Free semigroups in wreath powers of transformation semigroups
It is established a criterion when the infinite wreath power of a finite transformation semigroup contains a free subsemigroup. It is shown that the infinite wreath power of a transformation semigroup either contains no free non-commutative subsemigroups or most of its finitely generated subsemigr...
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| Цитувати: | Free semigroups in wreath powers of transformation semigroups / A. Oliynyk // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 96–106. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1548672019-06-17T01:31:29Z Free semigroups in wreath powers of transformation semigroups Oliynyk, A. It is established a criterion when the infinite wreath power of a finite transformation semigroup contains a free subsemigroup. It is shown that the infinite wreath power of a transformation semigroup either contains no free non-commutative subsemigroups or most of its finitely generated subsemigroups are free. 2010 Article Free semigroups in wreath powers of transformation semigroups / A. Oliynyk // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 96–106. — Бібліогр.: 8 назв. — англ. http://dspace.nbuv.gov.ua/handle/123456789/154867 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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It is established a criterion when the infinite wreath power of a finite transformation semigroup contains a free subsemigroup. It is shown that the infinite wreath power of a transformation semigroup either contains no free non-commutative subsemigroups or most of its finitely generated subsemigroups are free. |
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Oliynyk, A. Free semigroups in wreath powers of transformation semigroups Algebra and Discrete Mathematics |
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Oliynyk, A. |
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Oliynyk, A. |
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Free semigroups in wreath powers of transformation semigroups |
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Free semigroups in wreath powers of transformation semigroups |
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Free semigroups in wreath powers of transformation semigroups |
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Free semigroups in wreath powers of transformation semigroups |
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Free semigroups in wreath powers of transformation semigroups |
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free semigroups in wreath powers of transformation semigroups |
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Інститут прикладної математики і механіки НАН України |
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2010 |
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Free semigroups in wreath powers of transformation semigroups / A. Oliynyk // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 96–106. — Бібліогр.: 8 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT oliynyka freesemigroupsinwreathpowersoftransformationsemigroups |
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2025-07-14T06:56:08Z |
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2025-07-14T06:56:08Z |
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1837604451730325504 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 10 (2010). Number 2. pp. 96 – 106
c© Journal “Algebra and Discrete Mathematics”
Free semigroups in wreath powers
of transformation semigroups
Andriy Oliynyk
Communicated by V. I. Sushchansky
Abstract. It is established a criterion when the infinite
wreath power of a finite transformation semigroup contains a free
subsemigroup. It is shown that the infinite wreath power of a trans-
formation semigroup either contains no free non-commutative sub-
semigroups or most of its finitely generated subsemigroups are free.
1. Introduction
It is well-known that inverse limits of wreath products of permutation
groups are rich of free subgroups ([1], [2]). On the other hand the analogous
theorem for free subsemigroups in inverse limits of wreath products of
transformation semigroups is not true in general as this inverse limit do
not certainly contain such subsemigroups. Nevertheless, in the category
sense of Baire in some topological semigroups most finitely generated
subsemigroups
infinite wreath powers of transformation semigroups as inverse limits of
their finite wreath powers.
The work is organized as follows. In Section 2 we recall main definitions
and notations concerning wreath powers of transformation semigroups. In
Section 3 we prove a criterion in terms of finite transformation semigroup
when its infinite wreath power contains free subsemigroups. In Section 4
we prove that if the infinite wreath power of a transformation semigroup
contain free non-commutative subsemigroups then most of their finitely
generated subsemigroups are free.
A. Oliynyk 97
2. Wreath powers
For details on this section see [6] and [7].
Let X be a non-empty set. As usual, denote by TX the full transfor-
mation semigroup of X. In this note a transformation semigroup (T,X) is
a subsemigroup T of the TX acting on X. We will use the right actions of
transformations. The set of idempotents of a semigroup S will be denoted
by E(S). For subsets A,B ⊆ S let AB = {ab|a ∈ A, b ∈ B}. Analogously,
An = {an|a ∈ A} for n ≥ 1.
Denote by X(n) the n-th cartesian power of X, n ≥ 1, and by Xω its
countable cartesian power. Let X [n] = ∪n
i=0X
(i), n ≥ 0. Consider a free
monoid X∗ with a basis X. Then X∗ is naturally identified with the set
∪∞i=0X
(i), where X(0) = {Λ} and Λ is the identity of X∗, the empty word.
Being an associative operation, the wreath product of transformation
semigroups can be defined on arbitrary finite number of semigroups. Let
Wn(T,X) denotes the wreath product of n copies of the transformation
semigroup (T,X) acting of the set X(n), n ≥ 1. Each element t̄ of the
semigroup Wn(T,X) can be written in the form
t̄ = [t1; t2(x1); . . . ; tn(x1, . . . , xn−1)],
where t1 ∈ T , t2(x1) : X → T, . . . , tn(x1, . . . xn−1) : X
(n−1) → T . Such an
element acts on a point (a1, a2, . . . , an) ∈ X(n) by the rule
(a1, a2, . . . , an)
[t1;t2(x1);...;tn(x1,...,xn−1)] = (at11 , a
t2(a1)
2 , . . . , atn(a1,...,an−1)
n ).
The natural projection πn : Wn(T,X) → Wn−1(T,X) which erases the
last coordinate is an epimorphism, n ≥ 2. Hence, we obtain an inverse
limit
W = lim
←−
(Wn(T,X), πn)
acting on the set
∞∏
n=1
X(n).
The subset
{((x1), (x1, x2), . . . , (x1, x2, . . . , xn), . . .)|xi ∈ X, i ≥ 1},
which is naturally identified with Xω, is invariant under this action. We
call the transformation semigroup (W,Ω) the infinite wreath power of
(T,X), and denote it by W∞(T,X).
Each element of the infinite wreath power W∞(T,X) can be viewed
as an infinite sequence
t̄ = [t1; t2(x1); t3(x1, x2); . . .], (1)
98 Free semigroups in wreath powers
where t1 ∈ T , t2(x) : X → T, t3(x1, x2) : X
2 → T, . . .. The action of the
semigroup W∞(T,X) on the set Xω (or X(n), n ≥ 1,) is given by the rule
ut̄ = at11 a
t2(a1)
2 a
t3(a1,a2)
3 . . . (2)
for arbitrary u = a1a2a3 . . . ∈ Xω. For t̄, s̄ ∈ W∞(T,X), where t̄ is as
above, and
s̄ = [s1; s2(x1); s3(x1, x2); . . .],
their product t̄s̄ has the form
t̄s̄ = [t1s1; t2(x1)s2(x
t1
1 ); t3(x1, x2)s3(x
t1
1 , x
t2(x1)
2 ); . . .]. (3)
One may regard the infinite wreath power W∞(T,X) of (T,X) as a set
of infinite tuples of the form (1) with multiplication rule (3), acting on
Xω by (2).
We may use the following way to describe elements of wreath powers.
Let t̄ ∈ W∞(T,X) be of the form (1). For each u ∈ X∗ let us define an
element t̄u ∈ T by the rule
t̄u =
{
t1, if u = Λ
t|u|+1(a1, . . . , a|u|), if u = a1 . . . a|u|
,
where the length of the word u is denoted by |u|. Then the element
P(t̄) = {t̄u, u ∈ X∗} of the cartesian power TX∗
is called the portrait of
t̄. The correspondence P between sets W∞(T,X) and TX∗
is one-to-one.
This means that the infinite wreath power W∞(T,X) can be regarded as
the set of portraits TX∗
. For two portraits
P(t̄) = {t̄u, u ∈ X∗}, P(s̄) = {s̄u, u ∈ X∗}
by (3) their product is computed as
P(t̄ · s̄) = {t̄u · s̄ut̄ , u ∈ X∗}.
In the same way portraits of elements of finite wreath power Wn(T,X)
as elements of the cartesian power TX[n−1]
are defined, n ≥ 1.
The following proposition immediately follows from the definitions
above.
Proposition 1. Let (T,X) be a transformation semigroup.
1. For arbitrary k ≥ 1 the infinite wreath power W∞(T,X) admits a
decomposition W k(T,X) ≀W∞(T,X).
A. Oliynyk 99
2. For arbitrary sequence k1, k2, . . . of positive integers the infinite
wreath power W∞(T,X) contains as a subsemigroup the cartesian
product
∞∏
i=1
W ki(T,X).
Proof. (1) Let k ≥ 1 be fixed. For a tuple
t̄ = [t1; t2(x1); t3(x1, x2); . . .] ∈ W∞(T,X)
define a tuple
pk(t̄) ∈ W k(T,X)
and a mapping
sk(t̄) : X
k → W∞(T,X)
as follows
pk(t̄) = [t1; . . . tk(x1, . . . , xk−1)],
sk(t̄)(a1, . . . , ak) = [tk+1(a1, . . . , ak), tk+2(a1, . . . , ak, x1), . . .],
where a1, . . . , ak ∈ X. Then the rule
t̄ 7→ [pk(t̄); sk(t̄)]
is a required isomorphism.
(2) Let ki, i ≥ 1, be a fixed sequence of positive integers. Let k0 = 0.
Each element s of the cartesian product
∏∞
i=1W
ki(T,X) has the form
s = (s̄(1), s̄(2), . . .),
where s̄(i) ∈ W ki(T,X), i ≥ 1. Define an element ϕ(s) = s̄ ∈ W∞(T,X).
To do this it is sufficient to describe its portrait P(s̄). For any w ∈ X∗
there exist i ≥ 1 such that
k0 + k1 + . . .+ ki−1 ≤ |w| < k0 + k1 + . . .+ ki.
Denote by w1 the word obtained by deleting first k0 + k1 + . . . + ki−1
coordinates of w. Then we define the transformation s̄w to be equal to
the transformation s̄(i)w1 . It is easily verified that ϕ gives a required
isomorphic embedding.
100 Free semigroups in wreath powers
3. Existence of free subsemigroups
We start with some simple statements about finite transformation semi-
groups.
Lemma 1. For a finite semigroup T the following conditions are equiva-
lent.
1. The semigroup T is not nilpotent.
2. Not every element of T is nilpotent.
3. The semigroup T contains an idempotent which is not a zero element
of T .
Proof. The eqivalence of the first two conditions is well known (see, for
instance, [8, Proposition 8.1.2]).
Let T be not nilpotent. Each element of T in some power is an idem-
potent. If all these idempotents are equal to the same idempotent then
this idempotent is not a zero of T for (T,X) is not nilpotent. In other
case there are at least two different idempotents in T and at least one of
then is not a zero. In both cases there exist an idempotent in T which is
not a zero element.
Let T contains an idempotent e which is not a zero element. If e is
the unique idempotent of T then T does not contain a zero and is not
nilpotent. If T contains other idempotents then each power of e equals e
and is not equal to none of them. Therefore e is not nilpotent even if T
contains a zero. The proof is complete.
Lemma 2. Let in finite semigroup T each idempotent is a left zero. Then
T · E(T ) = E(T ) and T |T | = E(T ).
Proof. Let t ∈ T , e ∈ E(T ). Then (te)2 = t(ete) = te and we obtain the
inclusion T ·E(T ) ⊆ E(T ). The inverse inclusion is always true. Therefore
T · E(T ) = E(T ).
Assume that for some s1, . . . , sk ∈ T the product s1 . . . sk is not an
idempotent. If for some i, 1 ≤ i ≤ k, the element si is an idempotent then
s1 . . . sk = s1 . . . si for si is a left zero. Since T · E(T ) ⊆ E(T ) it would
implies s1 . . . si ∈ E(T ) which contradicts with the assumption. Then
elements s1, . . . , sk are not idempotents and each product ti = s1 . . . si,
1 ≤ i ≤ k, is not an idempotent as well for idempotents are left zeroes
in T . Assume that among elements t1, . . . , tk there are equal ones. Let
ti = tj , where i < j, and t = si+1 . . . sj . Then for arbitrary r ≥ 1 the
equalities tit
r = tjt
r−1 = tit
r−1 = . . . = ti hold. Since T is finite for some
r0 ≥ 1 the element tr0 is an idempotent. Then ti = tit
r0 ∈ E(T ). This is a
A. Oliynyk 101
contradiction. Hence, elements t1, . . . , tk are pairwise disjoint. Recall that
they are not idempotents. This means that k < |T |. Then T |T | ⊆ E(T ).
The inverse inclusion always holds. Therefore T |T | = E(T ) and the proof
is complete.
Lemma 3. For a finite transformation semigroup (T,X) the following
conditions are equivalent.
1. The semigroup (T,X) contains an idempotent which is not a left
zero element.
2. There exist e, s ∈ T and x ∈ X such that e is an idempotent and
xe = x 6= xs.
Proof. Let e ∈ T be an idempotent such that es 6= e for some s ∈ T . It
implies that there exist y ∈ X such that yes 6= ye. Take x = ye. Then
xe = ye
2
= ye = x and xs = yes 6= ye = x.
From the other hand, let e, s ∈ T and x ∈ X be such that e2 = e and
xe = x 6= xs. Then xes = xs 6= x = xe and es 6= e. Hence the idempotent
e is not a left zero element.
If a semigroup (T,X) satisfy conditions of Lemma 3 then it is neither
nilpotent by Lemma 1, nor contains a left zero.
Denote by Fk a free semigroup with basis {y1, . . . , yk}, k ≥ 1. Let
v ∈ Fk be a word of length n ≥ 1. Then
v = z1 . . . zn for some z1, . . . , zn ∈ {y1, . . . , yk}.
Define a mapping χv : {1, . . . n} → {1, . . . k} by the condition
χv(i) = j iff zi = yj , 1 ≤ i ≤ n.
Let vi = z1 . . . zi, 1 ≤ i ≤ n. In particular, vn = v. For a k-tuple (s1, . . . , sk)
over a semigroup S let us denote by vi(s1, . . . , sk) the value of the word
vi on this tuple. One obtains this value substituting sj instead of yj ,
1 ≤ j ≤ k, and calculating the product in S.
Lemma 4. Let a finite semigroup (T,X) contains an idempotent which is
not a left zero element. For arbitrary word v ∈ Fk of length n there exist a
tuple (t̄(1), . . . , t̄(k)) over the nth wreath power Wn(T,X) of (T,X) such
that for arbitrary word u ∈ Fk of length ≤ n, u 6= v, the inequality
u(t̄(1), . . . , t̄(k)) 6= v(t̄(1), . . . , t̄(k))
holds.
102 Free semigroups in wreath powers
Proof. By Lemma 3 we can fix e, s ∈ T and x, y ∈ X such that e ∈ E(T ),
xe = x, xs = y and x 6= y.
To define a required tuple (t̄(1), . . . , t̄(k)) over Wn(T,X) we describe
portraits of t̄(1), . . . , t̄(k) ∈ Wn(T,X). Let
t̄(i)w =
s, if χv(1) = i and |w| = 0
s, if χv(|w|+ 1) = i and w = w1y for some w1
e otherwise
for arbitrary w ∈ X [n−1] and 1 ≤ i ≤ k.
To avoid confusion in notations let us denote by ωk the word x . . . x
︸ ︷︷ ︸
k times
,
k ≥ 1, and let ω0 denotes the empty word Λ. We will show by induction
on i that
ωvi(t̄(1),...,t̄(k))
n = wiyωn−i
for some wi ∈ X(i−1), i ≤ n.
The case i = 1. Then v1(t̄(1), . . . , t̄(k)) = t(χv(1)). By the definition
of t(χv(1)) we obtain t(χv(1))ω0 = s and t(χv(1))ωl
= e for 1 ≤ l ≤ n− 1.
This means that
ωv1(t̄(1),...,t̄(k))
n = ωt(χv(1))
n = xsxe . . . xe = yωn−1.
Assume that for some i, 1 ≤ i < n, our claim is valid. Then
vi+1(t̄(1), . . . , t̄(k)) = vi(t̄(1), . . . , t̄(k)) · t(χv(i+ 1)).
By the definition of t(χv(i+ 1)) we obtain
t(χv(i+ 1))wiy = s and t(χv(i+ 1))wiyωl
= e for 1 ≤ l ≤ n− i.
Then we have
ω
vi+1(t̄(1),...,t̄(k))
n = (wiyωn−i)
t(χv(i+1)) =
(wiy)
t(χv(i+1))xsxe . . . xe = wi+1yωn−i−1,
where wi+1 = (wiy)
t(χv(i+1)) ∈ X(i+1), and the proof for i+ 1 is complete.
In particular, it implies, that the last letter of the word
ωv(t̄(1),...,t̄(k))
n
equals y.
Consider now a word u ∈ Fk of length ≤ n such that u 6= v. It is
sufficient to show that the last letter of the word
ωu(t̄(1),...,t̄(k))
n
A. Oliynyk 103
equals x. The definition of the elements t(1), . . . , t(k) implies that for
arbitrary i, j, 1 ≤ i ≤ n, 1 ≤ j ≤ k, and the word w ∈ X(n−i) such that
the last letter of w equals y (or w = Λ if i = n) equality
(wωi)
t(j) =
{
wt(j)yωi−1, if j = χv(n− i+ 1)
wt(j)ωi otherwise
holds. This means that to change the last letter of ωn by the product of
elements of the set {t(1), . . . t(k)} one has to take at least n multipliers. If
|u| < n, then the element u(t̄(1), . . . , t̄(k)) is a product of < n multipliers
and does not change the last letter of ωn. If |u| = n, but u 6= v then take
the smallest number j such that jth letters of u and v are different. Then
χu(j) 6= χv(j) and the last n− j + 1 letters of the word
ω
vj(t̄(1),...,t̄(k))
n
equal x. This implies that the rest n − j multipliers of u(t̄(1), . . . , t̄(k))
does not change the last letter of this word. The proof is complete.
Theorem 1. The infinite wreath power of a finite transformation semi-
group (T,X) contains a free subsemigroup of arbitrary finite rank if and
only if the semigroup (T,X) contains an idempotent which is not a left
zero element.
Proof. Let the semigroup (T,X) contains an idempotent which is not a
left zero element. Fix arbitrary words u, v of the free semigroup Fk of
rank k, u 6= v. Let n = max{|u|, |v|}. By Lemma 4 there exist k elements
t̄(1), . . . , t̄(k) ∈ Wn(T,X) such that
u(t̄(1), . . . , t̄(k)) 6= v(t̄(1), . . . , t̄(k)).
Hence for arbitrary increasing sequence n1, n2, . . . of positive integers the
cartesian product
∞∏
i=1
Wni(T,X)
contains a free subsemigroup of rank k. The required statement now
follows from part (2) of Proposition 1.
From the other hand, assume that all idempotents of the semigroup
(T,X) are left zero elements. By Lemma 2 it implies the equality T |T | =
E(T ). From the multiplication rule of elements of wreath powers written
in terms of their portraits it immediately follows that W∞(T,X)|T | ⊂
W∞(E(T ), X). Since idempotents of T are left zeros each element of the
last wreath power is an idempotent. This imply that each element of
W∞(T,X) generates finite subsemigroup. Therefore W∞(T,X) contains
no free subsemigroups.
104 Free semigroups in wreath powers
4. Most subsemigroups are free
Let X be a set, |X| > 1. The wreath power W∞(T (X), X) becomes a
complete metric semigroup via the metric
d(t̄, s̄) =
{
0, if t̄ = s̄
2−κ(t̄,s̄) otherwise,
where κ(t̄, s̄) is the least length of words w ∈ X∗ such that t̄w 6= s̄w,
t̄, s̄ ∈ W∞(T (X), X). It was established in [3] for finite X and for infinite
X the proof is the same.
For arbitrary transformation semigroup (T,X) the wreath power
W∞(T,X) is a closed subsemigroup in W∞(T (X), X) and hence is a met-
ric semigroup as well. For any k ≥ 1 the product topology on (W∞(T,X))k
can be defined by a metric dk such that
dk(t, s) = max{d(t̄i, s̄i) : 1 ≤ i ≤ k},
where t = (t̄1, . . . , t̄k), s = (s̄1, . . . , s̄k) ∈ (W∞(T,X))k. Therefore the
metric space (W∞(T,X))k is complete.
Define a subset Fk ⊂ (W∞(T,X))k as a set of elements (t̄1, . . . , t̄k) ∈
(W∞(T,X))k such that the subsemigroup 〈t̄1, . . . , t̄k〉 of W∞(T,X) is free
of rank k.
Theorem 2. Let (T,X) be a transformation semigroup, k ≥ 1. If the
wreath power W∞(T,X) contains a free subsemigroup of rank k then the
subset Fk is dense in (W∞(T,X))k.
Proof. Fix arbitrary t = (t̄1, . . . , t̄k) ∈ (W∞(T,X))k and positive integer
m. Let elements f̄1, . . . , f̄k ∈ W∞(T,X) generate a free semigroup of
rank k. It is sufficient to construct s = (s̄1, . . . , s̄k) ∈ Fk such that
dk(t, s) < 2−m.
We will use notations from the proof of Proposition 1. For arbitrary i,
1 ≤ i ≤ k, let an element s̄i ∈ W∞(T,X) satisfy equalities
pm(s̄i) = pm(t̄i)
and
sk(s̄i)(a1, . . . , am) = f̄i, a1, . . . , ak ∈ X.
It follows from the first equality that κ(t̄, s̄) > m and dk(t, s) < 2−m. For
arbitrary word v ∈ Fk denote the value v(s̄1, . . . , s̄k) by v(s). Then the
other equalities imply
sk(v(s))(a1, . . . , am) = v(f̄1, . . . , f̄k), a1, . . . , ak ∈ X.
This means that the semigroup generated by s̄1, . . . , s̄k is free of rank k
and s ∈ Fk.
A. Oliynyk 105
Recall that a subset of a topological space is called nowhere dense if
its closure has empty interior. A countable union of nowhere dense subsets
is called meagre. The complement of a meagre set is called co-meagre.
Theorem 3. Let (T,X) be a transformation semigroup. Then exactly one
of the following holds.
1. The infinite wreath power W∞(T,X) contains no free non-commu-
tative subsemigroups.
2. For each k ≥ 1 the subset Fk is co-meagre and not meagre in
(W∞(T,X))k.
Proof. Assume that W∞(T,X) contains a free non-commutative subsemi-
group. Then for each k ≥ 1 it contains a free subsemigroup of rank k and
the subset Fk is dense in (W∞(T,X))k by Theorem 2. Baire’s Category
Theorem implies that it is sufficient to prove that Fk is co-meagre.
For arbitrary words u, v ∈ Fk denote by Fk(u, v) the subset of elements
(t̄1, . . . , t̄k) ∈ (W∞(T,X))k such that u(t̄1, . . . , t̄k) = v(t̄1, . . . , t̄k). Since
multiplication is continuous in W∞(T,X) the subset Fk(u, v) is closed in
(W∞(T,X))k. It is easy to see that
(W∞(T,X))k \ Fk =
⋃
u,v∈Fk
u 6=v
Fk(u, v).
For arbitrary u, v ∈ Fk, u 6= v, the complement of the set Fk(u, v) in
(W∞(T,X))k contains Fk. This means that this complement is open and
dense. Hence the set Fk(u, v) is nowhere dense. Therefore (W∞(T,X))k \
Fk is meagre. The proof is complete.
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106 Free semigroups in wreath powers
Contact information
A. Oliynyk Department of Mechanics and Mathematics
Kyiv Taras Shevchenko University
Volodymyrska, 60
Kyiv 01033
E-Mail: olijnyk@univ.kiev.ua
URL: http://algebra.kiev.ua/oliynyk/
Received by the editors: 14.09.2010
and in final form 30.11.2010.
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