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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 13 (2012). Number 2. pp. 147 – 168
c© Journal “Algebra and Discrete Mathematics”
Algebra in superextensions
of inverse semigroups
Taras Banakh and Volodymyr Gavrylkiv
Communicated by I. V. Protasov
Abstract. We find necessary and sufficient conditions on
an (inverse) semigroup X under which its semigroups of maximal
linked systems λ(X), filters ϕ(X), linked upfamilies N2(X), and
upfamilies υ(X) are inverse.
1. Introduction
In this paper we investigate the algebraic structure of various exten-
sions of an inverse semigroup X and detect semigroups whose extensions
λ(X), N2(X), ϕ(X), υ(X) are inverse semigroups.
The thorough study of extensions of semigroups was started in [9] and
continued in [1]–[5]. The largest among these extensions is the semigroup
υ(X) of all upfamilies on X.
A family F of subsets of a set X is called an upfamily if each set F ∈ F
is not empty and for each set F ∈ F any subset E ⊃ F of X belongs to
F . The space of all upfamilies on X is denoted by υ(X). It is a closed
subspace of the double power-set P(P(X)) endowed with the compact
Hausdorff topology of the Tychonoff product {0, 1}P(X). Identifying each
point x ∈ X with the upfamily 〈x〉 = {A ⊂ X : x ∈ A}, we can identify
The first author has been partially financed by NCN means granted by decision
DEC-2011/01/B/ST1/01439
2010 MSC: 20M10, 20M14, 20M17, 20M18, 54B20.
Key words and phrases: inverse semigroup, regular semigroup, Clifford semi-
group, superextension, semigroup of filters.
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148 Algebra in superextensions of inverse semigroups
X with a subspace of υ(X). Because of that we call υ(X) an extension
of X.
The compact Hausdorff space υ(X) contains many other important
extensions of X as closed subspaces. In particular, it contains the spaces
N2(X) of linked upfamilies, λ(X) of maximal linked upfamilies, ϕ(X)
of filters, and β(X) of ultrafilters on X; see [8]. Let us recall that an
upfamily F ∈ υ(X) is called
• linked if A ∩B 6= ∅ for any sets A,B ∈ F ;
• maximal linked if F = F ′ for any linked upfamily F ′ ∈ υ(X) that
contains F ;
• a filter if A ∩B ∈ F for any A,B ∈ F ;
• an ultrafilter if F = F ′ for any filter F ′ ∈ υ(X) that contains F .
The family β(X) of all ultrafilters on X is called the Stone-Čech extension
and the family λ(X) of all maximal linked upfamilies is called the su-
perextension of X, see [12] and [15]. The arrows in the following diagram
denote the identity inclusions between various extensions of a set X.
X // β(X) //

λ(X)

ϕ(X) // N2(X) // υ(X).
Any map f : X → Y induces a continuous map
υf : υ(X) → υ(Y ), υf : F 7→ {A ⊂ Y : f−1(A) ∈ F},
such that υf(β(X)) ⊂ β(Y ), υf(λ(X)) ⊂ λ(Y ), υf(ϕ(X)) ⊂ ϕ(Y ), and
υf(N2(X)) ⊂ N2(Y ). If the map f is injective, then υf is a topological
embedding, which allows us to identify the extensions β(X), λ(X), ϕ(X),
N2(X), υ(X) with corresponding closed subspaces in β(Y ), λ(Y ), ϕ(Y ),
N2(Y ), and υ(Y ), respectively.
In [9] it was observed that any (associative) binary operation ∗ :
X × X → X can be extended to an (associative) binary operation ∗ :
υ(X)× υ(X) → υ(X) defined by the formula:
A ∗ B = 〈
⋃
a∈A
a ∗Ba : A ∈ A, {Ba}a∈A ⊂ B
〉
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T. Banakh, V. Gavrylkiv 149
for upfamilies A,B ∈ υ(X). Here for a family C of non-empty subsets of
X by
〈C〉 = {A ⊂ X : ∃C ∈ C with C ⊂ A}
we denote the upfamily generated by the family C.
According to [8], for each semigroup X, υ(X) is a compact Hausdorff
right-topological semigroup containing the subspaces β(X), λ(X), ϕ(X),
N2(X) as closed subsemigroups. Algebraic and topological properties of
these semigroups have been studied in [9], [1]–[5]. In particular, in [3]
we studied properties of extensions of groups while [4] was devoted to
extensions of semilattices. There are two important classes of semigroups
that include all groups and all semilattices. Those are the classes of inverse
and Clifford semigroups.
Let us recall that a semigroup S is inverse if for any element x ∈ S
there is a unique element x−1 (called the inverse of x) such that xx−1x = x
and x−1xx−1 = x−1. A semigroup S is regular if each element x ∈ S
is regular in the sense that x ∈ xSx. It is known [13, II.1.2] that a
semigroup S is inverse if and only if S is regular and idempotents of S
commute. For a semigroup S by E = {x ∈ S : xx = x} we denote the
set of idempotents of S. It follows that for an inverse semigroup, E is a
commutative subsemigroup of S and hence E is a maximal semilattice in
S. Let us recall that a semilattice is a set endowed with an associative
commutative idempotent operation. A semigroup S is called linear if
xy ∈ {x, y} for any points x, y ∈ X. Each linear commutative semigroup
is a semilattice.
A semigroup S is Clifford if it is a union of groups. An inverse semi-
group S is Clifford if and only if xx−1 = x−1x for all x ∈ X. A semigroup
S is called sub-Clifford if it is a union of cancellative semigroups. A semi-
group S is sub-Clifford if and only if for any positive integer numbers
n < m and any x ∈ S the equality xn+1 = xm+1 implies xn = xm. Each
subsemigroup of a Clifford semigroup is sub-Clifford and each finite sub-
Clifford semigroup is Clifford. A commutative semigroup is Clifford if and
only if it is inverse if and only if it is regular. A semigroup X is called
Boolean if x3 = x for all x ∈ X. Each Boolean semigroup is Clifford.
It is well-known that the class of inverse (Clifford) semigroups includes
all groups and all semilattices. Moreover, each inverse Clifford semigroup
S decomposes into the union S = ⋃e∈E He of maximal subgroups He =
{x ∈ S : xx−1 = e = x−1x} indexed by the idempotents, which commute
with all elements of S; see [13, II.2].
The algebraic structure of extensions of groups was studied in details
in [3]. Extensions of semilattices were investigated in [4]. In particular,
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150 Algebra in superextensions of inverse semigroups
in [4] it was shown that for a semigroup X the superextension λ(X) is a
semilattice if and only if the semigroup υ(X) is a semilattice if and only
if X is a finite linear semilattice.
In Theorems 1.1–1.4 below we shall list all semigroups X whose
extensions are (commutative) inverse semigroups.
For a natural number n by Cn = {z ∈ C : zn = 1} we denote the
cyclic group of order n and by Ln the linear semilattice {0, . . . , n− 1} of
order n, endowed with the operation of minimum. In particular, L0 is an
empty semigroup.
For two semigroups (X, ∗) and (Y, ⋆) by X ⊔ Y we denote the disjoint
union of these semigroups endowed with the semigroup operation
x ◦ y =











x ∗ y if x, y ∈ X,
x if x ∈ X and y ∈ Y ,
y if x ∈ Y and y ∈ X,
x ⋆ y if x, y ∈ Y .
The semigroup X ⊔ Y will be called the disjoint ordered union of the
semigroups X and Y . Observe that the operation of disjoint ordered
union is associative in the sense that (X ⊔ Y ) ⊔ Z = X ⊔ (Y ⊔ Z) for any
semigroups X,Y, Z.
We say that a subsemigroup X of a semigroup Y is regular in Y if
each element x ∈ X is regular in Y .
Theorem 1.1. For a semigroup X and its superextension λ(X) the
following conditions are equivalent:
(1) λ(X) is a commutative Clifford semigroup;
(2) λ(X) is an inverse semigroup;
(3) the idempotents of the semigroup λ(X) commute and λ(X) is sub-
Clifford or regular in N2(X);
(4) X is a finite commutative Clifford semigroup, isomorphic to one of
the following semigroups: C2, C3, C4, C2 × C2, L2 × C2, L1 ⊔ C2,
Ln, or C2 ⊔ Ln for some n ∈ ω.
Theorem 1.2. For a semigroup X and its semigroup of filters ϕ(X) the
following conditions are equivalent:
(1) ϕ(X) is a commutative Clifford semigroup;
(2) ϕ(X) is an inverse semigroup;
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T. Banakh, V. Gavrylkiv 151
(3) the idempotents of the semigroup ϕ(X) commute and ϕ(X) is sub-
Clifford or regular in N2(X);
(4) X is isomorphic to one of the semigroups: C2, Ln or Ln ⊔ C2 for
some n ∈ ω.
Theorem 1.3. For a semigroup X and its semigroup of linked upfamilies
N2(X) the following conditions are equivalent:
(1) N2(X) is a commutative Clifford semigroup;
(2) N2(X) is an inverse semigroup;
(3) the idempotents of the semigroup N2(X) commute and N2(X) is
sub-Clifford or regular;
(4) X is isomorphic to C2 or Ln for some n ∈ ω.
Theorem 1.4. For a semigroup X and its semigroup of upfamilies υ(X)
the following conditions are equivalent:
(1) υ(X) is a finite semilattice;
(2) υ(X) is an inverse semigroup;
(3) the idempotents of the semigroup υ(X) commute and υ(X) is sub-
Clifford or regular;
(4) X is a finite linear semilattice, isomorphic to Ln for some n ∈ ω.
Surprisingly, the following problem remains open.
Problem 1.5. Characterize semigroups X whose Stone-Čech extension
β(X) is an inverse semigroup. (Such semigroups have finite linear and
finite cyclic subsemigroups; see Proposition 2.1.)
Theorems 1.1, 1.2, 1.3, and 1.4 will be proved in Sections 5, 6, 7, and
8, respectively.
2. Commutativity in the Stone-Čech extension
In this section we establish some properties of semigroups whose
Stone-Čech extension has commuting idempotents. Let us recall that a
semigroup S is cyclic if S = {xn : n ∈ N} for some element x ∈ S, called
the generator of S.
Proposition 2.1. If for a semigroup X all idempotents of the Stone-Čech
extension β(X) commute, then all cyclic subsemigroups and all linear
subsemigroups of X are finite.
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152 Algebra in superextensions of inverse semigroups
Proof. First we show that each element x ∈ X generates a finite cyclic
subsemigroup {xn}n∈N. If {xn}n∈N is infinite, then it is isomorphic to
the semigroup (N,+). Then the Stone-Čech extension β(X) contains
a subsemigroup isomorphic to the Stone-Čech extension β(N) of the
semigroup (N,+). By Theorem 6.9 of [11], the semigroup β(N) contains
2c non-commuting idempotents and so does the semigroup β(X) which
is forbidden by our assumption. So, the cyclic subsemigroup {xn}n∈N is
finite.
Next, assume that X contains an infinite linear subsemigroup L. Then
xy ∈ {x, y} for any elements x, y ∈ L. Choose any injective sequence
{xn}n∈ω in L and define a 2-coloring χ : [ω]2 → {0, 1} of the set [ω]2 =
{(n,m) ∈ ω2 : n < m} letting
χ(n,m) =
{
0 if xnxm = xn
1 if xnxm = xm.
By Ramsey’s Theorem [14] (see also [10, Theorem 5]), there is an infinite
subset Ω ⊂ ω and a color k ∈ {0, 1} such that χ(n,m) = k for any pair
(n,m) ∈ [ω]2∩Ω2. Consider the infinite linear subsemigroup Z = {xn}n∈Ω
of X. By Theorem 1.1 of [4], each element of the semigroup β(Z) is an
idempotent. We claim that any two distinct free ultrafilters U ,V ∈ β(Z)
do not commute (which is forbidden by our assumption). If the color
k = 0, then xnxm = xn for any numbers n < m in Ω, which implies that
U ∗ V = U 6= V = V ∗ U . If k = 1, then xnxm = xm for any numbers
n < m in Ω and then U ∗ V = V 6= U = V ∗ U .
3. The regularity of extensions of semigroups
In this section we shall prove some results related to the regularity of
semigroups. Let us recall that an element x ∈ S is regular in a semigroup
S if x ∈ xSx.
Proposition 3.1. Let X be a semigroup. An element x ∈ X is regular
in X if and only if the ultrafilter 〈x〉 is regular in the semigroup υ(X).
Proof. The “if” part is trivial. To prove the “only if” part, assume that
〈x〉 is regular in υ(X) and find an upfamily F ∈ υ(X) such that 〈x〉 =
〈x〉 ∗ F ∗ 〈x〉. Then for some set F ∈ F we get x ∈ xFx ⊂ xSx, which
means that x is regular in X.
Corollary 3.2. A semigroup X is inverse if and only if X lies in some
inverse semigroup S ⊂ υ(X).
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T. Banakh, V. Gavrylkiv 153
Proof. The “only if” part is trivial (just take S = X). To prove the
“if” part, assume that a semigroup X lies in some inverse subsemigroup
S ⊂ υ(X). The inverse semigroup S is regular and has commuting
idempotents. Then the idempotents of the subsemigroup X ⊂ S also
commute. Each element x ∈ X ⊂ S is regular in S and hence is regular in
X by Proposition 3.1. Then the semigroup X is inverse, being a regular
semigroup with commuting idempotents; see [13, II.1.2].
Let us recall that a non-empty subset I of a semigroup X is called an
ideal in X if XI ∪ IX ⊂ I.
Lemma 3.3. Let X be a semigroup and Z ⊂ X be a subsemigroup whose
complement X\Z is an ideal in X. If for two upfamilies A ∈ υ(Z) ⊂ υ(X)
and B ∈ υ(X) we get A = A∗B∗A, then A = A∗BZ ∗A for the upfamily
BZ = {B ∈ B : B ⊂ Z} ∈ υ(Z).
Proof. It is clear that A ∗ BZ ∗ A ⊂ A ∗ B ∗ A = A. To prove the reverse
inclusion, take any set A ∈ A. It follows from A ∈ υ(Z) ⊂ υ(X) that
A ∩ Z ∈ A ⊂ (A ∗ B) ∗ A. So, we can find a set C ∈ A ∗ B and a family
{Ac}c∈C ⊂ A such that ⋃c∈C c∗Ac ⊂ A∩Z. For every c ∈ C the inclusion
c ∗Ac ⊂ A ∩ Z ⊂ Z implies c ∈ Z (because X \ Z is an ideal in X). So,
C ⊂ Z. Since C ∈ A ∗ B, there is a set A ∈ A and a family {Ba}a∈A ⊂ B
such that ⋃a∈A a ∗ Ba ⊂ C. Since X \ Z is an ideal in X, for every
a ∈ A the inclusion a ∗ Ba ⊂ C ⊂ Z implies Ba ⊂ Z which means that
{Ba}a∈A ⊂ BZ and hence A ⊂ A ∗ BZ ∗ A.
Corollary 3.4. Let X be a semigroup and S ∈ {β(X), λ(X), ϕ(X),
N2(X), υ(X)} be one of its extensions. If the semigroup S is regular, then
for any subsemigroup Z ⊂ X whose complement X \ Z is an ideal in X
the semigroup S ∩ υ(Z) is regular.
Proof. Fix any upfamily A ∈ S ∩ υ(Z) and by the regularity of the
semigroup S, find an upfamily B ∈ S such that A = A ∗ B ∗ A. By
Lemma 3.3, A = A ∗ BZ ∗ A for the upfamily BZ = {B ∈ B : B ⊂ Z} ∈
υ(Z). If S ∈ {ϕ(X), N2(X), υ(X)}, then BZ ∈ S ∩ υ(Z) and hence A is
regular in S ∩ υ(Z).
If S = β(X), then BZ is a filter on Z and we can enlarge it to an
ultrafilter B˜Z ∈ β(Z). Then A = A ∗ BZ ∗ A ⊂ A ∗ B˜Z ∗ A implies that
A = A∗ B˜Z ∗A by the maximality of the ultrafilter A. So, A in regular in
the semigroup β(Z). By analogy we can consider the case S = λ(X).
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154 Algebra in superextensions of inverse semigroups
4. The extensions of the exceptional semigroups from
Theorem 1.1
In this section we describe the structure of the extensions of the
exceptional semigroups from Theorem 1.1(4).
We start with studying the superextensions of these semigroups. First
note that for each set X of cardinality 1 ≤ |X| ≤ 2 the superextension
λ(X) coincides with β(X) = X. If a set X has cardinality |X| = 3, then
λ(X) = X∪{△} where △ = {A ⊂ X : |A| ≥ 2}. For a set X of cardinality
|X| = 4 the superextension λ(X) = {x,△x,x : x ∈ X} consists of 12
elements, where
△x = {A ⊂ X : |A \ {x}| ≥ 2} and
x = (X \ {x}) ∪ {A ⊂ X : x ∈ A, |A| ≥ 2} for x ∈ X.
Given two semigroups X,Y we shall write X ∼= Y if these semigroups are
isomorphic.
Proposition 4.1. For finite exceptional semigroups we have the following
isomorphisms:
(1) λ(C2) = C2.
(2) λ(C3) ∼= L1 ⊔ C3.
(3) λ(C4) ∼= (C2 ⊔ L1)× C4.
(4) λ(C2 × C2) ∼= (C2 ⊔ L1)× C2 × C2.
(5) λ(L1 ⊔ C2) ∼= L1 ⊔ L1 ⊔ C2.
(6) λ(L2 × C2) ∼=
(
L1 ⊔ (L2 × L2) ⊔ L1
)× C2.
Proof. 1–4. The first four statements were proved in [5, §6].
5. For the semigroup X = L1 ⊔ C2 = {0, 1,−1} the superextension
λ(X) = {0,△, 1,−1} has the structure of the ordered union {0} ⊔ {△} ⊔
{1,−1}, which is isomorphic to the semigroup L1 ⊔ L1 ⊔ C2.
6. The semigroup L2 × C2 = {0, 1} × {−1, 1} has two idempotents
e = (0, 1) and f = (1, 1) and two elements a = (0,−1) and b = (1,−1) of
order 2 such that a2 = e and b2 = f . The superextension
λ(L2 × C2) = {x,△x,x : x ∈ L2 × C2}
has the 6-element set of idempotents E = {e,e,△a,△b,f , f}, isomor-
phic to the semilattice L1 ⊔ (L2 ×L2)⊔L1. The semigroup λ(L2 ×C2) is
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T. Banakh, V. Gavrylkiv 155
isomorphic to the product E × C2 under the isomorphism h : E × C2 →
λ(L2 × C2) defined by
h : (x, g) 7→
{
x if g = 1,
xb if g = −1.
The following proposition was proved in [4, 3.1].
Proposition 4.2. For every n ∈ N the semigroup υ(Ln) is a finite
semilattice. Consequently, the semigroups λ(Ln), ϕ(Ln), N2(Ln) also are
finite semilattices.
We recall that a semigroup X is called Boolean if x = x3 for all x ∈ X.
It is clear that each Boolean semigroup is Clifford and each commutative
Boolean semigroup is inverse.
Proposition 4.3. For every n ∈ N and the semigroup X = C2 ⊔ Ln the
superextension λ(X) is a finite commutative Boolean semigroup whose
maximal semilattice E(λ(X)) coincides with the set λ(X)\{a} where a is
the unique element generating the subgroup C2 of X = C2⊔Ln. Moreover,
ea = a for any idempotent e of λ(X).
Proof. Observe that X \ {a} is a linear semilattice such that xa = a for
all x ∈ X \ {a}. We identify the point a with the principal ultrafilter 〈a〉
generated by a. For the convenience of the reader we divide the proof of
Proposition 4.3 into a series of claims.
Claim 4.4. For each F ∈ λ(X) \ {a} we get a ∗ F = F ∗ a = 〈a〉.
Proof. Since F 6= 〈a〉, there is a set F ∈ F with a /∈ F . Then a ∗ F =
F ∗ a = {a}, which implies a ∗ F = a ∗ F = 〈a〉.
Claim 4.5. Each element F ∈ λ(X) \ {a} is an idempotent.
Proof. Since the upfamilies F and F ∗ F are maximal linked, it suffices
to check that F ⊂ F ∗ F . Fix any set F ∈ F and consider two cases. If
a /∈ F , then F = F ∗ F ∈ F ∗ F . So, assume that a ∈ F . Since F 6= 〈a〉,
there is a non-empty set Fa ∈ F that does not contain the point a. Then
a ∗ Fa = {a} ⊂ F . For each x ∈ F \ {a}, let Fx = F and observe that
x ∗ F ⊂ {x} ∪ F ⊂ F . Then ⋃x∈F x ∗ Fx ⊂ {a} ∪ F = F and hence
F ∈ F ∗ F .
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156 Algebra in superextensions of inverse semigroups
Claim 4.6. U ∗ V = V ∗ U for any maximal linked systems U ,V ∈ λ(X).
Proof. The equality U ∗ V = V ∗ U is trivial if U or V belongs to β(X) =
X. So, we assume that the maximal linked systems U ,V /∈ X are not
ultrafilters.
First we prove that U ∗ V ⊂ V ∗ U . Fix any set W ∈ U ∗ V. Without
loss of generality, it is of the basic form W = ⋃u∈U u ∗ Vu for some set
U ∈ U and a family {Vu}u∈U ⊂ V. Since X \ {a} is a linear semilattice,
(the proof of Theorem 2.5) [4] guarantees that:
(U \ {a}) ∗ (Vu \ {a}) ⊂ W for some point u ∈ U \ {a}.
We consider three cases.
1) a /∈ Vu and a /∈ U . Then V ∗ U ∋ Vu ∗ U = U ∗ Vu = (U \ {a}) ∗
(Vu \ {a}) ⊂ W and hence W ∈ V ∗ U .
2) a /∈ Vu and a ∈ U . Then a ∗ Vu = {a}. Since V 6= 〈a〉, the
set Va \ {a} is not empty and hence contains some idempotent. Then
aVu = {a} ⊂ a ∗ Va ⊂ W and
V ∗ U ∋ Vu ∗ U = U ∗ Vu = (a ∗ Vu) ∪ (U \ {a}) ∗ Vu =
{a} ∪ (U \ {a}) ∗ (Vu \ {a}) ⊂ {a} ∪W ⊂ W
and again W ∈ V ∗ U .
3) a ∈ Vu. In this case a ∈ u ∗ Vu ⊂ W . It follows from U 6= 〈a〉 that
a /∈ Ua for some set Ua ∈ U . Let Uv = U for all v ∈ Vu \ {a} and observe
that
V ∗ U ∋
⋃
v∈Vu
v ∗ Uv = a ∗ Ua ∪
((Vu \ {a}) ∗ U) =
{a} ∪ ((Vu \ {a}) ∗ (U \ {a})
) ∪ ((Vu \ {a}) ∗ a
) ⊂ {a} ∪W ∪ {a} ⊂ W
and hence W ∈ V ∗ U . Therefore, U ∗ V ⊂ V ∗ U .
The inclusion V ∗ U ⊂ U ∗ V can be proved by analogy.
Next, we study the structure of the space of filters ϕ(X) of the finite
exceptional groups from Theorem 1.2.
Proposition 4.7. (1) ϕ(C2) = N2(C2) ∼= L1 ⊔ C2;
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T. Banakh, V. Gavrylkiv 157
(2) ϕ(L1 ⊔ C2) is a commutative Boolean semigroup isomorphic to the
subsemigroup
{(e, x) ∈ (L1 ⊔ (L2×L2))×C2 : e ∈ L1 ⊔{(0, 0), (0, 1)} ⇒ (x = 1)}
of the commutative Boolean semigroup (L1 ⊔ (L2 × L2)
)× C2.
Proof. 1. The semigroup ϕ(C2) contains two ultrafilters and one filter
Z = 〈C2〉 generated by the set C2. The filter Z is the zero of the semigroup
ϕ(C2) and hence ϕ(C2) is isomorphic to {Z} ⊔ C2.
2. For the semigroup X = L1 ⊔ C2 = {0, 1,−1} the semigroup ϕ(X)
contains 7 filters generated by all non-empty subsets of X. So, we can
identify filters with their generating sets. Among these 7 filters there
are 5 idempotents: {0}, {0, 1,−1}, {0, 1}, {1,−1}, and {1} which form a
semilattice E
{1}
{0, 1}
99
{−1, 1}
ff
{0, 1,−1}
ee 88
{0}
OO
isomorphic to L1 ⊔ (L2 × L2). Two filters {−1} and {0,−1} generate
2-element subgroups with idempotents {1} and {0, 1}, respectively. Since
{−1} ∗ {0, 1} = {0,−1}, the semigroup λ(X) is isomorphic to the sub-
semigroup {(e, x) ∈ E × C2 : e ∈
{{0}, {0, 1,−1}, {1,−1}} ⇒ (x = 1)} of
the commutative Boolean semigroup E × C2.
Now we consider the the semigroups ϕ(Ln) and ϕ(Ln ⊔C2). We shall
show that the latter semigroup has the structure of the reduced product
of a semilattice and a group.
Let X,Y be two semigroups and I be an ideal in X. The reduced
product X ×I Y is the set I ∪
((X \ I)× Y ) endowed with the semigroup
operation
a ∗ b =
{
pX(a) ∗ pX(b) if pX(a) ∗ pX(b) ∈ I,
(pX(a) ∗ pX(b), pY (a) ∗ pY (b)) if pX(a) ∗ pX(b) /∈ I.
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158 Algebra in superextensions of inverse semigroups
Here by pX : X ×I Y → X and pY : (X \ I) × Y → Y we denote the
natural projections. Let us recall that by Proposition 4.7(1), the semigroup
ϕ(C2) is isomorphic to the commutative Boolean semigroup L1 ⊔ C2.
Proposition 4.8. For every n ∈ N the semigroup
(1) ϕ(Ln) is a finite semilattice, and
(2) ϕ(Ln ⊔ C2) is a commutative Boolean semigroup isomorphic to the
reduced product ϕ(Ln+1)×ϕ(Ln) ϕ(C2).
Proof. By Proposition 4.2, the semigroup ϕ(Ln) is a finite semilattice.
Now consider the semigroup X = Ln ⊔ C2. Since X is finite we can
identify the semigroup ϕ(X) with the commutative semigroup of all non-
empty subsets of X. Let a be the generator of the cyclic group C2 and
e = a2 be its idempotent. The idempotent semilattice E = Ln ⊔ {e} of X
is isomorphic to the linear semilattice Ln+1. So, we shall identify E with
Ln+1. Observe that ϕ(X) \ ϕ(Ln) = {F ⊂ X : F ∩ C2 6= ∅} and the map
h : ϕ(Ln+1)×ϕ(Ln) ϕ(C2) → ϕ(X) defined by
h(A) =
{
A if A ⊂ ϕ(Ln)
(A \ C2) ∪B if (A,B) ∈ (ϕ(Ln+1) \ ϕ(Ln))× ϕ(C2)
is a required isomorphism between the semigroups ϕ(X) and
ϕ(Ln+1)×ϕ(Ln) ϕ(C2).
5. Proof of Theorem 1.1
Given a semigroup X, we need to prove the equivalence of the following
statements:
(1) λ(X) is a commutative Clifford semigroup;
(2) λ(X) is an inverse semigroup;
(3) the idempotents of λ(X) commute and λ(X) is sub-Clifford or
regular in N2(X);
(4) X is a finite commutative inverse semigroup, isomorphic to one of
the following semigroups: C2, C3, C4, C2 × C2, L2 × C2, L1 ⊔ C2,
Ln, or C2 ⊔ Ln for some n ∈ ω.
We shall prove the implications (4) ⇒ (1) ⇒ (2) ⇒ (3) ⇒ (4).
The implication (4) ⇒ (1) follows from Propositions 4.1—4.3 while
(1) ⇒ (2) ⇒ (3) are trivial or well-known; see [13, II.1.2].
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T. Banakh, V. Gavrylkiv 159
To prove that (3) ⇒ (4), assume that the idempotents of the semigroup
λ(X) commute and λ(X) is sub-Clifford or regular in N2(X). Then
the idempotents of the semigroup X also commute and hence the set
E = {e ∈ X : ee = e} of idempotents of X is a semilattice. For the
convenience of the reader we divide the further proof into a series of
claims.
Claim 5.1. The semigroup λ(X) is sub-Clifford or regular.
Proof. If λ(X) is not sub-Clifford, then it is regular in the semigroup
N2(X) according to our assumption. We claim that λ(X) is regular. Given
any maximal linked system A ∈ λ(X), use the regularity of λ(X) in N2(X)
to find a linked upfamily B ∈ N2(X) such that A = A ∗ B ∗A. Enlarge B
to a maximal linked upfamily B˜ ∈ λ(X). Then A = A∗B ∗A ⊂ A∗ B˜ ∗A
implies that A = A ∗ B˜ ∗ A by the maximality of the linked family A.
Therefore A is regular in λ(X).
Claim 5.2. The semigroup X is inverse.
Proof. Since the idempotents of the semigroup X commute, it suffices to
check that X is regular. By our assumption, the semigroup X is regular or
sub-Clifford. If λ(X) is regular, then X is regular by Proposition 3.1. Now
assume that λ(X) is sub-Clifford. Then so is the semigroup X. Since the
idempotents of the semigroup β(X) ⊂ λ(X) commute, by Proposition 2.1,
each cyclic subsemigroup {xn}n∈N of S is finite and hence is a group by
the sub-Clifford property of X. Then the semigroup X is Clifford and
hence regular.
Claim 5.3. The semilattice E ⊂ X is linear and finite.
Proof. Assuming that E is not linear, we can find two idempotents x, y ∈
E such that xy /∈ {x, y}. Now consider the maximal linked system L =
〈{x, y}, {x, xy}, {y, xy}〉. It can be shown that L 6= L ∗ L = 〈{xy}〉 =
L ∗ L ∗ L, which is not possible if the semigroup λ(X) is sub-Clifford.
Next, we show that the element L is not regular in the semigroup
υ(X), which is not possible if the semigroup λ(X) is regular. Assuming
that L is regular, find an upfamily A ∈ υ(X) such that L ∗ A ∗ L = L. It
follows from {x, y} ∈ L = L∗A∗L that {x, y} ⊃ ⋃u∈L u∗Bu for some set
L ∈ L and some family {Bu}u∈L ⊂ A ∗ L. The linked property of family
L implies that the intersection L ∩ {x, xy} contains some point u. Now
for the set Bu ∈ A ∗ L find a set A ∈ A and a family {La}a∈A ⊂ L such
that Bu ⊃ ⋃a∈A a ∗La. Fix any point a ∈ A and a point v ∈ La ∩{y, xy}.
Then uav ∈ uaLa ⊂ uBu ⊂ {x, y}. Since u ∈ {x, xy} and v ∈ {y, xy},
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160 Algebra in superextensions of inverse semigroups
the element uav is equal to xby for some element b ∈ {a, ya, ax, yax}.
So, xby ∈ {x, y}. If xby = x, then xy = xbyy = xby = x ∈ {x, y}. If
xby = y, then xy = xxby = xby = y ∈ {x, y}. In both cases we obtain a
contradiction with the choice of the idempotents x and y.
Since the idempotents of the semigroup β(X) ⊂ λ(X) commute, the
linear semilattice E is finite according to Proposition 2.1.
Since X is an inverse semigroup with finite linear semilattice E, we
can apply Theorem 7.5 of [6] to derive our next claim.
Claim 5.4. The semigroup X is inverse and Clifford.
Since the semigroup X is inverse and Clifford, the idempotents of X
commute with all elements of X; see Theorem II.2.6 in [13].
Claim 5.5. Each subgroup H in X has cardinality |H| ≤ 4.
Proof. We lose no generality assuming that H coincides with the maximal
group He containing the idempotent e of the group H. An upfamily
A ∈ υ(H) is called left invariant if xA = A for any point x ∈ H. By
↔
N2(H) denote the family of all left invariant linked systems on H and
by
↔
λ(H) = max ↔N2(H) the family of all maximal elements of
↔
N2(H).
Elements of
↔
λ(H) are called maximal invariant linked systems. Zorn’s
Lemma guarantees that the set
↔
λ(H) is not empty.
We claim that
↔
λ(H) is a singleton. Assuming the opposite, fix two
distinct maximal invariant linked systems A1,A2 ∈
↔
λ(H). By Proposition
1 of [2], for every i ∈ {1, 2} the set
↑Ai = {L ∈ λ(H) : L ⊃ Ai}
is a left ideal in the compact right-topological semigroup λ(H). By Ellis’
Theorem [7] (see also [11, 2.5]), this left ideal contains an idempotent
Ei ⊃ Ai. The idempotents E1, E2 do not commute because the products
E1∗E2 and E2∗E1 belong to the disjoint left ideals ↑A2 and ↑A1, respectively
(the left ideals ↑A1 and ↑A2 are disjoint by the maximality of the invariant
linked systems A1 and A2).
Since |
↔
λ(H)| = 1, we can apply Theorems 2.2 and 2.6 of [5] and
conclude that the group H either has cardinality |H| ≤ 5 or else H is
isomorphic to the dihedral group D6 or to the group (C2)3.
To finish the proof of Claim 5.5 it remains to show that H is not
isomorphic to the groups C5, D6 or C32 .
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T. Banakh, V. Gavrylkiv 161
C5: If H is isomorphic to the 5-element cyclic group C5, then the
superextension λ(X) contains an isomorphic copy of the semigroup λ(C5).
By [5, §6.4], the semigroup λ(H) ∼= λ(C5) contains two distinct elements
Z,Θ such that L ∗Θ = Z for any maximal linked system L ∈ λ(H). This
implies that the element Θ is not regular in λ(H). We claim that this
element is not regular in λ(X). Assuming the converse, find a maximal
linked system L ∈ λ(X) such that Θ = Θ ∗ L ∗Θ.
Let e be the idempotent of the maximal subgroup H = He of X.
Since X is inverse and Clifford, the idempotent e lies in the center of
the semigroup X, so the shift se : X → eX, se : x 7→ xe = ex, is a
well-defined homomorphism from the semigroup X onto its principal ideal
eX = Xe. Since eΘ = Θ, for the maximal linked system eL ∈ λ(eX) we
get Θ = Θ ∗ eL ∗ Θ, which means that Θ is regular in the semigroup
λ(eX). Since eX \He is an ideal in eX, Corollary 3.4 implies that the
element Θ is regular in the semigroup λ(He), which contradicts the choice
of Θ. So, Θ is not regular in λ(X) and the semigroup λ(X) is not regular.
On the other hand, the property of Θ guarantees that Θ 6= Z = Θ∗Θ =
Θ ∗Θ ∗Θ, which means that the semigroup λ(X) is not sub-Clifford. In
both cases we obtain a contradiction with Claim 5.1.
D6: Next, assume that H is isomorphic to the dihedral group D6.
In this case H contains an element a of order 3 and element b of or-
der 2 such that ba = a2b. Consider the maximal linked systems ∆ =
〈{e, a}, {e, a2}, {a, a2}〉 and Λ = 〈{e, b}, {e, ab}, {e, a, a2}, {a, b, ab},
{a2, b, ab}〉. It is easy to check that ∆ and Λ are two non-commuting idem-
potents in λ(X) (because {e, a, ab} ∈ ∆ ∗ Λ and {e, a, ab} /∈ Λ ∗∆). So,
the semigroups λ(X) ⊃ λ(H) contains two non-commuting idempotents,
which is a contradiction.
C32 : In the case H ∼= C32 fix three elements a, b, c ∈ C32 generating
the group C32 . Consider two maximal linked systems b = 〈{e, a}, {e, b},
{e, ab}, {a, b, ab}〉 and c = 〈{e, a}, {e, c}, {e, ac}, {a, c, ac}〉 and observe
that they are non-commuting idempotents of λ(H) (because {e, c, b, ab} ∈
b ∗c and {e, c, b, ab} /∈ c ∗b).
Claim 5.6. The semigroup λ(X) is inverse.
Proof. By Claim 5.1, the semigroup λ(X) is regular or sub-Clifford. We
claim that λ(X) is regular. If not, then λ(X) is sub-Clifford. Claims 5.3—
5.5 imply that the semigroup X is finite and so is its superextension λ(X).
Being finite and sub-Clifford, the semigroup λ(X) is Clifford and hence
regular. Taking into account that the idempotents of λ(X) commute, we
conclude that the semigroup λ(X) is inverse.
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162 Algebra in superextensions of inverse semigroups
Since the semilattice E is linear, we can write it as E = {e1, . . . , en}
where eiej = ei 6= ej for all 1 ≤ i < j < n. For every i ≤ n by Hi = Hei
denote the maximal subgroup of X that contains the idempotent ei.
By Claim 5.5, each subgroup Hi has cardinality |Hi| ≤ 4 and hence
is commutative. We claim that the semigroup X also is commutative.
Indeed, given any points x, y ∈ X we can find numbers i, j ≤ n such that
x ∈ Hi and y ∈ Hj . We lose no generality assuming that i ≤ j. Then
xy = (xei)y = x(eiy) = (yei)x = yx ∈ Hi, so X is commutative.
Claim 5.7. For any 1 < i < n the maximal subgroup Hi is trivial.
Proof. Assume conversely that the subgroup Hi is not trivial and take
any element a ∈ Hi \ E. Next, consider the maximal linked system
∆ = 〈{ei−1, a}, {a, ei+1}, {ei−1, ei+1}〉. We claim that ∆ is not regular in
λ(X). Assume conversely that ∆ is regular in λ(X). Using Corollary 3.4,
we can show that ∆ is regular in the semigroup λ(Hi−1 ∪ Hi ∪ Hi+1).
Then we can find a maximal linked system F ∈ λ(Hi−1 ∪ Hi ∪ Hi+1)
such that ∆ = ∆ ∗ F ∗ ∆. For the set {a, ei+1} ∈ ∆, find a set A ∈
∆ ∗ F with A ⊂ Hi−1 ∪Hi ∪Hi+1 and a family {Da}a∈A ⊂ ∆ such that
{a, ei+1} ⊃
⋃
a∈A a ∗Da. Observe that such an inclusion is possible only
if Da = {a, ei+1} for all a ∈ A. But then A ∗ {a, ei+1} ⊂ {a, ei+1} implies
A = {ei+1} and ∆ ∗ F = 〈ei+1〉 which is not possible.
Claim 5.8. If n ≥ 2, then |Hn| ≤ 2.
Proof. Assume conversely that |Hn| > 2. Then two cases are possible.
1. The group Hn is cyclic. Fix a generator a of the cyclic group
Hn and consider the maximal linked system ∆ = 〈{a, en−1}, {a, en},
{en−1, en}〉. We claim that the element ∆ is not regular in the semigroup
λ(X). Assuming the opposite, we can find a maximal linked system
F ∈ λ(X) with ∆ ∗ F ∗ ∆ = ∆. Then for the set {en, a} ∈ ∆ we can
find a set A ∈ ∆ ∗ F and a family {Da}a∈A of minimal subsets of ∆ such
that {en, a} ⊃ ⋃a∈A a∗Da. This inclusion is possible only if {a, en} = Da
for all a ∈ A. The inclusion A ∗ {a, en} ⊂ {a, en} implies that A = {en}.
Now for the set A ∈ ∆ ∗ F , find a minimal set D ∈ ∆ and a family
{Fd}d∈D ⊂ F such that
⋃
d∈D d ∗ Fd ⊂ A = {en}. This inclusion is
possible only if {a, en} ⊂ D ⊂ Hn and Ad = {d−1} ⊂ Hn for each d ∈ D.
Then the family A contains two disjoint sets {en} and {a−1} which is not
possible as A is linked.
2. The group Hn is isomorphic to the group C2 × C2. Then we can
take two distinct elements a, b generating the group Hn, and consider the
maximal linked system  = 〈{en−1, a}, {en−1, b}, {en−1, ab}, {a, b, ab}〉.
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T. Banakh, V. Gavrylkiv 163
We claim that the element  is not regular in the semigroup λ(X).
Assuming the opposite, find a maximal linked system F ∈ λ(X) with
∗F ∗ = . Then for the set {a, b, ab} ∈  we can find a set A ∈ ∗F
and a family {Dx}x∈A of minimal subsets of  such that {a, b, ab} ⊂
⋃
x∈A x ∗ Dx. This inclusion is possible only if Dx = {a, b, ab} for all
x ∈ A. The inclusion A∗{a, b, ab} = ⋃x∈A x∗Dx ⊂ {a, b, ab} implies that
A = {en}. Now for the set A ∈  ∗ F , find a minimal set S ∈  and a
family {Fs}s∈S ⊂ F such that ⋃s∈S s ∗ Fs ⊂ A = {en}. This inclusion is
possible only if S = {a, b, ab} and Fs = {s} ⊂ Hn for each d ∈ D. Then
the family F contains disjoint sets {s}, s ∈ S, which is not possible as F
is linked.
Claim 5.9. If n ≥ 3, then the group Hn is trivial.
Proof. Assume that Hn is not trivial. By Claim 5.8, |Hn| = 2. Fix a
generator a of the cyclic group Hn and observe that the maximal linked
systems  = 〈{en−2, a}, {en−2, en−1}, {en−2, en}, {a, en−1, en}〉 and ∆ =
〈{en−1, a}, {en−1, en}, {a, en}〉 are non-commuting idempotents of the se-
migroup λ(X) because
 ∗∆ = 〈{en−1, en−2}, {en−1, a}, {en−1, en}, {en−2, a, en}〉 6=  = ∆ ∗.
Claim 5.10. If n ≥ 2, then |H1| ≤ 2.
Proof. Assume that |H1| > 2 and chose two distinct elements a, b ∈ H1\E.
We claim that the maximal linked system
∆ = 〈{a, b}, {a, e2}, {b, e2}
〉
is not regular element of λ(X). Assuming the opposite, find a maximal
linked system F ∈ λ(X) such that ∆ ∗ F ∗ ∆ = ∆. Replacing F by
e2 ∗ F , if necessary, we can assume that F ∈ λ(H1 ∪ H2). For the set
{a, e2} ∈ ∆, find a set A ∈ ∆ ∗ F and a family {Dx}x∈A ⊂ ∆ such that
⋃
x∈A x ∗ Dx ⊂ {a, e2}. This inclusion implies that A = {e2} ∈ ∆ ∗ F ,
which is not possible.
Now we are able to finish the proof of Theorem 1.1. If n = |E| ≥ 3,
then the semigroup X is isomorphic to Ln or to C2⊔Ln−1 by Claims 5.7—
5.10. If n = |E| = 1, then X is a group of cardinality |X| ≤ 4, isomorphic
to one of groups: L1, C2, C3, C4, C2 × C2.
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164 Algebra in superextensions of inverse semigroups
It remains to consider the case |E| = 2. By Claims 5.8, 5.10,
max{|H1|, |H2|} ≤ 2. If |H1| = |H2| = 1, then X ∼= L2. If |H1| = 1 and
|H2| = 2, then X ∼= L1 ⊔C2. If |H1| = 2 and |H2| = 1, then X ∼= C2 ⊔L1.
Finally assume that |H1| = |H2|. For i ∈ {1, 2} let ai be the unique
generator of the 2-element cyclic group Hi.
Claim 5.11. a2 ∗ e1 = a1.
Proof. Assuming that a2 ∗ e1 6= a1, we get a2 ∗ e1 = e1. Then the maximal
linked systems
e = 〈{e1, a1}, {e1, a2}, {e1, e2}, {a1, a2, e2}〉 and
a = 〈{a1, e1}, {a1, e2}, {a1, a2}, {e1, e2, a2}〉
are not commuting idempotents of λ(X) (because e ∗ a = a while
a ∗e = e).
Claim 5.11 implies that the semigroup X = H1 ∪H2 is isomorphic to
L2 × C2.
6. Proof of Theorem 1.2
Given a semigroup X, we need to check the equivalence of the following
statements:
(1) ϕ(X) is a commutative Clifford semigroup;
(2) ϕ(X) is an inverse semigroup;
(3) the idempotents of ϕ(X) commute and ϕ(X) is sub-Clifford or
regular in N2(X);
(4) X is isomorphic to C2, Ln, or Ln ⊔ C2 for some n ∈ ω.
We shall prove the implications (4) ⇒ (1) ⇒ (2) ⇒ (3) ⇒ (4).
The implication (4) ⇒ (1) follows from Propositions 4.7, 4.8 while
(1) ⇒ (2) ⇒ (3) are trivial or well-known; see [13, II.1.2].
To prove that (3) ⇒ (4), assume that idempotents of the semigroup
ϕ(X) commute and ϕ(X) is sub-Clifford or regular in N2(X). Then the
idempotents of the semigroup X commute and thus the set E = {e ∈ X :
ee = e} is a commutative subsemigroup of X. By analogy with Claim 5.2
we can prove:
Claim 6.1. The semigroup X is inverse.
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Claim 6.2. The semilattice E is linear and finite.
Proof. Assuming that the semilattice E is not linear, we can find two
non-commuting idempotents x, y ∈ E. By (the proof of) Theorem 1.1
of [4], the filter F = 〈{x, y}〉 is not a regular element in υ(X) and
F 6= 〈{x, y, xy}〉 = F ∗ F = F ∗ F ∗ F , which is not possible if the
semigroup ϕ(X) is sub-Clifford or regular in N2(X). So, the semilattice
E is linear. Since β(X) ⊂ ϕ(X), Proposition 2.1 implies that the linear
semilattice E is finite.
Since X is an inverse semigroup with finite linear semilattice E, we
can apply Theorem 7.5 of [6] to derive our next claim.
Claim 6.3. The semigroup X is inverse and Clifford.
Since the semigroup X is inverse and Clifford, the idempotents of X
commute with all elements of X; see [13, II.2.6].
Claim 6.4. Each subgroup H in X has cardinality |H| ≤ 2.
Proof. Assume X contains a subgroup H of cardinality |H| > 2. We
lose no generality assuming that the subgroup H coincides with the
maximal subgroup He containing the idempotent e of H. Take any subset
F ⊂ H with |H \ F | = 1 and consider the filter F = 〈F 〉. It follows that
F 6= 〈H〉 = F ∗ F = F ∗ F ∗ F , which is forbidden if the semigroup λ(X)
is sub-Clifford. Next, we show that F is not regular in the semigroup
N2(X) ⊃ ϕ(X). Assuming the opposite, find an upfamily A ∈ N2(X)
such that F = F ∗A∗F . Replacing A by the linked upfamily eA, we can
assume that A ∈ N2(eX). Since eX \He is an ideal in eX, Corollary 3.4
implies that the F is a regular element of the semigroup N2(H) and hence
we can assume that A ∈ N2(H). Then F ∗ A ∈ N2(H) ⊂ N2(X). The
inclusion F ∈ F = F ∗ A ∗ F implies the existence of a set B ∈ F ∗ A,
B ⊂ H, and a family {Fb}b∈B ⊂ F such that
⋃
b∈B b ∗ Fb ⊂ F . Replacing
Fb by the smallest possible set F generating the filter F , we can assume
that Fb = F for all b ∈ B. Then we get B∗F =
⋃
b∈B b∗Fb ⊂ F and hence
B = {e}. Since B ∈ F ∗ A, for the smallest set F ∈ F and each point
x ∈ F we can find a set Ax ⊂ A, Ax ⊂ H, such that ⋃x∈F x ∗Ax ⊂ {e}.
It follows that Ax = {x−1} and hence the family A ⊃ {Ax : x ∈ F} is not
linked, which is a desired contradiction. So, F is not regular in N2(X)
and F 6= F ∗ F = F ∗ F ∗ F , which is not possible if the semigroup λ(X)
is sub-Clifford or regular in N2(X).
By analogy with Claim 5.6 we can prove:
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166 Algebra in superextensions of inverse semigroups
Claim 6.5. The semigroup ϕ(X) is regular in N2(X).
Since the semilattice E is linear, we can write it as E = {e1, . . . , en}
where eiej = ei 6= ej for all 1 ≤ i < j ≤ n. For every i ≤ n by Hi = Hei
denote the maximal subgroup of X that contains the idempotent ei.
By Claim 6.4, each subgroup Hi has cardinality |Hi| ≤ 2 and hence is
commutative. Then the inverse Clifford semigroup X also is commutative.
Claim 6.6. For any 1 ≤ i < n the maximal subgroup Hi is trivial.
Proof. Assume conversely that for some i < n the subgroup Hi is not
trivial and take any element a ∈ Hi \E. Next, consider the filter F = 〈F 〉
generated by the doubleton F = {a, ei+1}. We claim that F is a non-
regular element in the semigroup N2(X), which will contradict Claim 6.5.
Assuming that F is regular in N2(X), we can find a linked upfamily
A ∈ N2(X) such that F = F∗A∗F . Replacing A by ei+1A, we can assume
that A ∈ N2(ei+1X). For the set F = {a, ei+1} ∈ F , find a set B ∈ F ∗A
such that B ∗ F ⊂ F . The latter inclusion implies that B ⊂ Hi ∪Hi+1.
The inclusion B ∈ F ∗ A implies that the intersection B ∩ Hi is not
empty and the inclusion B ∗ F ⊂ F implies that B ∩Hi = {ei} and then
{a, ei+1} = F ⊃ B ∗ F ⊃ {ei} ∗ {a, ei+1} = {a, ei}, which is a desired
contradiction.
Now we are able to finish the proof of Theorem 1.2. By Claim 6.6, all
maximal subgroups Hi, i < n, are trivial. If the group Hn is trivial, then
X = E is isomorphic to the linear semilattice Ln. If Hn is not trivial, then
Hn ∼= C2 by Claim 6.4 and X is isomorphic to the semigroup Ln−1 ⊔ C2.
For n = 1 we get L0 = ∅ and L0 ⊔ C2 = C2.
7. Proof of Theorem 1.3
Given a semigroup X we need to prove the equivalence of the following
statements:
(1) N2(X) is a finite commutative Clifford semigroup;
(2) N2(X) is an inverse semigroup;
(3) idempotents of N2(X) commute and N2(X) is sub-Clifford or regu-
lar;
(4) X is isomorphic to C2 or Ln for some n ∈ ω.
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T. Banakh, V. Gavrylkiv 167
We shall prove the implications (4) ⇒ (1) ⇒ (2) ⇒ (3) ⇒ (4).
The implication (4) ⇒ (1) follows from Propositions 4.7(1) and 4.2
while (1) ⇒ (2) ⇒ (3) are trivial or well-known; see [13, II.1.2].
To prove that (3) ⇒ (4), assume that idempotents of the semigroup
N2(X) commute and N2(X) is sub-Clifford or regular. Then the idempo-
tents of the subsemigroups λ(X) and ϕ(X) of N2(X) also commute and
these semigroups are regular in the semigroup N2(X). By Theorems 1.1
and 1.2, the semigroup X is isomorphic to one of the semigroups C2,
L1 ⊔ C2 or Ln for some n ∈ ω. It remains to prove that X cannot be
isomorphic to L1 ⊔ C2 = {0, 1,−1}.
This follows from the fact that the semigroup N2({0, 1,−1}) contains
two idempotents ∆ = {A ⊂ {0, 1,−1} : |A| ≥ 2} and F = 〈{{0, 1,−1}〉,
which do not commute because ∆ ∗ F = F 6= 〈{0, 1}, {0,−1}〉 = F ∗∆.
8. Proof of Theorem 1.4
Given a semigroup X we need to prove the equivalence of the following
statements:
(1) υ(X) is a finite commutative Clifford semigroup;
(2) υ(X) is an inverse semigroup;
(3) idempotents of υ(X) commute and υ(X) is sub-Clifford or regular;
(4) X is a finite linear semilattice, isomorphic to Ln for some n ∈ ω.
We shall prove the implications (4) ⇒ (1) ⇒ (2) ⇒ (3) ⇒ (4).
The implication (4) ⇒ (1) follows from Proposition 4.2 while (1) ⇒
(2) ⇒ (3) are trivial.
To prove that (3) ⇒ (4), assume that the idempotents of the semigroup
υ(X) commute and υ(X) is sub-Clifford or regular. Then the idempotents
of the semigroup X commute and thus the set E = {e ∈ X : ee = e} is a
commutative subsemigroup of X. By analogy with Claims 5.2—5.4 we
can prove that the semigroup X is inverse and Clifford and the semilattice
E is finite and linear.
Next, we show that each subgroup H of X is trivial. Assume conversely
that X contains a non-trivial subgroup H. Then the filter F = 〈H〉 and
the upfamily U = {A ⊂ H : A 6= ∅} are two non-commuting idempotents
in the semigroup υ(H) ⊂ υ(X) (because F ∗ U = U 6= F = U ∗ F).
Now we see that the inverse Clifford semigroup X contains no non-
trivial subgroups and hence coincides with its maximal semilattice E,
which is finite and linear.
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168 Algebra in superextensions of inverse semigroups
References
[1] T. Banakh, V. Gavrylkiv, Algebra in superextension of groups, II: cancelativity
and centers, Algebra Discr. Math. (2008), No.4, 1–14.
[2] T. Banakh, V. Gavrylkiv, Algebra in superextension of groups: minimal left ideals,
Mat. Stud. 31 (2009), 142–148.
[3] T. Banakh, V. Gavrylkiv, Algebra in the superextensions of twinic groups, Dissert.
Math. 473 (2010), 74pp.
[4] T. Banakh, V. Gavrylkiv, Algebra in superextensions of semilattices, Algebra Discr.
Math. 13 (2012), No.1, 26–42.
[5] T. Banakh, V. Gavrylkiv, O. Nykyforchyn, Algebra in superextensions of groups,
I: zeros and commutativity, Algebra Discr. Math. (2008), No.3, 1–29.
[6] A.H. Clifford, G.B. Preston, The algebraic theory of semigroups. Vol. II., Mathe-
matical Surveys. 7. AMS, Providence, RI, 1967.
[7] R. Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–405.
[8] V. Gavrylkiv, The spaces of inclusion hyperspaces over noncompact spaces, Mat.
Stud. 28:1 (2007), 92–110.
[9] V. Gavrylkiv, Right-topological semigroup operations on inclusion hyperspaces,
Mat. Stud. 29:1 (2008), 18–34.
[10] R. Graham, B. Rothschild, J. Spencer, Ramsey theory, John Wiley & Sons, Inc.,
New York, 1990.
[11] N. Hindman, D. Strauss, Algebra in the Stone-Čech compactification, de Gruyter,
Berlin, New York, 1998.
[12] J. van Mill, Supercompactness and Wallman spaces, Math. Centre Tracts. 85.
Amsterdam: Math. Centrum., 1977.
[13] M. Petrich, Inverse semigroups, John Wiley & Sons, Inc., New York, 1984.
[14] F. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930),
264–286.
[15] A. Verbeek, Superextensions of topological spaces, MC Tract 41, Amsterdam,
1972.
Contact information
T. Banakh Ivan Franko University of Lviv, Ukraine and
Jan Kochanowski University, Kielce, Poland
E-Mail: t.o.banakh@gmail.com
URL: http://www.franko.lviv.ua/faculty/
mechmat/Departments/Topology/bancv.html
V. Gavrylkiv Vasyl Stefanyk Precarpathian National
University, Ivano-Frankivsk, Ukraine
E-Mail: vgavrylkiv@yahoo.com
URL: http://gavrylkiv.pu.if.ua
Received by the editors: 05.10.2011
and in final form 22.12.2011.