The structure of automorphism groups of semigroup inflations
In this paper we prove that the automorphism group of a semigroup being an inflation of its proper subsemigroup decomposes into a semidirect product of two groups one of which is a direct sum of full symmetric groups.
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irk-123456789-1573472019-06-21T01:29:43Z The structure of automorphism groups of semigroup inflations Kudryavtseva, G. In this paper we prove that the automorphism group of a semigroup being an inflation of its proper subsemigroup decomposes into a semidirect product of two groups one of which is a direct sum of full symmetric groups. 2007 Article The structure of automorphism groups of semigroup inflations / G. Kudryavtseva // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 1. — С. 61–66. — Бібліогр.: 7 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20M10. http://dspace.nbuv.gov.ua/handle/123456789/157347 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we prove that the automorphism
group of a semigroup being an inflation of its proper subsemigroup
decomposes into a semidirect product of two groups one of which
is a direct sum of full symmetric groups. |
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Article |
| author |
Kudryavtseva, G. |
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Kudryavtseva, G. The structure of automorphism groups of semigroup inflations Algebra and Discrete Mathematics |
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Kudryavtseva, G. |
| author_sort |
Kudryavtseva, G. |
| title |
The structure of automorphism groups of semigroup inflations |
| title_short |
The structure of automorphism groups of semigroup inflations |
| title_full |
The structure of automorphism groups of semigroup inflations |
| title_fullStr |
The structure of automorphism groups of semigroup inflations |
| title_full_unstemmed |
The structure of automorphism groups of semigroup inflations |
| title_sort |
structure of automorphism groups of semigroup inflations |
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Інститут прикладної математики і механіки НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/157347 |
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The structure of automorphism groups of semigroup inflations / G. Kudryavtseva // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 1. — С. 61–66. — Бібліогр.: 7 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT kudryavtsevag thestructureofautomorphismgroupsofsemigroupinflations AT kudryavtsevag structureofautomorphismgroupsofsemigroupinflations |
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2025-07-14T09:47:31Z |
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2025-07-14T09:47:31Z |
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1837615233852506112 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2007). pp. 61 – 66
c© Journal “Algebra and Discrete Mathematics”
The structure of automorphism groups of
semigroup inflations
Ganna Kudryavtseva
Communicated by B. V. Novikov
Abstract. In this paper we prove that the automorphism
group of a semigroup being an inflation of its proper subsemigroup
decomposes into a semidirect product of two groups one of which
is a direct sum of full symmetric groups.
1. Introduction
In the study of a specific semigroup the description of all its automor-
phisms is one of the most important questions. The automorphism groups
of many important specific semigroups are described (see, for example, [7]
and references therein). It can be observed that for two types of semi-
groups of rather diverse nature the automorphism groups have similarities
in their structure: each of them decomposes into a semidirect product of
two groups one of which being a direct sum of full symmetric groups.
The mentioned two types of semigroups are variants of some semigroups
of mappings (see [4], [6]) and maximal nilpotent subsemigroups of some
transformation semigroups (see [2], [5]).
In the present paper we establish a general result, which, in particu-
lar, unifies all the above-mentioned examples. We deal with an abstract
semigroup being an inflation of its proper subsemigroup, constructed as
follows. Let S be a semigroup. First introduce an equivalence relation
h on S via (a, b) ∈ h if and only if ax = bx and xa = xb for all x ∈ S,
that is a and b are not distinguished by multiplication from either side.
2000 Mathematics Subject Classification: 20M10.
Key words and phrases: semigroup inflation, semidirect product, automor-
phism group.
62 The structure of automorphism groups...
We would like to remark that the history of the study of h stems back at
least to [6], where it was considered for the variants of certain semigroups
of mappings. Set further
ψ = (h ∩ ((S \ S2) × (S \ S2))) ∪ {(a, a) : a ∈ S2}.
Denote by T any transversal of ψ. Then T is a subsemigroup of S, and
S is an inflation of T . An automorphism τ of T will be called extendable
provided that τ coincides with the restriction to T of a certain auto-
morphism of S. Clearly, all extendable automorphisms of T constitute a
subgroup, H, of the group AutT of all automorphisms of T .
We state our result as the following theorem.
Theorem 1. The group AutS is isomorphic to a semidirect product of
two groups one of which (the one which is normal) is the direct sum of
the full symmetric groups on the ψ-classes and the other one is the group
H consisting of all extendable automorphisms of T .
All the papers [2], [4], [5], [6] deal precisely with the "semigroup infla-
tion construction" introduced in this paper (though this construction is
not mentioned explicitly in any of these papers), and their main results
concerning automorphism groups can be deduced from our present gen-
eral result — Theorem 1. Recently Theorem 1 was applied to describe
automorphism groups of certain partition semigroups (see [3], Section 11
for details).
2. Construction
A semigroup S is called an inflation of its subsemigroup (see [1], Section
3.2) T provided that there is an onto map θ : S → T such that:
• θ2 = θ;
• aθbθ = ab for all a, b ∈ S.
In the described situation S is often referred to as an inflation of T with
an associated map θ (or just with a map θ).
It is immediate that if S is an inflation of T then T is a retract
of S (that is the image under an idempotent homomorphism) and that
S2 ⊂ T .
Lemma 2. Suppose that S is an inflation of T with the map θ. Then
kerθ ⊂ h.
G. Kudryavtseva 63
Proof. Let (a, b) ∈ kerθ and s ∈ S. Then
as = aθsθ = bθsθ = bs; sa = sθaθ = sθbθ = sb,
which implies that (a, b) ∈ h.
Lemma 3. The equivalence ψ, defined in the Introduction, is a congru-
ence on S.
Proof. Obviously, ψ is an equivalence relation. Prove that ψ is left and
right compatible. Let (a, b) ∈ ψ and a 6= b. Then (a, b) ∈ (h ∩ ((S \
S2) × (S \ S2))). Let c ∈ S. Since (a, b) ∈ h we have that ac = bc and
ca = cb for each c ∈ S. It follows that (ac, bc) ∈ ψ and (ca, cb) ∈ ψ as ψ
is reflexive.
Fix an arbitrary transversal of ψ and denote it by T .
Lemma 4. T is a subsemigroup of S, and S is an inflation of T .
Proof. T is a subsemigroup of S as T ⊃ S2. Let θ be the map S → T
which sends any element x from S to the unique element of the ψ-class
of x, belonging to T . The construction implies that S is an inflation of
T with the map θ.
Let S = ∪a∈TXa be a decomposition of S into the union of ψ-classes,
where Xa denotes the ψ-class of a. Set Ga to be the full symmetric group
acting on Xa and G = ⊕a∈TGa.
Lemma 5. π is an automorphism of S, for every π ∈ G.
Proof. It is enough to show that (xy)π = xπyπ whenever x, y ∈ S. Sup-
pose first that x, y ∈ S \ S2. Since xy ∈ S2 it follows that π stabilizes
xy, so that (xy)π = xy. Now, the inclusions (x, xπ) ∈ h and (y, yπ) ∈ h
imply xπyπ = xπy = xy. This yields xyπ = xπyπ, and the proof is
complete.
The following proposition gives a characterization of extendable au-
tomorphisms of T .
Proposition 6. An automorphism τ of T is extendable if and only if the
following condition holds:
(∀a, b ∈ T ) aτ = b ⇒ |Xa| = |Xb|. (1)
64 The structure of automorphism groups...
Proof. Suppose τ ∈ AutT is extendable and a ∈ T . In the case when
a ∈ S \ S2 we have
Xa = {b ∈ S | (a, b) ∈ h and b ∈ S \ S2}.
Clearly, (a, b) ∈ h ⇐⇒ (aτ, bτ) ∈ h and b ∈ S \ S2 ⇐⇒ bτ ∈ S \ S2 for
all a, b ∈ S. It follows that
Xaτ = {bτ | b ∈ Xa},
which implies (1). The inclusion a ∈ S2 is equivalent to aτ ∈ S2. But
then |Xa| = |Xaτ | = 1, which also implies (1).
Suppose now that (1) holds for certain τ ∈ AutT . Then one can
extend τ to τ ∈ AutS as follows.
Fix a collection of sets Ia, a ∈ T , and bijections fa : Ia → Xa, a ∈ T ,
satisfying the following conditions:
• |Ia| = |Xa|;
• Ia = Ib whenever |Xa| = |Xb|;
• Ia ∩ Ib = ∅ whenever |Xa| 6= |Xb|;
• if a, b ∈ T and |Xa| = |Xb| then af−1
a = bf−1
b
.
It is straightforward that such collections Ia, a ∈ T , and fa, a ∈ T , exist.
Consider x ∈ S \ T . Since T is a transversal of ψ, there is a ∈ T
such that x ∈ Xa. By the hypothesis we have |Xa| = |Xaτ |. Set τ on
Xa to be the map from Xa to Xaτ defined via x 7→ xf−1
a faτ . In this way
we define a permutation τ of S such that τ |T = τ . It will be called an
extension of τ to S. To complete the proof, we are left to show that τ is
a homomorphism. Let x, y ∈ S, x ∈ Xa, y ∈ Xb. Then
(xy)τ = (ab)τ = aτbτ = xτyτ ,
as required.
Let τ ∈ H. Of course, τ , constructed in the proof of Proposition 6,
depends not only on τ , but also on the sets Ia and the maps fa, so that
τ may have several extentions to S. Fix any extention τ of τ .
Lemma 7. τ 7→ τ is an embedding of H into AutS.
Proof. Proof is immediate by the construction of τ .
Denote by H the image of H under the embedding of H into AutS
from Lemma 7.
G. Kudryavtseva 65
3. Proof of Theorem 1
Proposition 8. H acts on G by automorphisms via πτ = τ−1πτ , τ ∈
H,π ∈ G.
Proof. Let π ∈ G and τ ∈ H. Show first that πτ ∈ G. Take any
x ∈ S. Let Xa be the block which contains x. We consequently have
xτ−1 ∈ Xaτ−1 , xτ−1π ∈ Xaτ−1 and xτ−1πτ ∈ Xaτ−1τ = Xa. Hence
xπτ ∈ Xa. It follows that πτ ∈ G.
That π 7→ πτ is one-to-one, onto and homomorphic immediately fol-
lows from its definition. We are left to show that the map sending τ ∈ H
to π 7→ πτ ∈ AutG is homomorphic. The latter follows from the equalities
πτ1τ2 = (τ1τ2)
−1π(τ1τ2) = (πτ1)τ2 . The proof is complete.
In the following two lemmas we show that G and H intersect by the
identity automorphism and generate AutS.
Lemma 9. G ∩H = {id}, where id is the identity automorphism of S.
Proof. The proof follows from the observation that the decomposition
S = ∪a∈TXa is fixed by each element of G, while only by the identity
element of H.
Lemma 10. AutS = H ·G.
Proof. Let ϕ ∈ AutS. The definition of ψ implies that ϕ maps each ψ-
class onto some other ψ-class. Define a bijection τ : T → T via aτ = b
provided that Xaϕ = Xb. Show that τ is an extendable automorphism of
T . The definition of τ assures that (1) holds, and thereby τ is extendable
in view of Proposition 6. Let τ ∈ AutS be an extension of τ . The
construction implies that ϕ(τ)−1 ∈ G.
Now the proof of Theorem 1 follows from Proposition 8 and Lemmas 9
and 10.
4. Acknowledgements
The author expresses her gratitude to Prof. B.V. Novikov for the careful
reading of the initial version of this paper.
References
[1] A.H. Clifford, G.B. Preston, The algebraic theory of semigroups, Amer. Math.
Soc. Surveys 7 (Providence, R.I.) 1961, 1967.
[2] O. Ganyushkin, S. Temnikov and H. Shafranova, Automorphism groups of
maximal nilpotent subsemigroups of the semigroup IS(M), Math. Stud., 13
(2000), 1, 11-22.
66 The structure of automorphism groups...
[3] G. Kudryavtseva and V. Maltcev, On the structure of two generalizations of
the full inverse symmetric semigroup, math.GR/0602623.
[4] G. Kudryavtseva and G. Tsyaputa, The automorphism group of the sandwich
inverse symmetric semigroup, Visnyk of Kyiv University, 13-14 (2005), 101-
105 (in Ukrainian), English translation: math.GR/0509678.
[5] A. Stronska, On the automorphisms for the nilpotent subsemigroups of the
order-decreasing transformation semigroup, Proceedings of the F. Skorina
Gomel University, 2 (2007), 38-49.
[6] J.S.V. Symons, On a generalization of the transformation semigroup, J. Aus-
tral Math. Soc., 19 (1975), 47-61.
[7] F. Szechtman, On the automorphism group of the centralizer of an idempotent
in the full transformation monoid, Semigroup Forum, 70 (2005), 238-242.
Contact information
G. Kudryavtseva Department of Mechanics and Mathemat-
ics, Kyiv Taras Shevchenko University,
Volodymyrs’ka, 60, Kyiv, 01033, Ukraine
E-Mail: akudr@univ.kiev.ua
Received by the editors: 19.11.2005
and in final form 28.05.2007.
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