On closures in semitopological inverse semigroups with continuous inversion
We study the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse semigroups with continuous inversion. In particularly we show that a topological group G is H-closed in the class of semitopological inverse semigroups with continuous inversion if...
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Інститут прикладної математики і механіки НАН України
2014
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| Цитувати: | On closures in semitopological inverse semigroups with continuous inversion / O. Gutik // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 59–85. — Бібліогр.: 33 назв. — англ. |
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irk-123456789-1533472019-06-15T01:30:35Z On closures in semitopological inverse semigroups with continuous inversion Gutik, O. We study the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse semigroups with continuous inversion. In particularly we show that a topological group G is H-closed in the class of semitopological inverse semigroups with continuous inversion if and only if G is compact, a Hausdorff linearly ordered topological semilattice E is H-closed in the class of semitopological semilattices if and only if E is H-closed in the class of topological semilattices, and a topological Brandt λ⁰-extension of S is (absolutely) H-closed in the class of semitopological inverse semigroups with continuous inversion if and only if so is S. Also, we construct an example of an H-closed non-absolutely H-closed semitopological semilattice in the class of semitopological semilattices. 2014 Article On closures in semitopological inverse semigroups with continuous inversion / O. Gutik // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 59–85. — Бібліогр.: 33 назв. — англ. 1726-3255 2010 MSC:22A05, 22A15, 22A26; 20M18, 20M15. http://dspace.nbuv.gov.ua/handle/123456789/153347 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We study the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse semigroups with continuous inversion. In particularly we show that a topological group G is H-closed in the class of semitopological inverse semigroups with continuous inversion if and only if G is compact, a Hausdorff linearly ordered topological semilattice E is H-closed in the class of semitopological semilattices if and only if E is H-closed in the class of topological semilattices, and a topological Brandt λ⁰-extension of S is (absolutely) H-closed in the class of semitopological inverse semigroups with continuous inversion if and only if so is S. Also, we construct an example of an H-closed non-absolutely H-closed semitopological semilattice in the class of semitopological semilattices. |
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Gutik, O. |
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Gutik, O. On closures in semitopological inverse semigroups with continuous inversion Algebra and Discrete Mathematics |
| author_facet |
Gutik, O. |
| author_sort |
Gutik, O. |
| title |
On closures in semitopological inverse semigroups with continuous inversion |
| title_short |
On closures in semitopological inverse semigroups with continuous inversion |
| title_full |
On closures in semitopological inverse semigroups with continuous inversion |
| title_fullStr |
On closures in semitopological inverse semigroups with continuous inversion |
| title_full_unstemmed |
On closures in semitopological inverse semigroups with continuous inversion |
| title_sort |
on closures in semitopological inverse semigroups with continuous inversion |
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Інститут прикладної математики і механіки НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/153347 |
| citation_txt |
On closures in semitopological inverse semigroups with continuous inversion / O. Gutik // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 59–85. — Бібліогр.: 33 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT gutiko onclosuresinsemitopologicalinversesemigroupswithcontinuousinversion |
| first_indexed |
2025-07-14T04:33:58Z |
| last_indexed |
2025-07-14T04:33:58Z |
| _version_ |
1837595509178499072 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 18 (2014). Number 1, pp. 59 – 85
© Journal “Algebra and Discrete Mathematics”
On closures in semitopological inverse
semigroups with continuous inversion
Oleg Gutik
Communicated by M. Ya. Komarnytskyj
Abstract. We study the closures of subgroups, semilattices
and different kinds of semigroup extensions in semitopological inverse
semigroups with continuous inversion. In particularly we show that
a topological group G is H-closed in the class of semitopological
inverse semigroups with continuous inversion if and only if G is
compact, a Hausdorff linearly ordered topological semilattice E is
H-closed in the class of semitopological semilattices if and only if E
is H-closed in the class of topological semilattices, and a topological
Brandt λ0-extension of S is (absolutely) H-closed in the class of
semitopological inverse semigroups with continuous inversion if
and only if so is S. Also, we construct an example of an H-closed
non-absolutely H-closed semitopological semilattice in the class of
semitopological semilattices.
1. Introduction and preliminaries
We shall follow the terminology of [2, 8, 12,27,30].
A subset A of an infinite set X is called cofinite in X if X \A is finite.
Given a semigroup S, we shall denote the set of idempotents of S
by E(S). A semilattice is a commutative semigroup of idempotents. For
2010 MSC: 22A05, 22A15, 22A26; 20M18, 20M15.
Key words and phrases: semigroup, semitopological semigroup, topological
Brandt λ
0-extension, inverse semigroup, quasitopological group, topological group,
semilattice, closure, H-closed, absolutely H-closed.
60 On closures in semitopological inverse semigroups
a semilattice E the semilattice operation on E determines the partial
order 6 on E:
e 6 f if and only if ef = fe = e.
This order is called natural. An element e of a partially ordered set X
is called minimal if f 6 e implies f = e for f ∈ X. An idempotent
e of a semigroup S without zero (with zero 0S) is called primitive if e
is a minimal element in E(S) (in (E(S)) \ {0S}). A maximal chain of
a semilattice E is a chain which is properly contained in no other chain
of E. The Axiom of Choice implies the existence of maximal chains in
any partially ordered set.
A semigroup S with the adjoined unit [zero] will be denoted by S1
[S0] (cf. [8]). Next, we shall denote the unit (identity) and the zero of
a semigroup S by 1S and 0S , respectively. Given a subset A of a semigroup
S, we shall denote by A∗ = A \ {0S} and |A| = the cardinality of A. A
semigroup S is called inverse if for any x ∈ S there exists a unique y ∈ S
such that xyx = x and yxy = y. Such an element y is called inverse of x
and it is denoted by x−1.
If h : S → T is a homomorphism (or a map) from a semigroup S
into a semigroup T and if s ∈ S, then we denote the image of s under h
by (s)h. A semigroup homomorphism h : S → T is called annihilating if
(s)h = (t)h for all s, t ∈ S.
Let S be a semigroup with zero and λ a cardinal > 1. We define the
semigroup operation on the set Bλ(S) = (λ× S × λ) ∪ {0} as follows:
(α, a, β) · (γ, b, δ) =
{
(α, ab, δ), if β = γ;
0, if β 6= γ,
and (α, a, β)·0 = 0·(α, a, β) = 0·0 = 0, for all α, β, γ, δ ∈ λ and a, b ∈ S. If
S = S1 then the semigroup Bλ(S) is called the Brandt λ-extension of the
semigroup S [13]. Obviously, if S has zero then J = {0} ∪ {(α, 0S , β) | 0S
is the zero of S} is an ideal of Bλ(S). We put B0
λ(S) = Bλ(S)/J and the
semigroup B0
λ(S) is called the Brandt λ0-extension of the semigroup S
with zero [19].
Next, if A ⊆ S then we shall denote Aαβ = {(α, s, β) | s ∈ A} if A
does not contain zero, and Aα,β = {(α, s, β) | s ∈ A \ {0}} ∪ {0} if 0 ∈ A,
for α, β ∈ λ.
We shall denote the semigroup of λ×λ-matrix units by Bλ and the
subsemigroup of λ×λ-matrix units of the Brandt λ0-extension of a monoid
S with zero by B0
λ(1). We always consider the Brandt λ0-extension only
O. Gutik 61
of a monoid with zero. Obviously, for any monoid S with zero we have
B0
1(S) = S. Note that every Brandt λ-extension of a group G is isomor-
phic to the Brandt λ0-extension of the group G0 with adjoined zero. The
Brandt λ0-extension of the group with adjoined zero is called a Brandt
semigroup [8, 27]. A semigroup S is a Brandt semigroup if and only if
S is a completely 0-simple inverse semigroup [7, 25] (cf. also [27, Theo-
rem II.3.5]). We also observe that the semigroup Bλ of λ× λ-matrix units
is isomorphic to the Brandt λ0-extension of the two-element monoid with
zero S = {1S , 0S} and the trivial semigroup S (i. e. S is a singleton set)
is isomorphic to the Brandt λ0-extension of S for every cardinal λ > 1.
Let {Sι : ι ∈ I} be a disjoint family of semigroups with zero such that
0ι is zero in Sι for any ι ∈ I. We put S = {0} ∪
⋃
{S∗
ι : ι ∈ I}, where
0 /∈
⋃
{S∗
ι : ι ∈ I}, and define a semigroup operation “ · ” on S in the
following way
s · t =
{
st, if st ∈ S∗
ι for some ι ∈ I;
0, otherwise.
The semigroup S with the operation “ · ” is called an orthogonal sum of
the semigroups {Sι : ι ∈ I} and in this case we shall write S =
∑
ι∈I Sι.
A non-trivial inverse semigroup is called a primitive inverse semi-
group if all its non-zero idempotents are primitive [27]. A semigroup S
is a primitive inverse semigroup if and only if S is an orthogonal sum of
Brandt semigroups [27, Theorem II.4.3].
In this paper all topological spaces are Hausdorff. If Y is a subspace
of a topological space X and A ⊆ Y , then by clY (A) we denote the
topological closure of A in Y .
A (semi)topological semigroup is a Hausdorff topological space with
a (separately) continuous semigroup operation. A topological semigroup
which is an inverse semigroup is called an inverse topological semigroup.
A topological inverse semigroup is an inverse topological semigroup with
continuous inversion. We observe that the inversion on a (semi)topological
inverse semigroup is a homeomorphism (see [10, Proposition II.1]). A
semitopological group is a Hausdorff topological space with a separately
continuous group operation. A semitopological group with continuous
inversion is a quasitopological group. A paratopological group is called
a group with a continuous group operation. A paratopological group with
continuous inversion is a topological group.
Let STSG0 be a class of semitopological semigroups. A semigroup
S ∈ STSG0 is called H-closed in STSG0, if S is a closed subsemigroup
62 On closures in semitopological inverse semigroups
of any topological semigroup T ∈ STSG0 which contains S both as
a subsemigroup and as a topological space. The H-closed topological
semigroups were introduced by Stepp in [31], and there they were called
maximal semigroups. A semitopological semigroup S ∈ STSG0 is called
absolutely H-closed in the class STSG0, if any continuous homomorphic
image of S into T ∈ STSG0 is H-closed in STSG0. An algebraic
semigroup S is called:
• algebraically complete in STSG0, if S with any Hausdorff topology
τ such that (S, τ) ∈ STSG0 is H-closed in STSG0;
• algebraically h-complete in STSG0, if S with discrete topology d
is absolutely H-closed in STSG0 and (S, d) ∈ STSG0.
Absolutely H-closed topological semigroups and algebraically h-complete
semigroups were introduced by Stepp in [32], and there they were called
absolutely maximal and algebraic maximal, respectively.
Recall [1], a topological group G is called absolutely closed if G is
a closed subgroup of any topological group which containsG as a subgroup.
In our terminology such topological groups are called H-closed in the
class of topological groups. In [28] Raikov proved that a topological
group G is absolutely closed if and only if it is Raikov complete, i.e. G is
complete with respect to the two-sided uniformity. A topological group
G is called h-complete if for every continuous homomorphism h : G → H
the subgroup f(G) of H is closed [9]. In our terminology such topological
groups are called absolutely H-closed in the class of topological groups.
The h-completeness is preserved under taking products and closed central
subgroups [9]. H-closed paratopological and topological groups in the
class of paratopological groups studied in [29].
In [32] Stepp studied H-closed topological semilattice in the class of
topological semigroups. There he proved that an algebraic semilattice E
is algebraically h-complete in the class of topological semilattices if and
only if every chain in E is finite. In [23] Gutik and Repovš established
the closure of a linearly ordered topological semilattice in a topological
semilattice. They proved the criterium ofH-closedness of a linearly ordered
topological semilattice in the class of topological semilattices and showed
that every H-closed topological semilattice is absolutely H-closed in the
class of topological semilattices. Also, such semilattices studied in [6, 14].
In [3] the structure of closures of the discrete semilattices (N,min) and
(N,max) is described. Here the authors constructed an example of an H-
closed topological semilattice in the class of topological semilattices which
O. Gutik 63
is not absolutely H-closed in the class of topological semilattices. The
constructed example gives a negative answer on Question 17 from [32].
Definition 1.1 ([19]). Let STSG0 be a class of semitopological semi-
groups. Let λ > 1 be a cardinal and (S, τ) ∈ STSG0. Let τB be a topology
on B0
λ(S) such that
a)
(
B0
λ(S), τB
)
∈ STSG0;
b) the topological subspace (Sα,α, τB|Sα,α
) is naturally homeomorphic
to (S, τ) for some α ∈ λ.
Then
(
B0
λ(S), τB
)
is called a topological Brandt λ0-extension of (S, τ) in
STSG0.
In the paper [24] Gutik and Repovš established homomorphisms of
the Brandt λ0-extensions of monoids with zeros. They also described
a category whose objects are ingredients in the constructions of the
Brandt λ0-extensions of monoids with zeros. Here they introduced fi-
nite, compact topological Brandt λ0-extensions of topological semigroups
and countably compact topological Brandt λ0-extensions of topological
inverse semigroups in the class of topological inverse semigroups, and
established the structure of such extensions and non-trivial continuous
homomorphisms between such topological Brandt λ0-extensions of topo-
logical monoids with zero. There they also described a category whose
objects are ingredients in the constructions of finite (compact, count-
ably compact) topological Brandt λ0-extensions of topological monoids
with zeros. These investigations were continued in [20–22], where estab-
lished countably compact topological Brandt λ0-extensions of topological
monoids with zeros and pseudocompact topological Brandt λ0-extensions
of semitopological monoids with zeros their corresponding categories. In
the papers [4,15,16,19,26] where studies H-closed and absolutely H-closed
topological Brandt λ0-extensions of topological semigroups in the class of
topological semigroups.
In Section 2 we study the closure of a quasitopological group in a semi-
topological inverse semigroup with continuous inversion. In particularly
we show that a topological group G is H-closed in the class of semitopo-
logical inverse semigroups with continuous inversion if and only if G is
compact.
Section 3 is devoted to the closure of a semitopological semilattice in
a semitopological inverse semigroup with continuous inversion. We show
that a Hausdorff linearly ordered topological semilattice E is H-closed
64 On closures in semitopological inverse semigroups
in the class of semitopological semilattices if and only if E is H-closed
in the class of topological semilattices. Also, we construct an example of
an H-closed semitopological semilattice in the class of semitopological
semilattices which is not absolutelyH-closed in the class of semitopological
semilattices.
In Section 4 we show that a topological Brandt λ0-extension of S is
(absolutely) H-closed in the class of semitopological inverse semigroups
with continuous inversion if and only if so is S. Also, we study the
preserving of (absolute) H-closedness in the class of semitopological
inverse semigroups with continuous inversion by orthogonal sums.
2. On the closure of a quasitopological group in a semi-
topological inverse semigroup with continuous inver-
sion
Proposition 2.1. Every left topological inverse semigroup with continu-
ous inversion is semitopological semigroup.
Proof. We write an arbitrary right translation ρa : S → S : x 7→ xa of a left
topological inverse semigroup S with continuous inversion inv : S → S
on three steps in the following way:
ρa(x) = xa =
(
a−1x−1
)
−1
= (inv ◦λa−1 ◦ inv) (x).
This implies the continuity of right translations i S.
It is well known that the closure of an inverse subsemigroup of a topo-
logical inverse semigroup is again a topological inverse semigroup (see:
[10, Proposition II.1]). The following proposition extends this result to
semitopological inverse semigroups with continuous inversion.
Proposition 2.2. The closure of an inverse subsemigroup T in a semi-
topological inverse semigroup S with continuous inversion is an inverse
semigroup.
Proof. By Proposition 1.8(ii) from [30, Chapter I, Proposition 1.8(ii)]
the closure clS(T ) of T in a semitopological semigroup S is a semitopo-
logical semigroup. Then the continuity of the inversion inv : S → S and
Theorem 1.4.1 from [11] imply that inv(clS(T )) ⊆ clS(inv(T )) = clS(T )
and hence we get that inv(clS(T )) = clS(T ). This implies that clS(T ) is
an inverse subsemigroup of S.
O. Gutik 65
We observe that the statement of Proposition 2.2 is not true in the
case of inverse topological semigroup. It is complete to consider the set
R+ = [0,+∞) of non-negative real numbers with usual topology and
usual multiplication of real numbers. This implies that in Proposition 2.2
the condition that S has continuous inversion is essential.
In a compact topological semigroup the closure of a subgroup is
a topological subgroup (see: [5, Vol. 1, Theorems 1.11 and 1.13]). Also,
since for a topological inverse semigroup S the map f : S → S : x → xx−1
is continuous, the maximal subgroup of S is closed, and hence the closure
of a subgroup of a topological inverse semigroup is a subgroup. The
previous observation implies that this is not true in the general case of
topological semigroups. Also, the following example shows that the closure
of a subgroup in a semitopological inverse semigroup with continuous
inversion is not a subgroup.
Example 2.3. Let Z be the discrete additive group of integers. We put
A(Z) is the one point Alexandroff compactification of the space Z with
the remainder ∞. We extend the semigroup operation from Z onto A(Z)
in the following way:
n+ ∞ = ∞ + n = ∞ + ∞ = ∞, for every n ∈ Z.
It is well known that A(Z) with such defined operation is a semitopological
inverse semigroup with continuous inversion and Z is not a closed subgroup
of A(Z) [30].
A quasitopological group G is called precompact if for every open
neighbourhood U of the neutral element of G there exists a finite subset
F of G such that UF = G [2].
The following proposition gives examples quasitopological groups
which are non-closed subgroups of some semitopological inverse semigroups
with continuous inversion.
Proposition 2.4. For every non-precompact regular quasitopological
group (G, τ) there exists a regular semitopological inverse semigroup with
continuous inversion which contains (G, τ) as a non-closed subgroup.
Proof. Since the quasitopological group (G, τ) is non-precompact there
exists an open neighbourhood U of the neutral element e of the group
G such that FU 6= G and UF 6= G for every finite subset F in G. Let
Be be a base of the topology τ at the neutral element e of (G, τ). Since
the inversion is continuous in (G, τ), without loss of generality we may
66 On closures in semitopological inverse semigroups
assume that all elements of the family Be are symmetric, i.e., V = V −1
for every V ∈ Be. We put
BU = {V ∈ Be : clG(V ) ⊆ U} .
Since the quasitopological group (G, τ) is not precompact we have that
FV 6= G and V F 6= G for every V ∈ BU and for every finite subset F
in G.
By G0 we denote the group G with a joined zero 0. Now, we put
P0 = {Wg,V = {0} ∪G \ clG(gV ) : V ∈ BU , g ∈ G}
∪ {WV,g = {0} ∪G \ clG(V g) : V ∈ BU , g ∈ G}
and τ ∪ P0 is a subbase of a topology τ0 on G0.
Since (G, τ) a quasitopological group, it is sufficient to show that the
semigroup operation on (G0, τ0) is separately continuous in the following
two cases: h · 0 = 0 and 0 · h = 0, for h ∈ G. Then for arbitrary subbase
neighbourhoods Wg1,V1
, . . . ,Wgn,Vn
and WV1,g1
, . . . ,WVn,gn
we have that
h · (Wg1,V1
∩ · · · ∩Wgn,Vn
) ⊆ Whg1,V1
∩ · · · ∩Whgn,Vn
and
(WV1,g1
∩ · · · ∩WVn,gn
) · h ⊆ WV1,g1h ∩ · · · ∩WVn,gnh.
Also, since translations in the quasitopological group (G, τ) are home-
omorphisms, for every open subbase neighbourhood V ∈ BU of the
neutral element of G and every g ∈ G we have that (Wg,V )−1 ⊆ WV −1,g−1 .
Therefore (G0, τ0) is a quasitopological inverse semigroup with continuous
inversion.
Now for every open subbase neighbourhoods V1, V2 ∈ BU of the
neutral element of G such that clG(V1) ⊆ V2 and every g ∈ G the following
conditions holds:
clG(Wg,V2
) ⊆ Wg,V1
and clG(WV2,g) ⊆ WV1,g.
Hence we get that the topological space (G0, τ0) is regular.
Theorem 2.5. A topological group G is H-closed in the class of semi-
topological inverse semigroups with continuous inversion if and only if G
is compact.
O. Gutik 67
Proof. The implication (⇐) is trivial.
(⇒) Let a topological group G be H-closed in the class of semitopo-
logical inverse semigroups with continuous inversion. Suppose to the
contrary: the space G is not compact. Then G is H-closed in the class of
topological groups and hence it is Rǎıkov complete. If G is precompact
then by Theorem 3.7.15 of [2], G is compact. Hence the topological group
G is not precompact. This contradicts Proposition 2.4. The obtained
contradiction implies the statement of our theorem.
Theorem 2.5 implies the following two corollaries:
Corollary 2.6. A topological group G is absolutely H-closed in the class
of semitopological inverse semigroups with continuous inversion if and
only if G is compact.
Corollary 2.7. A topological group G is H-closed in the class of semi-
topological semigroups if and only if G is compact.
The following example shows that there exists a non-compact qua-
sitopological group with adjoined zero which H-closed in the class of
semitopological inverse semigroups with continuous inversion.
Example 2.8. Let R be the additive group of real numbers with usual
topology. We put G is the direct quare of R with the product topology.
It is well known that G is a topological group. Let G0 be the group G
with the adjoined zero 0. We define the topology τ on G0 in the following
way. For every non-zero element x of G0 the base of the topology τ
at x coincides with base of the product topology at x in G. For every
(x0, y0) ∈ R2 and every ε > 0 we denote by
Oε(x0, y0) =
{
(x, y) ∈ R2 :
√
(x− x0)2 + (y − y0)2 6 ε
}
the usual closed ε-ball with the center at the point (x0, y0). We denote
A(x0, y0) =
{
(x0, y) ∈ R2 : y ∈ R
}
∪
{
(x, y0) ∈ R2 : x ∈ R
}
and
Uε(x0, y0) = G0 \ (Oε(x0, y0) ∪A(x0, y0)) .
Now we put P(0) =
{
Uε(x, y) : (x, y) ∈ R2, ε > 0
}
and P(0) ∪ BG is
a subbase of the topology τ on G0, where BG is a base of the topology
of the topological group G. Simple verifications show that (G0, τ) is
68 On closures in semitopological inverse semigroups
a Hausdorff semitopological inverse semigroup with continuous inversion
and (G0, τ) is not a regular space.
Then for any finitely many points (x1, y1), . . . , (xn, yn) ∈ R2 and
finitely many ε1, . . . , εn > 0 the following conditions hold:
(a) Oε1
(x1, y1) ∪ · · · ∪ Oεn(xn, yn) is a compact subset of the space
(G0, τ);
(b) clG0(Uε1
(x1, y1)∩· · ·∩Uεn(xn, yn))∪Oε1
(x1, y1)∪· · ·∪Oεn(xn, yn)=
G0.
This implies that (G0, τ) is an H-closed topological space and hence
the semigroup (G0, τ) is H-closed in the class of semitopological inverse
semigroups with continuous inversion.
3. On the closure of a semilattice in a semitopological
inverse semigroup with continuous inversion
It is well known that the subset of idempotent E(S) of a topological
semigroup S is a closed subset of S (see: [5, Vol. 1, Theorem 1.5]). We ob-
serve that for semitopological semigroups this statement does not hold [30].
Amassing, but the subset of all idempotent E(S) of a semitopological
inverse semigroup S with continuous inversion is a closed subset of S.
Proposition 3.1. The subset of idempotents E(S) of a semitopological
inverse semigroup S with continuous inversion is a closed subset of S.
Proof. First we observe that for any topological space X and any con-
tinuous map f : X → X the set Fix(f) of fixed point of f is closed
subset of X (see: [5, Vol. 1, Theorem 1.4] or [11, Theorem 1.5.4]). Since
e−1 = e for every idempotent e ∈ S, the continuity of inversion implies
that E(S) ⊆ Fix(inv). Let be x ∈ S such that x ∈ Fix(inv). Since S
is an inverse semigroup we obtain that xx = xx−1 ∈ E(S) and hence
Fix(inv) ⊆ E(S). This completes the proof of the proposition.
Proposition 3.1 implies the following
Corollary 3.2. The closure of a subsemilattice in a semitopological
inverse semigroup S with continuous inversion is a subsemilattice of S.
Since the closure of a subsemilattice in a Hausdorff topological semi-
group is again a topological semilattice, an (absolutely) H-closed topo-
logical semilattice in the class of topological semilattices is (absolutely)
O. Gutik 69
H-closed in the class of topological semigroups [16]. In [32] Stepp proved
that an algebraic semilattice E is algebraically h-complete in the class
of topological semilattices if and only if every chain in E is finite. The
following example shows that for every infinite cardinal λ there exists
an algebraically h-complete semilattice E(λ) in the class of topological
semilattices of cardinality λ such that E(λ) with the discrete topology is
not H-closed in the class of semitopological semigroups.
Example 3.3. Let λ be any infinite cardinal. We fix an arbitrary a0 ∈ λ
and define the semigroup operation on λ by the formula:
xy =
{
x, if x = y;
a0, if x 6= y.
The cardinal λ with so defined semigroup operation we denote by E(λ).
It is obvious that E(λ) is a semilattice such that a0 is zero of E(λ) and
any two distinct non-zero elements of E(λ) are incomparable with respect
to the natural partial order on E(λ). Let be a /∈ E(λ). We extend the
semigroup operation from E(λ) onto S = E(λ) ∪ {a} in the following
way:
aa = ax = xa = a0, for any x ∈ E(λ).
It is obvious that S with so defined operation is not a semilattice.
We define a topology τ on S in the following way. Fix an arbitrary
sequence of distinct points {xn : n ∈ N} from E(λ) and put Un(a) =
{a} ∪ {xi : i > n}. Put all elements of the set E(λ) are isolated points of
the space (S, τ) and the family B(a) = {Un(a) : n ∈ N} is a base of the
topology τ at the point a ∈ S. Simple verifications show that (S, τ) is
a metrizable 0-dimensional semitopological semigroup and E(λ) is a dense
subsemilattice of (S, τ). Also, we observe that by Theorem 9 from [32]
the semilattice E(λ) is algebraically h-complete in the class of topological
semilattices.
Remark 3.4. We observe that for every infinite cardinal λ and every
Hausdorff topology τ on E(λ) such that (E(λ), τ) is a semitopological
semilattice we have that all non-zero idempotents of (E(λ), τ) are isolated
points and moreover (E(λ), τ) is a topological semilattice. Also, a simple
modification of the proof in the Example 3.3 shows that a semitopological
semilattice (E(λ), τ) is H-closed in the class of semitopological semigroups
if and only if the space (E(λ), τ) is compact.
Suppose that E is a Hausdorff semitopological semilattice. If L is
a maximal chain in E, then by Proposition IV-1.13 of [12] we have that
70 On closures in semitopological inverse semigroups
L =
⋂
e∈L(↑e ∪ ↓e) is a closed subset of E and hence we proved the
following proposition:
Proposition 3.5. The closure of a linearly ordered subsemilattice of
a Hausdorff semitopological semilattice E is a linearly ordered subsemilat-
tice of E.
It is well known that the natural partial order on a Hausdorff semi-
topological semilattice is semiclosed (see [12, Proposition IV-1.13]). Also,
by Lemma 3 of [33] a semiclosed linear order is closed, and hence every
linearly ordered set with a closed order admits the structure of a Hausdorff
topological semilattice. This implies the following proposition:
Proposition 3.6. Every linearly ordered Hausdorff semitopological semi-
lattice is a topological semilattice.
Propositions 3.5 and 3.6 imply
Theorem 3.7. A Hausdorff linearly ordered topological semilattice E is
H-closed in the class of semitopological semilattices if and only if E is
H-closed in the class of topological semilattices.
Theorem 3.7 and results obtained in the paper [23] imply Corollar-
ies 3.8—3.12.
A linearly ordered semilattice E is called complete if every non-empty
subset of S has inf and sup.
Corollary 3.8. A linearly ordered semitopological semilattice E is H-
closed in the class of semitopological semilattices if and only if the following
conditions hold:
(i) E is complete;
(ii) x = supA for A = ↓A \ {x} implies x ∈ clE A, whenever A 6= ∅;
and
(iii) x = inf B for B = ↑B \ {x} implies x ∈ clE B, whenever B 6= ∅
Corollary 3.9. Every linearly ordered H-closed semitopological semilat-
tice in the class of semitopological semilattices is absolutely H-closed in
the class of semitopological semilattices.
Corollary 3.10. Every linearly ordered H-closed semitopological semi-
lattice in the class of semitopological semilattices contains maximal and
minimal idempotents.
O. Gutik 71
Corollary 3.11. Let E be a linearly ordered H-closed semitopological
semilattice in the class of semitopological semilattices and e ∈ E. Then
↑e and ↓e are (absolutely) H-closed topological semilattices in the class
of semitopological semilattices.
Corollary 3.12. Every linearly ordered semitopological semilattice is
a dense subsemilattice of an H-closed semitopological semilattice in the
class of semitopological semilattices.
Remark 3.13. Theorem 3.7, Example 7 and Proposition 8 from [23] imply
that there exists a countable linearly ordered σ-compact 0-dimensional
scattered locally compact metrizable topological semilattice which does
not embeds into any compact Hausdorff semitopological semilattice.
At the finish of this section we construct an H-closed semitopologi-
cal semilattice in the class of semitopological semilattices which is not
absolutely H-closed in the class of semitopological semilattices.
A filter F on a set X is called free if
⋂
F = ∅.
Example 3.14 ([3]). Let N denote the set of positive integers. For each
free filter F on N consider the topological space NF = N ∪ {F} in which
all points x ∈ N are isolated while the sets F ∪ {F}, F ∈ F, form
a neighbourhood base at the unique non-isolated point F.
The semilattice operation min of N extends to a continuous semilat-
tice operation min on NF such that min{n,F} = min{F, n} = n and
min{F,F} = F for all n ∈ N. By NF,min we shall denote the topological
space NF with the semilattice operation min. Simple verifications show
that NF,min is a topological semilattice. Then by Theorem 2(i) of [3]
the topological semilattice NF,min is H-closed in the class of topological
semilattices and hence by Theorem 3.7 it is H-closed in the class of
semitopological semilattices.
Later by E2 = {0, 1} we denote the discrete topological semilattice
with the semilattice operation min.
Theorem 3.15. Let F be a free filter on N and F ∈ F be a set
with infinite complement N \ F . Then the closed subsemilattice E =
(
NF,min × {0}
)
∪ ((N \ F ) × {1}) of the direct product NF,min ×E2 is H-
closed not absolutely H-closed in the class of semitopological semilattices.
72 On closures in semitopological inverse semigroups
Proof. The definition of the topological semilattice NF,min ×E2 implies
that E is a closed subsemilattice of NF,min × E2.
Suppose the contrary: the topological semilattice E is not H-closed
in the class of semitopological semilattices. Since the closure of a sub-
semilattice in a semitopological semilattice is a semilattice (see [30, Chap-
ter I, Proposition 1.8(ii)]) we conclude that there exists a semitopological
semilattice S which contains E as a dense subsemilattice and S \ E 6= ∅.
We fix an arbitrary a ∈ S \E. Then for every open neighbourhood U(a) of
the point a in S we have that the set U(a)∩E is infinite. By Theorem 2(i)
of [3] and Theorem 3.7, the subspace NF,min × {0} of E with the induced
semilattice operation from E is an H-closed in the class of semitopological
semilattices. Therefore there exists an open neighbourhood U(a) of the
point a in S such that U(a) ∩ E ⊆ (N \ F ) × {1} and hence the set
U(a) ∩ ((N \ F ) × {1}) is infinite.
Since the subset NF,min × {0} is an ideal of E, the H-closedness of
NF,min × {0} in the class of semitopological semilattices implies that
NF,min ×{0} is a closed ideal in S and hence we have that x ·a ∈ NF,min ×
{0} for every x ∈ NF,min × {0}. Since for every open neighbourhood U(a)
of the point a in S the set U(a) ∩ ((N\F ) × {1}) is infinite the semilattice
operation in E implies that for every x ∈ (NF,min × {0}) \ {(F, 0)}
the set x · U(a) is infinite and hence we have that x · a /∈ N × {0} =
(NF,min × {0}) \ {(F, 0)}. Therefore we obtain that x · a = (F, 0). Now,
since in NF,min the sets F∪{F}, F ∈ F, form a neighbourhood base at the
unique non-isolated point F, we conclude that x ·U(a) * (F ∪{F})×{0},
which contradicts the separate continuity of the semilattice operation
on S. Hence we get that S \ E = ∅. This implies that the topological
semilattice E is H-closed in the class of semitopological semilattices.
Now, by Theorem 3 of [3] the topological semilattice E is not absolutely
H-closed in the class of topological semilattices, and hence E is not
absolutely H-closed in the class of semitopological semilattices.
Remark 3.16. Corollary 3.2 implies that the topological semilattice E
determined in Theorem 3.15 is an example a topological inverse semi-
group which is H-closed but is not absolutely H-closed in the class of
semitopological semigroups with continuous inversion.
Remark 3.17. Proposition 3.6 and Theorem 3.7 imply that Theorem 2
of [3] describes all H-closed semilattices in the class of semitopological
semilattices which contain the discrete semilattice (N,min) or the discrete
semilattice (N,max) as a dense subsemilattice.
O. Gutik 73
4. On the closure of topological Brandt λ-extensions in
a semitopological inverse semigroup with continuous
inversion
In this section we study the preserving of H-closedness and absolute
H-closedness by topological Brandt λ0-extensions and orthogonal sums
of semitopological semigroups.
Theorem 4.1. Let S be a Hausdorff semitopological inverse monoid
with zero and continuous inversion. Then the following conditions are
equivalent:
(i) S is absolutely H-closed in the class of semitopological inverse
semigroups with continuous inversion;
(ii) there exists a cardinal λ > 2 such that every topological Brandt λ0-
extension of S is absolutely H-closed in the class of semitopological
inverse semigroups with continuous inversion;
(iii) for each cardinal λ > 2 every topological Brandt λ0-extension of S is
absolutely H-closed in the class of semitopological inverse semigroups
with continuous inversion.
Proof. (i) ⇒ (iii). Suppose that the semigroup S is absolutely H-closed in
the class of semitopological inverse semigroups with continuous inversion.
We fix an arbitrary cardinal λ > 2. Let B0
λ(S) be a topological Brandt
λ0-extension of S in the class of semitopological inverse semigroups with
continuous inversion, T be a semitopological inverse semigroup with
continuous inversion and h : B0
λ(S) → T be a continuous homomorphism.
First we observe that by Proposition 2.3 of [24], either h is an annihi-
lating homomorphism or the image (B0
λ(S))h is isomorphic to the Brandt
λ0-extension B0
λ((Sα,α)h) of the semigroup (Sα,α)h for some α ∈ λ. If h is
an annihilating homomorphism then (Sα,α)h is a singleton, and therefore
we have that (Sα,α)h is a closed subset of T . Hence, later we assume that
h is a non-annihilating homomorphism.
Next we show that for any γ, δ ∈ λ the set (Sγ,δ)h is closed in the
space T . By Definition 1.1 there exists α ∈ λ such that (Sα,α)h is a closed
subset of T . We define the maps ϕh, ψh : T → T by the formulae (x)ϕh =
(α, 1S , γ)h·(x)h·(δ, 1S , α)h and (x)ψh = (γ, 1S , α)h·(x)h·(α, 1S , δ)h. Then
the maps ϕh and ψh are continuous because left and right translations in
T and homomorphism h : B0
λ(S) → T are continuous maps. Thus, the full
preimage A = ((Sα,α)h)ϕ−1
h is a closed subset of T . Then the restriction
74 On closures in semitopological inverse semigroups
map (ϕh ◦ψh)|A : A → (Sγ,δ)h is a retraction, and therefore the set (Sγ,δ)h
is a retract of A. This implies that (Sγ,δ)h is a closed subset of T .
Suppose to the contrary that (B0
λ(S))h is not a closed subsemigroup
of T . By Lemma II.1.10 of [27], (B0
λ(S))h is an inverse subsemigroup of T .
Since by Proposition 2.2 the closure of an inverse subsemigroup (B0
λ(S))h
in a semitopological inverse semigroup T with continuous inversion is
an inverse semigroup, without loss of generality we may assume that
(B0
λ(S))h is a dense proper inverse subsemigroup of T .
We fix an arbitrary x ∈ clT ((B0
λ(S))h) \ (B0
λ(S))h. Then only one of
the following cases holds:
a) x is an idempotent of the semigroup T ;
b) x is a non-idempotent element of T .
Suppose that case a) holds. By the previous part of the proof we have
that every open neighbourhood U(x) of the point x in the topological space
T intersects infinitely sets of the form (Sα,β)h,α, β ∈ λ. By Proposition 2.3
of [24], (B0
λ(S))h is isomorphic to the Brandt λ0-extension B0
λ((Sα,α)h)
of the semigroup (Sα,α)h for some α ∈ λ, and since (B0
λ(S))h is a dense
subsemigroup of semitopological semigroup T , the zero 0 of the semigroup
(B0
λ(S))h is zero of T (see [20, Lemma 23]). Then the semigroup operation
of B0
λ((Sα,α)h) implies that either 0 ∈ (α, e, α)h · U(x) or 0 ∈ U(x) ·
(α, e, α)h for every non-zero idempotent (α, e, α) of B0
λ(S), e ∈ E(S), α ∈
λ. Now by the Hausdorffness of the space T and the separate continuity
of the semigroup operation of T we have that either (α, e, α)h · x = 0
or x · (α, e, α)h = 0 for every non-zero idempotent (α, e, α) of B0
λ(S),
e ∈ E(S), α ∈ λ. Since in an inverse semigroup any two idempotents
commute we conclude that (α, e, α)h · x = x · (α, e, α)h = 0 for every
non-zero idempotent (α, e, α) of the semigroup B0
λ(S).
We fix an arbitrary non-zero element (α, s, β)h of the semigroup
(B0
λ(S))h, where α, β ∈ λ and s ∈ S∗. Then by the previous part of the
proof we obtain that
x · (α, s, β)h = x · (α, ss−1s, β)h = x · ((α, ss−1, α)(α, s, β))h
= x · (α, ss−1, α)h · (α, s, β)h = 0 · (α, s, β)h = 0
and
(α, s, β)h · x = (α, ss−1s, β)h · x = ((α, s, β)(β, s−1s, β))h · x
= (α, s, β)h · (β, s−1s, β)h · x = (α, s, β)h · 0 = 0.
O. Gutik 75
This implies that for every open neighbourhood U(x) of the point x
in the space T we have that 0 ∈ x · U(x) and 0 ∈ U(x) · x. Then by
Hausdorffness of the space T and the separate continuity of the semigroup
operation in T we get that x · x = 0, and hence x = 0. This implies that
E(T ) = E((B0
λ(S))h).
Suppose that case b) holds. If xx−1 = 0, then x = xx−1x = 0 · x = 0,
and similarly if x−1x = 0, then x = xx−1x = x · 0 = 0. This implies that
xx−1, x−1x ∈ E((B0
λ(S))h) \ {0}.
Then by Lemma I.7.10 of [27] there exist idempotents (α, e, α),
(β, f, β) ∈ B0
λ(S) such that xx−1 = (α, e, α)h and x−1x = (β, f, β)h,
where e, f ∈ (E(S))∗ and α, β ∈ λ. Then we have that x · (β, f, β)h =
(α, e, α)h · x = x. Since x ∈ clT ((B0
λ(S))h) \ (B0
λ(S))h, every open neigh-
bourhood U(x) of the point x in the space T intersects infinitely many sets
(Sγ,δ)h, γ, δ ∈ λ, and hence we obtain that either U(x) · (β, f, β)h ∋ 0 or
(α, e, α)h ·U(x) ∋ 0. Then the Hausdorffness of the space T and the sepa-
rate continuity of the semigroup operation on T imply that x·(β, f, β)h = 0
or (α, e, α)h·x = 0. If x·(β, f, β)h = 0 then x = x·xx−1 = x·(β, f, β)h = 0
and if (α, e, α)h · x = 0 then x = xx−1x = (α, e, α)h · x = 0. All these
two cases imply that x = 0, and hence we get that T = (B0
λ(S))h, which
completes the proof of our theorem.
The implication (iii) ⇒ (ii) is trivial.
(ii) ⇒ (i). Suppose to the contrary: there exists semigroup S such that
S is not absolutely H-closed semigroup S in the class of semitopological
inverse semigroups with continuous inversion and condition (ii) holds
for S. Then there exists a semitopological inverse semigroup T with
continuous inversion and continuous homomorphism h : S → T such
that (S)h is non-closed subset of T . Now, by Proposition 2.2 without
loss of generality we may assume that (S)h is a proper dense inverse
subsemigroup of T .
Next, for the cardinal λ we define topologies τB
T and τB
S on Brandt
λ0-extensions B0
λ(T ) and B0
λ(S), respectively, in the following way. We
put
BT
(α,t,β) = {(U(t))α,β : 0 /∈ U(t) ∈ BT (t)}
and Bs
(α,s,β) = {(U(s))α,β : 0 /∈ U(s) ∈ BS(s)}
are bases of topologies τB
T and τB
S at non-zero elements (α, t, β) ∈ B0
λ(T )
and (α, s, β) ∈ B0
λ(S), respectively, α, β ∈ λ, where BT (t) and BS(s) are
bases of topologies of spaces T and S at non-zero elements t ∈ T and
s ∈ S, respectively. Also, if BT (0T ) and BS(0S) are bases at zeros 0T ∈ T
76 On closures in semitopological inverse semigroups
and 0S ∈ S then we define
BT
0 =
{
{0} ∪
⋃
α,β∈λ
(U(0T ))∗
α,β : U(0T ) ∈ BT (0T )
}
and BS
0 =
{
{0} ∪
⋃
α,β∈λ
(U(0S))∗
α,β : U(0S) ∈ BS(0S)
}
to be the bases of topologies τB
T and τB
S at zeros 0 ∈ B0
λ(T ) and 0 ∈ B0
λ(S),
respectively.
Simple verifications show that if T and S are semitopological in-
verse semigroups with continuous inversion, then so are (B0
λ(T ), τB
T ) and
(B0
λ(S), τB
S ). Also the continuity of homomorphism h : S → T implies
that the map hB : B0
λ(S) → B0
λ(T ) defined by the formulae
(α, s, β)hB =
{
(α, (s)h, β), if (s)h 6= 0T ;
0, otherwise,
s ∈ S∗, α, β ∈ λ, and (0)hB = 0 is continuous. Also, by Theorem 3.10
of [24] so defines map hB : B0
λ(S) → B0
λ(T ) is a homomorphism. The
definition of the topology τB
T on B0
λ(T ) implies that the homomorphic
image (B0
λ(S))hB is a dense proper subsemigroup of the semitopological
inverse semigroup (B0
λ(T ), τB
T ) with continuous inversion, which contra-
dicts to statement (ii). The obtained contradiction implies the requested
implication.
Now, if we put h is a topological isomorphic embedding of semitopo-
logical semigroups with continuous inversions in the proof of Theorem 4.1,
then we get the proof of the following theorem:
Theorem 4.2. Let S be a Hausdorff semitopological inverse monoid
with zero and continuous inversion. Then the following conditions are
equivalent:
(i) S is H-closed in the class of semitopological inverse semigroups with
continuous inversion;
(ii) there exists a cardinal λ > 2 such that every topological Brandt
λ0-extension of S is H-closed in the class of semitopological inverse
semigroups with continuous inversion;
(iii) for each cardinal λ > 2 every topological Brandt λ0-extension of S
is H-closed in the class of semitopological inverse semigroups with
continuous inversion.
O. Gutik 77
Theorem 4.1 implies Corollary 4.3 which generalizes Corollary 20
from [17].
Corollary 4.3. For any cardinal λ > 2 the semigroup of λ×λ-units Bλ is
algebraically h-complete in the class of semitopological inverse semigroups
with continuous inversion.
Also, Theorems 4.1 and 4.2 imply the following corollary:
Corollary 4.4. For an inverse monoid S with zero the following condi-
tions are equivalent:
(i) S is algebraically complete (algebraically h-complete) in the class of
semitopological inverse semigroups with continuous inversion;
(ii) there exists a cardinal λ > 2 such the Brandt λ0-extension of S
is algebraically complete (algebraically h-complete) in the class of
semitopological inverse semigroups with continuous inversion;
(iii) for each cardinal λ > 2 the Brandt λ0-extension of S is algebraically
complete (algebraically h-complete) in the class of semitopological
inverse semigroups with continuous inversion.
Theorems 4.5, 4.6 and 4.7 give a method of the construction of abso-
lutely H-closed and H-closed semigroups in the class of semitopological
inverse semigroups with continuous inversion.
Theorem 4.5. Let S =
⋃
α∈A Sα be a semitopological inverse semigroup
with continuous inversion such that
(i) Sα is an absolutely H-closed semigroup in the class of semitopological
inverse semigroups with continuous inversion for any α ∈ A; and
(ii) there exists an ideal T of S which is absolutely H-closed in the class
of semitopological inverse semigroups with continuous inversion such
that Sα · Sβ ⊆ T for all α 6= β, α, β ∈ A.
Then S is an absolutely H-closed semigroup in the class of semitopological
inverse semigroups with continuous inversion.
Proof. Suppose to the contrary that there exists a semitopological inverse
semigroup K with continuous inversion and continuous homomorphism
h : S → K such that the image (S)h is not a closed subsemigroup of K.
By Lemma II.1.10 of [27], (S)h is an inverse subsemigroup of K. Since by
Proposition 2.2 the closure clK((S)h) of an inverse subsemigroup (S)h in
78 On closures in semitopological inverse semigroups
a semitopological inverse semigroup K with continuous inversion is an
inverse semigroup, without loss of generality we may assume that (S)h is
a dense proper inverse subsemigroup of K.
We observe that the assumption of the theorem states that T is an
ideal of S. This implies that (T )h is an ideal in (S)h. Then by Proposi-
tion I.1.8(iii) of [30] the closure of an ideal of a semitopological semigroup
is again an ideal, and hence we get that (T )h is a closed ideal of the
semigroup K.
We fix an arbitrary x ∈ K \ (S)h. Then only one of the following cases
holds:
a) x is an idempotent of the semigroup K;
b) x is a non-idempotent element of K.
First we show that x · y, y · x ∈ (T )h for every y ∈ (S)h. We fix
an arbitrary open neighbourhood U(x) of the point x in the space
K. Since U(x) intersects infinitely many subsemigroups of K from the
family {(Sα)h : α ∈ A} we conclude that U(x) · y ∩ (T )h 6= ∅ and
y · U(x) ∩ (T )h 6= ∅ for every y ∈ (S)h. Then the separate continuity of
the semigroup operation in K implies that any open neighbourhoods
W (x · y) and W (y · x) of the points x · y and y · x in K, respectively,
intersect the ideal (T )h. This implies that x · y, y · x ∈ clK((T )h). Since
the ideal (T )h is closed in K we conclude that x · y, y · x ∈ (T )h.
Suppose that case a) holds. Then there exists an open neighbourhood
U(x) of the point x in the space K such that U(x) ∩ (T )h = ∅ and the
neighbourhood U(x) intersects infinitely many semigroups from the family
{(Sα)h : α ∈ A}. By the separate continuity of the semigroup operation
in K we have that for every open neighbourhood U(x) of the point x in K
such that U(x) ∩ (T )h = ∅ there exists an open neighbourhood V (x) of x
in K such that x·V (x) ⊆ U(x) and V (x)·x ⊆ U(x). Now, the previous part
of proof implies that x · V (x) ∩ (T )h 6= ∅ and V (x) · x∩ (T )h 6= ∅, which
contradict the assumption U(x) ∩ (T )h = ∅. The obtained contradiction
implies that E((S)h) = E(K).
Suppose that case b) holds. Then there exist idempotents e and f
in (S)h such that xx−1 = e and x−1x = f . We observe that e, f /∈ (T )h.
Indeed, if e ∈ (T )h or f ∈ (T )h, then we have that
x = xx−1x = ex ∈ (T )h and x = xx−1x = xf ∈ (T )h,
because (T )h is an ideal of the semigroup K. Since x ∈ clK((S)h),
every open neighbourhood of the point x in K intersects infinitely many
O. Gutik 79
semigroups from the family {(Sα)h : α ∈ A}, and hence we get that
(U(x) · f) ∩ (T )h 6= ∅ and (e · U(x)) ∩ (T )h 6= ∅.
Then the Hausdorffness of K and the separate continuity of the semigroup
operation in K imply that x = xx−1x = x · f = e · x ∈ (T )h. This
contradicts the assumption that x 6= (T )h. The obtained contradiction
implies the statement of our theorem.
The proof of Theorem 4.6 is similar to the proof of Theorem 4.5.
Theorem 4.6. Let S =
⋃
α∈A Sα be a semitopological inverse semigroup
with continuous inversion such that
(i) Sα is an H-closed semigroup in the class of semitopological inverse
semigroups with continuous inversion for any α ∈ A; and
(ii) there exists an ideal T of S which is H-closed in the class of semi-
topological inverse semigroups with continuous inversion such that
Sα · Sβ ⊆ T for all α 6= β, α, β ∈ A.
Then S is an H-closed semigroup in the class of semitopological inverse
semigroups with continuous inversion.
Theorem 4.7. Let a semitopological semigroup S with continuous inver-
sion be the orthogonal sum of a family {Sα : α ∈ I} of semitopological
inverse semigroups with zeros. Then S is an (absolutely) H-closed semi-
group in the class of semitopological inverse semigroups with continuous
inversion if and only if so is any element of the family {Sα : α ∈ I}.
Proof. First we observe that if S is a semitopological semigroup with con-
tinuous inversion then so is every semigroup from the family {Sα : α ∈ I}.
The implication (⇐) follows from Theorems 4.5 and 4.6.
First we shall prove the implication (⇒) in the case of absolute H-
closedness.
Suppose to the contrary that there exists an absolute H-closed semi-
group S in the class of semitopological inverse semigroups with continuous
inversion which is an orthogonal sum of a family {Sα : α ∈ I} of semi-
topological inverse semigroups and there exists a semigroup Sα0
in this
family such that Sα0
is not absolute H-closed in the class of semitopo-
logical inverse semigroups with continuous inversion. Then there exists
a semitopological inverse semigroup K with continuous inversion and
continuous homomorphism h : Sα0
→ K such that the image (Sα0
)h is not
80 On closures in semitopological inverse semigroups
a closed subsemigroup of K. By Lemma II.1.10 of [27], (Sα0
)h is an inverse
subsemigroup of K. Since by Proposition 2.2 the closure clK((Sα0
)h) of
an inverse subsemigroup (Sα0
)h in a semitopological inverse semigroup K
with continuous inversion is an inverse semigroup, without loss of gener-
ality we may assume that (Sα0
)h is a dense proper inverse subsemigroup
of K. Also, the semigroup K has zero because (Sα0
)h contains zero.
We define a map f : S → K by the formula
(x)f =
{
0K , if x ∈ S \ S∗
α0
;
(x)h, if x ∈ S∗
α0
,
where 0K is zero of the semigroup K. Simple verifications show that so de-
fined map f is a continuous homomorphism, but the image (S)f = (Sα0
)h
is a dense proper subsemigroup of K. This contradicts the assumption
that the semigroup S is absolutely H-closed semigroup in the class of
semitopological inverse semigroups with continuous inversion.
Now, we suppose that there exists an H-closed semigroup S in the
class of semitopological inverse semigroups with continuous inversion
which is an orthogonal sum of a family {Sα : α ∈ I} of semitopological
inverse semigroups and there exists a semigroup Sα0
in this family such
that Sα0
is not H-closed in the class of semitopological inverse semigroups
with continuous inversion. Then there exists a semitopological inverse
semigroup K with continuous inversion such that Sα0
is not a closed
subsemigroup of K. Since by Proposition 2.2 the closure clK(Sα0
) of an
inverse subsemigroup Sα0
in a semitopological inverse semigroup K with
continuous inversion is an inverse semigroup, without loss of generality
we may assume that Sα0
is a dense proper inverse subsemigroup of K.
Next, we put S′ be the orthogonal sum of the family {Sα :α ∈ I\{α0}}
and the semigroup K. We determine a topology τ on S′ in the following
way.
First we observe if the orthogonal sum T =
∑
i∈J Tj is an inverse
Hausdorff semitopological semigroup, then for every non-zero element
t ∈ Tj ⊂ T there exists an open neighbourhood U(t) of t in T such that
U(t) ⊆ T ∗
j . Indeed, for every open neighbourhood W (t) 6∋ 0 of t in T there
exists an open neighbourhood U(t) of t in T such that tt−1 ·U(t) ⊆ W (t).
The neighbourhood U(t) is requested.
We put that the bases of topologies at any point s of S \ Sα0
and of
S′ \K coincide in S and in S′, respectively. Also the bases at any point
s of subspace K∗ ⊆ S′ coincide with the base at the point s of K∗. The
following family determines the base of the topology τ at zero of the
O. Gutik 81
semigroup S′:
B0 =
U ⊆ S′ :
there exist an element V of the base at zero
of the topology of S and an element W of
the base at zero of the topology of K such
that U ∩ S′ \K = V ∩ S \ Sα0
, U ∩K = W
and U ∩ Sα0
= W ∩ Sα0
.
Simple verifications show that (S′, τ) is a Hausdorff semitopological inverse
semigroup with continuous inversion and moreover S is a dense proper
inverse subsemigroup of (S′, τ), which contradicts the assumption of
our theorem. The obtained contradiction implies the statement of the
theorem.
Theorem 4.7 implies the following corollary:
Corollary 4.8. A primitive Hausdorff semitopological inverse semigroup
S is (absolutely) H-closed in the class of semitopological inverse semi-
groups with continuous inversion if and only if so is every its maximal
subgroup G with adjoined zero with an induced topology from S.
Remark 4.9. We observe that the statements of Theorems 4.5, 4.6 and 4.7
hold for H-closed and absolute H-closed semitopological semilattices in
the class of semitopological semilattices.
Theorem 4.10. An infinite semitopological semigroup of λ× λ-matrix
units Bλ id H-closed in the class of semitopological semigroups if and
only if the space Bλ is compact.
Proof. Implication (⇐) is trivial.
(⇒). Suppose to the contrary that there exists a Hausdorff non-
compact topology τB on the semigroup Bλ such that (Bλ, τB) is an H-
closed semigroup in the class of semitopological semigroups. By Lemma 2
of [18] every non-zero element of Bλ is an isolated point in (Bλ, τB). Then
there exists an infinite open-and-closed subset A ⊆ Bλ \ {0}.
Then we have that at least one of the following cases holds:
1) there exist finitely many i1, . . . , in ∈ λ such that if (i, j) ∈ A then
i ∈ {i1, . . . , in};
2) there exist finitely many j1, . . . , jn ∈ λ such that if (i, j) ∈ A then
i ∈ {j1, . . . , jn};
3) cases 1) and 2) don’t hold.
82 On closures in semitopological inverse semigroups
Suppose case 1) holds. Then there exists an element i0 ∈ {i1, . . . , in}
such that the set {(i0, j) : j ∈ λ} ∩ A is infinite. We denote Ai0
=
{(i0, j) ∈ Bλ : (i0, j) ∈ A}. It is obvious that Ai0
is infinite subset of
the semigroup Bλ. By Lemma 2 of [18] every non-zero element of
Bλ is an isolated point in (Bλ, τB) and hence Ai0
is an open-and-
closed subset in the topological space (Bλ, τB). Since the left shift
l(i0,i) : Bλ → Bλ : x 7→ (i0, i) · x is a continuous map for any i ∈ λ,
Ai = {(i, j) ∈ Bλ : (i0, j) ∈ A} is an infinite open-and-closed subset in
(Bλ, τB) for every i ∈ λ. This implies that the set Bλ \ {Ai1
∪ · · · ∪Aik
}
is an open neighbourhood of the zero in (Bλ, τB) for every finite subset
{i1, . . . , ik} ⊂ λ.
Now. for every i ∈ λ we put ai /∈ Bλ. We extend the semigroup
operation from Bλ onto the set S = Bλ ∪ {ai : i ∈ λ} in the following way:
(i) ai · aj = ai · 0 = 0 · ai = 0 for all i, j ∈ λ;
(ii) (s, p) · ai =
{
as, if p = i;
0, if p 6= i
for all (s, p) ∈ Bλ \ {0} and i ∈ λ;
(iii) ai · (s, p) = 0 for all (s, p) ∈ Bλ \ {0} and i ∈ λ.
Simple verifications show that so defines binary operation on S is asso-
ciative, and hence S is a semigroup.
Next, we define a topology τS on the semigroup S in the following
way. For every element x ∈ Bλ we put that bases of topologies τB and τS
at the point x coincide. Also, for every i ∈ λ we put
BS(ai) = {{ai} ∪ Ci : Ci is a cofinite subset of Ai}
is a base of the topology τS at the point ai ∈ S. It is obvious that (S, τS)
is a Hausdorff topological space. The separate continuity of the semigroup
operation in (S, τS) follows from the cofinality of the set Ci in Ai for each
i ∈ λ. Therefore we get that the semitopological semigroup (Bλ, τB) is
a dense proper subsemigroup of (S, τS), which contradicts the assumption
of the theorem.
In case 2) the proof is similar.
Suppose that cases 1) and 2) don’t hold. By induction we construct an
infinite sequence {(xi, yi)}i∈N in Bλ in the following way. First we fix an
arbitrary element (x, y) ∈ A and denote (x1, y1) = (x, y). Suppose that for
some positive integer n we construct the finite sequence {(xi, yi)}i=1,...,n.
Since the set A is infinite and cases 1) and 2) don’t hold, there exists
(x, y) ∈ A such that x /∈ {y1, . . . , yn} and y /∈ {x1, . . . , xn}. Then we put
(xn+1, yn+1) = (x, y).
O. Gutik 83
Let a /∈ Bλ. We put T = Bλ ∪{a} and extend the semigroup operation
from Bλ onto T in the following way:
a · x = x · a = a · a = 0, for every x ∈ Bλ.
Next, we define a topology τT on the semigroup T in the following
way. For every element x ∈ Bλ we put that bases of topologies τB and τT
at the point x coincide. Also, we put
BT (a) = {{a} ∪ C : C is a cofinite subset of the set {(xi, yi) : i ∈ N}}
is a base of the topology τT at the point a ∈ T . It is obvious that (T, τT )
is a Hausdorff topological space, the semigroup operation in (T, τT ) is
separately continuous, and Bλ is a dense subsemigroup of (T, τT ). This
contradicts the assumption of the theorem.
The obtained contradictions imply the statement of our theorem.
Remark 4.11. By Theorem 2 [18] for every infinite cardinal λ there
exists a unique Hausdorff pseudocompact topology τc on the semigroup
Bλ such that (Bλ, τc) is a semitopological semigroup. This topology is
compact and it is described in Example 1 of [18].
Acknowledgements
The author acknowledges Oleksandr Ravskyi for his comments and
suggestions.
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Contact information
O. Gutik Faculty of Mechanics and Mathematics, National
University of Lviv, Universytetska 1, Lviv, 79000,
Ukraine
E-Mail: o_gutik@franko.lviv.ua,
ovgutik@yahoo.com
Received by the editors: 17.09.2014
and in final form 17.09.2014.
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