. Here we want to explain how
codecomposition of a transformation semigroup is partial approach to a system, and how multitrans-
formation semigroup’s concept helps us in this process (suppose all phase semigroups are discrete).
1. Consider X1 := [0, 1] with induced topology of R, X2 := {(x, y) ∈ R2 : x2 + y2 = 1}
with induced topology of R2, X3 := {(x, y, z) ∈ R3 : x2 + y2 + z2 = 1} with induced topology
of R3, and X as disjoint union of X1, X2, X3. For i = 1, 2, 3 suppose Si is the collection of all
homeomorphisms f : X → X such that f(x) = x for all x ∈ X \Xi. Also consider S as the group
of all homeomorphisms of X. Then:
for all i 6= j, f ∈ Si and g ∈ Sj we have fg = gf,
the semigroup S is generated by S1 ∪ S2 ∪ S3.
In particular regarding our future topics, ((X,Si) : i ∈ {1, 2, 3}) is a multitransformation semigroup
and a codecomposition of (X,S).
2. Consider X := {(x, y) ∈ R2 : x2 + y2 = 1} with induced topology of R2, and for prime
natural number q, Sq is the group of all rotations under a integer multiple of
π
q
. Moreover suppose
S is the group of rotations under a rational multiple of π. Then:
for all i 6= j, s ∈ Si and t ∈ Sj we have st = ts,
the semigroup S is generated by
⋃
{Sq : q is a prime number}.
In particular regarding our future topics, ((X,Si) : i is a prime number) is a multitransformation
semigroup and a codecomposition of (X,S).
3. Consider X := [0, 1] with induced topology of R, S as the group of all homeomorphisms
of X, n ≥ 2, and 0 ≤ θ1 < θ2 < . . . < θn ≤ 1. Moreover for i = 1, . . . , n suppose Si := {f ∈
∈ S : f(θi) = θi}, then:
for all i 6= j there exists f ∈ Si and g ∈ Sj such that fg 6= gf,
the semigroup S is generated by S1 ∪ . . . ∪ Sn.
Then ((X,Si) : i ∈ {1, . . . , n}) is not a multitransformation semigroup, since there are elements of
different Si’s such that it is important which one act first on an element of X, although S is generated
by S1 ∪ . . . ∪ Sn.
In the following text in the transformation semigroup (X,S), S acts effectively on X, i.e., for
each s, t ∈ S with s 6= t there exists x ∈ X such that xs 6= xt.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1508 M. SABBAGHAN, F. AYATOLLAH ZADEH SHIRAZI, A. HOSSEINI
Definition 2.1. If Γ 6= ∅ and for each α ∈ Γ, (X,Sα) is a transformation semigroup with
eα as the identity element of Sα, such that for each y ∈ X, α1, . . . , αn ∈ Γ which are distinct,
sα1 ∈ Sα1 , . . . , sαn ∈ Sαn , and σ ∈ Sn (where Sn is the group of all permutations on {1, . . . , n})
we have (. . . ((ysα1)sα2) . . .)sαn = (. . . ((ysασ(1))sασ(2)) . . .)sασ(n) . We call (X, (Sα : α ∈ Γ)) a
multitransformation semigroup (it is sufficient to consider this definition for n = 2).
Now we have the main tool to introduce codecomposition of a transformation semigroup, since
codecompositions are certain multitransformation semigroups.
3. Codecompositions in transformation semigroups with discrete phase semigroup and com-
pact Hausdorff phase space. Codecompositions of transformation semigroup (X,S) are multitrans-
formation semigroups (X, (Sα : α ∈ Γ)) with certain properties. Since the idea of codecomposition
of a transformation semigroup, when phase semigroup is discrete brings the best motivation, so in
this section all phase semigroups considered discrete. Moreover in this section all phase spaces are
compact Hausdorff.
Remark 3.1. In the transformation semigroup (X,S) for compact Hausdorff X and discrete S,
for each s ∈ S, define the continuous map πs : X → X by xπs = xs (∀x ∈ X), then E(X,S) (or
simply E(X)) is the closure of {πs| s ∈ S} in XX with pointwise convergence topology, moreover
it is called the enveloping semigroup (or Ellis semigroup) of (X,S). We used to write s instead
of πs (s ∈ S). E(X,S) has a semigroup structure [3] (Chapter 3). A nonempty subset Z of X is
called invariant if ZS ⊆ Z. Let P(X,S) = {(x, y) ∈ X × X : ∃p ∈ E(X) : xp = yp}. (X,S) is
called distal if E(X) is a group (or equivalently P(X,S) = ∆X [3] (Chapter 5)), and it is called
equicontinuous if for all element α of the uniformity on X, there exists an element β of the uniformity
on X with βS ⊆ α and E(X) is a group of continuous functions (see [3] (Proposition 4.4)). (X,S)
is called proximal if P(X,S) = X × X (for more about proximal transformation groups see [6]).
(X,S) is called point transitive if ∃w ∈ X wS = X, it is called minimal if ∀w ∈ X wS = X.
Definition 3.1 (Discrete phase semigroup case). Multitransformation semigroup (X, (Sα : α ∈
∈ Γ)) is a codecomposition of transformation semigroup (X,S), where S is the semigroup generated
by
⋃
α∈Γ
Sα and Sαs are distinct subsemigroups of S.
3.1. Elementary properties.
Lemma 3.1. Let multitransformation semigroup (X, (Sα : α ∈ Γ)) be a codecomposition of
transformation semigroup (X,S), then we have
E(X,Sα) ⊆ E(X,S), α ∈ Γ . . . ;
ps = sp, p ∈ E(X,Sα), s ∈ Sβ, α 6= β.
Proof. If α 6= β, p ∈ E(X,Sα), and s ∈ Sβ, then there exists a net (tλ)λ∈Λ in Sα converges to
p (in E(X,Sα)), thus for each x ∈ X, (xtλ)λ∈Λ converges to xp and by continuity of s, (xtλs)λ∈Λ
converges to xps. Now replacing x by xs leads us to the fact that (xstλ)λ∈Λ converges to xsp, but
for each λ, tλs = stλ; thus xsp = xps for all x ∈ X and sp = ps.
Theorem 3.1. Let multitransformation semigroup (X, (Sα : α ∈ Γ)) be a codecomposition of
transformation semigroup (X,S), then we have:
1. If (X,S) is distal (resp. equicontinuous), then for each α ∈ Γ, (X,Sα) is distal (resp.
equicontinuous).
2. If for some α ∈ Γ, (X,Sα) is minimal (resp. point transitive, proximal), then (X,S) is
minimal (resp. point transitive, proximal).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
CODECOMPOSITION OF A TRANSFORMATION SEMIGROUP 1509
Proof. 1. Use Lemma 3.1, and the fact that every transformation semigroup is distal if and only
if its enveloping semigroup has at most one idempotent element [3] (Propositions 3.5 and 5.3).
2. If (X,Sβ) is minimal, then for each x ∈ X, xSβ = X, thus X = xSβ ⊆ xS ⊆ X, and (X,S)
is minimal.
Definition 3.2. Transformation semigroup (X,S) is called codecomposable to distal (resp.
proximal, equicontinuous, minimal, point transitive) transformation semigroups if there exists a code-
composition of (X,S) like (X, (Sα : α ∈ Γ)) such that for each α ∈ Γ, (X,Sα) is distal (resp.
proximal, equicontinuous, minimal, point transitive).
Theorem 3.2. In the transformation semigroup (X,S), (X,S) is codecomposable to distal
(resp. minimal, equicontinuous, proximal) transformation semigroups if and only if for each closed
nonempty invariant subset Z ofX, (Z, S) is codecomposable to distal (resp. minimal, equicontinuous,
proximal) transformation semigroups.
Proof. Use the fact that for each nonempty closed invariant subset Z of X, if (X, (Sα : α ∈ Γ))
is a codecomposition of (X,S), then (Z, (Sα : α ∈ Γ)) is a codecomposition of (Z, S), and in
distal (resp. equicontinuous, proximal) transformation semigroup (Y, T ) if W is a nonempty closed
invariant subset of Y then (W,T ) is distal (resp. equicontinuous, proximal).
Theorem 3.3. In the transformation group (X,S), if (X,S) is codecomposable to distal (resp.
point transitive, minimal, proximal) transformation groups and < is a clopen invariant equivalence
relation on X, then (X/<, S) is codecomposable to distal (resp. point transitive, minimal, proximal)
transformation groups.
Proof. Use Definition 3.1 [1] (Section 1.3.2) or [9] (Exercise 1.11), and similar methods de-
scribed in Theorem 3.2.
Theorem 3.4. Let ((Xθ, S) : θ ∈ Θ) be a nonempty collection of transformation semigroups,
such that
(∏
θ∈Θ
Xθ, S
)
is codecomposable to distal (resp. point transitive, minimal, proximal) trans-
formation semigroups. Then for each θ ∈ Θ, (Xθ, S) is codecomposable to distal (resp. point tran-
sitive, minimal, proximal) transformation semigroups.
Proof. Use Definition 3.1, and similar methods described in Theorem 3.2.
3.2. Counterexamples. Since the following counterexamples play important roles to find out
main results during discussion on compact Hausdorff phase space with discrete phase semigroup
case, so we devote them a special subsection.
Counterexample 3.1. Let X =
{
1
n
: n ∈ N
}
∪ {0} (with induced topology of R) and S =
= {ϕn : n ≥ 0} (with the discrete topology) where 0ϕ = 0,
1
n
ϕ =
1
n+ 1
(for n ∈ N). Let
(X, (Sα : α ∈ Γ)) be a codecomposition of (X,S), thus there exists α ∈ Γ such that Sα is infinite,
which leads to the fact that constant function 0 belongs to E(X,Sα) and E(X,Sα) is not a group,
therefore (X,Sα) is not distal. Therefore transformation semigroup (X,S) is not codecomposable to
distal transformation semigroups.
Counterexample 3.2. Let Y =
⋃
n∈N
{2n} × Z2n . For each (2n, x), (2m, y) ∈ Y define (2n,
x) ∼ (2m, y) if and only if (n ≥ m ∧ 2n−my = x) ∨ (m ≥ n ∧ 2m−nx = y), ∼ is an equivalence
relation on Y. Consider quotient space Y
∼ with discrete topology and let X := Y
∼ ∪ {∞} be its one
point compactification, moreover denote the equivalence class of (2m, y) ∈ Y under ∼ by [2m, y].
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1510 M. SABBAGHAN, F. AYATOLLAH ZADEH SHIRAZI, A. HOSSEINIt
{2} × Z2
{4} × Z4
{8} × Z8
(2,0) (2,1)
(4,0) (4,1) (4,2)
(8,0) (8,1) (8,2)(8,3) (8,4) (8,5) (8,6)(8,7)
∞
(4,3)
An imagination of X: vertical bold lines show equivalence classes in Y, however X is homeomorph with
{
1
n
: n ∈
∈ N
}
∪ {0} (with induced topology of R), i.e., ∞ is the unique limit point of X, X is infinite countable, and each open
neighborhood of∞ contains all points of X except finitely.
Let n ∈ N. (X,Z2n) is a transformation group, where for each l ∈ Z2n = {0, 1, . . . , 2n − 1} and
[2m, y] ∈ X with m ≥ n we have [2m, y]l = [2m, y + l2m−n] and ∞l =∞ (note to the fact that for
each [2k, y] ∈ X we have [2k, y] = [2k+n, 2ny] and k + n > n so [2k, y]l = [2k+n, 2ny + l2k]).
For each n ∈ N, let Tn =
{
(xi)i∈N ∈
∏
i∈N
Z2i ∀i 6= n(xi = 0)
}
. (X,Tn) is a transformation
semigroup where for each (xi)i∈N ∈ Tn and z ∈ X we define z(xi)i∈N := zxn. (X, (Tn : n ∈ N))
is a multitransformation semigroup and a codecomposition of (X,T ) when T is the subgroup of∏
i∈N
Z2i generated by
⋃
{Tn : n ∈ N}
(
thus T =
⊕
n∈N
Z2n (with discrete topology)
)
.
Claim. We have the following facts:
1) (X,T ) is niether distal nor equicontinuous;
2) (X,T ) is point transitive;
3) For each n ∈ N, (X,Tn) is equicontinuous, and distal;
4) For each n ∈ N, (X,Tn) is not point transitive.
Proof. 1. For each n ∈ N let tn = (δmn)m∈N ∈ Tn with δnn = 1 and δmn = 0 for m 6= n.
The sequence {tn}n∈N in E(X,T ) converges to constant function ∞; since for each x ∈ X − {∞}
and n 6= m we have xtn 6= xtm, and if U is an open neighborhood of ∞ then X − U is finite
and there exists m ∈ N such that xtn ∈ U for all n ≥ m, i.e., {tn}n∈N converges to ∞, moreover
for each n ∈ N we have ∞tn = ∞. So the constant function ∞ belongs to E(X,T ) (with no
inverse) thus E(X,T ) is not a group, therefore (X,T ) is neither distal nor equicontinuous (see [3],
Proposition 4.4).
2. (X,T ) is point transitive since [2, 1]T = X.
3. Tn is a finite group and E(X,Tn) = Tn is a (finite) group of continuous functions on X,
which shows distality and equicontinuity of (X,Tn).
4. For each x ∈ X, xTn = xTn is a finite subspace of infinite space X, thus xTn 6= X and
(X,Tn) is not point transitive.
Therefore in transformation semigroup (X,T ) we have
(X,T ) is a non-distal transformation semigroup codecomposable to distal ones;
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
CODECOMPOSITION OF A TRANSFORMATION SEMIGROUP 1511
(X,T ) is a non-equicontinuous transformation semigroup codecomposable to equicontinuous
ones;
(X,T ) is a point transitive transformation semigroup codecomposable to non-point transitive
ones.
Counterexample 3.3. Let X := {eiθ : θ ∈ [0, 2π]} with the induced topology of R2, and
Sn = {z ∈ X : zn = 1} (n ∈ N), then Sn acts on X as one of its subsemigroups. For each n ∈ N,
let Tn =
{
(xi)i∈N ∈
∏
i∈N
Sn : ∀i 6= n(xi = 1)
}
. (X,Tn) is a transformation semigroup where
for each (xi)i∈N ∈ Tn and z ∈ X we define z(xi)i∈N := zxn. (X, (Tn : n ∈ N)) is a multitransfor-
mation semigroup and a codecomposition of (X,T ) when T is the subgroup of
∏
i∈N
Sn generated by
⋃
{Tn : n ∈ N}
(
thus T =
⊕
n∈N
Sn (with discrete topology)
)
. (X,T ) is minimal, but (X,Tn)
is not minimal (∀n ∈ N), thus (X,T ) is a minimal transformation semigroup codecomposable to
non-minimal ones.
Counterexample 3.4. In this counterexample we show that there are examples of multitransfor-
mation semigroup (X, (Sα : α ∈ Γ)) in which there exist α 6= β, p ∈ E(X,Sα) and q ∈ E(X,Sβ)
with pq 6= qp (compare with the second item in Theorem 3.1). Let X =
{
(−1)i
(
1 − 1
n
)
: n ∈
N, i = ±1
}
∪ {1,−1} with induced topology from R. Define q, p, ϕ : X → X with:
xϕ :=
x, x = ±1,
1− 1
n
, x = 1− 1
n+ 1
, n ∈ N,
−1 +
1
n+ 1
, x = −1 +
1
n
, n ∈ N,
xp :=
−1, x 6= 1,
1, x = 1,
xq :=
1, x 6= −1,
−1, x = −1.
For S = {ϕn : n ∈ Z}, S1 = {ϕn : n ∈ N ∪ {0}}, S2 = {ϕ−n : n ∈ N ∪ {0}}, and S3 = {ϕ2n :
n ∈ Z}; (X, (Si : i = 1, 2, 3)) is a codecomposition of (X,S). p ∈ E(X,S1) and q ∈ E(X,S2)
moreover pq 6= qp.
3.3. Main achievements.
Theorem 3.5 (Main theorem). We have the following classifications (where “⊂” means strict
inclusion):
1. Class of all distal transformation semigroups ⊂ Class of all transformation semigroups code-
composable to distal ones ⊂ Class of all transformation semigroups.
2. Class of all equicontinuous transformation semigroups ⊂ Class of all transformation semi-
groups codecomposable to equicontinuous ones ⊂ Class of all transformation semigroups.
Proof. Use Counterexamples 3.1 and 3.2. One may consider the following diagram for (1):
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1512 M. SABBAGHAN, F. AYATOLLAH ZADEH SHIRAZI, A. HOSSEINI
Class of all transformation
semigroups codecomposable to distal ones
(see Counterexample 3.2)
Class of all distal
transformation semigroups
Class of all
transformation semigroups
(see Counterexample 3.1)
and a similar diagram for (2).
Theorem 3.6. We have the following classifications (where “⊂” means strict inclusion):
1. Class of all point transitive transformation semigroups codecomposable to non-point transitive
ones ⊂ Class of all point transitive transformation semigroups.
2. Class of all minimal transformation semigroups codecomposable to non-minimal ones ⊂ Class
of all minimal transformation semigroups.
Proof. Use Counterexamples 3.2 and 3.3. One may consider the following diagram for (2):
Class of all transformation semigroups
codecomposable to non-minimal ones
(Counterexample 3.3)
(Counterexample 3.4
or any other
Class of all minimal
transformation semigroups
(one point transformation semigroup)
non-minimal)
and a similar diagram for (1).
Corollary 3.1. We have the following diagram in the class of all transformation groups (with
compact Hausdorff phase space and discrete phase semigroup) (use [3] (Proposition 4.4)):
EQUICONTINUOUS ⇒ codecomposABLE TO EQUICONTINUOUS ONES
⇓ ⇓
DISTAL ⇒ codecomposABLE TO DISTAL ONES.
In order to complete the reverse implications of the above diagram see Counterexample 3.2 (which
shows the negative answer for horizontal reverse implications), see also [5] (which shows the negative
answer for first vertical reverse implication by an example).
Now the next question arises.
Question. Is there any transformation group decomposable to distal ones, and non-decomposable
to equicontinuous ones?
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CODECOMPOSITION OF A TRANSFORMATION SEMIGROUP 1513
4. General case. In this section our main aim is to define codecomposition of a transformation
semigroup in general case.
In multi transformation semigroup (X, (Sα : α ∈ Γ)) if⊕
α∈Γ
Sα =
{
(sα)α∈Γ : ∃α1, . . . , αn ∈ Γ ∀α 6= α1, . . . , αn(sα = eα)
}
,
then
⊕
α∈Γ
Sα acts on X (not necessarily continuously), by
w(sα)α∈Γ = wsα1 . . . sαn(
for all w ∈ X and (sα)α∈Γ ∈
⊕
α∈Γ
Sα with distinct α1, . . . , αn ∈ Γ and sα = eα for all
α ∈ Γ− {α1, . . . , αn}
)
.
In this section we will use the above defined action, moreover for more convenient the element
(sα)α∈Γ ∈
⊕
α∈Γ
Sα with sα = eα for all α 6= β, is denoted by sβxβ.
Moreover if
(
X,
⊕
α∈Γ
Sα
)
is a transformation semigroup and for each α ∈ Γ the semigroup
Sα has at least two elements, also for each β ∈ Γ, Tβ :=
{
(sα)α∈Γ ∈
⊕
α∈Γ
Sα : ∀α 6= βsα = eα
}
,
then (X, (Tα : α ∈ Γ)) is a multitransformation semigroup.
The above statements leads us to the following definition.
Definition 4.1 (General case). Multi transformation semigroup (X, (Sα : α ∈ Γ)) is called a
codecomposition of transformation semigroup (X,S) if S is the semigroup generated by
⋃
α∈Γ
Sα,
Sαs are distinct subsemigroups of S, and there exists a topology on
⊕
α∈Γ
Sα which makes a
topological semigroup and:(
X,
⊕
α∈Γ
Sα
)
is a (topological) transformation semigroup;
for each β ∈ Γ the map ιβ : Sβ →
⊕
α∈Γ
Sα with ιβ(s) = sxβ, be an embedding;
under the map sα1x
α1 +. . .+sαnx
αn 7→ sα1 . . . sαn , S is a continuous semigroup homomorphism
image of
⊕
α∈Γ
Sα.
Note 4.1 (Compatibility of Definitions 3.1 and 4.1). In Definition 4.1, whenever S is discrete,
discrete topology on
⊕
α∈Γ
Sα shows that if (X, (Sα : α ∈ Γ)) is a codecomposition of (X,S) in the
sense of Definition 3.1, then it is a codecomposition of (X,S) in the sense of Definition 4.1 too.
On the other hand if S is discrete and (X, (Sα : α ∈ Γ)) is a codecomposition of (X,S) in the
sense of Definition 4.1, then it is clear that (X, (Sα : α ∈ Γ)) is a codecomposition of (X,S) in the
sense of Definition 3.1 too. In multitransformation semigroup (X, (Sα : α ∈ Γ)), let T :=
⊕
α∈Γ
Sα
with the induced product topology from
∏
α∈Γ
Sα, then T is a topological semigroup, however in
the following example T does not act continuously on X.
Example 4.1. Let {pn : n ∈ N} be a Q-linearly independent sequence in R such that lim
n→∞
pn = 2
and p1 = 1. In the multitransformation semigroup (R, (pnQ : n ∈ N)) (pnQs act as subgroups of R
on topological group (R,+)) as above let T :=
⊕
n∈N
pnQ with the induced product topology from∏
n∈N
pnQ. The sequence (1, 2x1 + pnx
n+1)n∈N (where 2x1 + pnx
n+1 = (amn )m∈N for a1
n = 2,
an+1
n = pn and amn = 0 for m 6= n+1, 1) converges to (1, 2x1) in R×T, but (1(2x1 +pnx
n+1))n∈N
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1514 M. SABBAGHAN, F. AYATOLLAH ZADEH SHIRAZI, A. HOSSEINI
converges to 5 (since for all n ∈ N we have 1(2x1 + pnx
n+1) = 3 + pn) and it does not converges
to 3(= 1(2x1)), thus (R, T ) is not a topological transformation semigroup.
Theorem 4.1. If ((Xθ, S) : θ ∈ Θ) is a nonempty collection of transformation semigroups,
then
(∏
θ∈Θ
Xθ, (Sα : α ∈ Γ)
)
is a codecomposition of
(∏
θ∈Θ
Xθ, S
)
if and only if for each
θ ∈ Θ, (Xθ, (Sα : α ∈ Γ)) is a codecomposition of (Xθ, S).
Proof. Use Definitions 2.1 and 4.1.
Theorem 4.2. In the transformation group (X,S) if < is an open invariant equivalence re-
lation on X, Z is a nonempty invariant subset of X, and (X, (Sα : α ∈ Γ)) is a codecomposition
of (X,S), then (X/<, (Sα : α ∈ Γ)) is a codecomposition of (X/<, S) and (Z, (Sα : α ∈ Γ)) is a
codecomposition of (Z, S).
Proof. Use Definitions 2.1, 4.1 and [1] (Section 1.3.2).
Remark 4.1. It is possible to define multitransformation group, and codecomposition of a trans-
formation group in a similar way.
Remark 4.2. In all sections of this paper except the last one, we dealt with discrete phase
semigroup. However one may be interested in the following cases: We call a codecomposition
(X, (Sα : α ∈ Γ)) of (X,S) a degenerated codecomposition if there exists α ∈ Γ with Sα = S.
It is clear that if S = Zp, when p is a prime, then all codecompositions of (X,S) are degenerated. Is
there any transformation semigroup (X,S) with just degenerated codecompositions, but (X,Sd) has
non-degenerated codecompositions, where Sd is S with discrete topology?
Under which topologies on
⊕
α∈Γ
Sα, the multitransformation semigroup (X, (Sα : α ∈ Γ)) is a
codecomposition of
(
X,
⊕
α∈Γ
Sα
)
(is there any?) moreover if Λ denotes the set of such topologies,
what are the properties of minimal and maximal elements of (Λ,⊆)?
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P. 317 – 332.
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Received 11.10.11,
after revision — 16.06.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
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