Algebra in superextensions of semilattices
Given a semilattice X we study the algebraic properties of the semigroup υ(X) of upfamilies on X. The semigroup υ(X) contains the Stone-ˇCech extension β(X), the superextension λ(X), and the space of filters φ(X) on X as closed subsemigroups. We prove that υ(X) is a semilattice iff λ(X) is a semilat...
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| Cite this: | Algebra in superextensions of semilattices / T. Banakh, V. Gavrylkiv // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 26–42. — Бібліогр.: 14 назв. — англ. |
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| citation_txt | Algebra in superextensions of semilattices / T. Banakh, V. Gavrylkiv // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 26–42. — Бібліогр.: 14 назв. — англ. |
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| description | Given a semilattice X we study the algebraic properties of the semigroup υ(X) of upfamilies on X. The semigroup υ(X) contains the Stone-ˇCech extension β(X), the superextension λ(X), and the space of filters φ(X) on X as closed subsemigroups. We prove that υ(X) is a semilattice iff λ(X) is a semilattice iff φ(X) is a semilattice iff the semilattice X is finite and linearly ordered. We prove that the semigroup β(X) is a band if and only if X has no infinite antichains, and the semigroup λ(X) is commutative if and only if X is a bush with finite branches.
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 13 (2012). Number 1. pp. 26 – 42
c© Journal “Algebra and Discrete Mathematics”
Algebra in superextensions of semilattices
Taras Banakh1 and Volodymyr Gavrylkiv
Communicated by I. V. Protasov
Abstract. Given a semilattice X we study the algebraic
properties of the semigroup υ(X) of upfamilies on X. The semigroup
υ(X) contains the Stone-Čech extension β(X), the superextension
λ(X), and the space of filters ϕ(X) on X as closed subsemigroups.
We prove that υ(X) is a semilattice iff λ(X) is a semilattice iff ϕ(X)
is a semilattice iff the semilattice X is finite and linearly ordered.
We prove that the semigroup β(X) is a band if and only if X has
no infinite antichains, and the semigroup λ(X) is commutative if
and only if X is a bush with finite branches.
Introduction
One of the most powerful tools in the modern Combinatorics of Num-
bers is the method of ultrafilters based on the fact that each (associa-
tive) binary operation ∗ : X × X → X defined on a discrete topo-
logical space X extends to a right-topological (associative) operation
∗ : β(X)× β(X) → β(X) on the Stone-Čech compactification β(X) of X,
see [9], [11]. The Stone-Čech extension β(X) is the space of ultrafilters on
X. The extension of the operation from X to β(X) can be defined by the
simple formula:
U ∗ V =
〈
⋃
x∈U
x∗Vx : U ∈ U , (Vx)x∈U ∈ VU
〉
, (1)
1The first author has been partially financed by NCN means granted by decision
DEC-2011/01/B/ST1/01439
2010 Mathematics Subject Classification: 06A12, 20M10.
Key words and phrases: semilattice, band, commutative semigroup, the space
of upfamilies, the space of filters, the space of maximal linked systems, superextension.
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T. Banakh, V. Gavrylkiv 27
where 〈B〉 = {A ⊂ X : ∃B ∈ B B ⊂ A} is the upper closure of a family
B. In this case B is called a base of 〈B〉. Identifying each point x ∈ X
with the principal ultrafilter 〈x〉 = {U ⊂ X : x ∈ U}, we identify X with
a subspace of the Stone-Čech compactification.
Endowed with this extended operation, the Stone-Čech compactifica-
tion β(X) becomes a compact right-topological semigroup. The algebraic
properties of this semigroup (for example, the existence of idempotents
or minimal left ideals) have important consequences in combinatorics of
numbers, see [9], [11].
In [8] it was observed that the binary operation ∗ extends not only to
β(X) but also to the space υ(X) of all upfamilies on X. By definition, a
family F of non-empty subsets of a discrete space X is called an upfamily
if for any sets A ⊂ B ⊂ X the inclusion A ∈ F implies B ∈ F . The space
υ(X) is a closed subspace of the double power-set P(P(X)) endowed with
the compact Hausdorff topology of the Tychonoff power {0, 1}P(X). In the
papers [7], [8], [1]–[3] the space υ(X) was denoted by G(X) and its elements
were called inclusion hyperspaces2. The extension of a binary operation ∗
from X to υ(X) can be defined in the same way as for ultrafilters, i.e.,
by the formula (1) applied to any two upfamilies U ,V ∈ υ(X). If X is a
semigroup, then υ(X) is a compact Hausdorff right-topological semigroup
containing β(X) as a closed subsemigroup. The algebraic properties of
this semigroups were studied in detail in [8].
The space υ(X) of upfamilies over a discrete space X contains many
interesting subspaces. First we recall some definitions. An upfamily A ∈
υ(X) is defined to be
• a filter if A1 ∩A2 ∈ A for all sets A1, A2 ∈ A;
• an ultrafilter if A = A′ for any filter A′ ∈ υ(X) containing A;
• linked if A ∩B 6= ∅ for any sets A,B ∈ A;
• maximal linked if A = A′ for any linked upfamily A′ ∈ υ(X) con-
taining A.
By ϕ(X), β(X), N2(X), and λ(X) we denote the subspaces of υ(X)
consisting of filters, ultrafilters, linked upfamilies, and maximal linked
upfamilies, respectively. The space λ(X) is called the superextension of
X, see [10], [14]. In [8] it was observed that for a discrete semigroup X
2We decided to change the terminology and notation after discovering the paper
[12, 2.7.4] that discusses monadic properties of the up-set functor υ.
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28 Algebra in superextensions of semilattices
the subspaces ϕ(X), β(X), N2(X), λ(X) are closed subsemigroups of the
semigroup υ(X). The following diagram describes the inclusion relations
between these subspaces of υ(X) (an arrow A → B indicates that A is a
subset of B).
X // β(X)
��
// λ(X)
��
ϕ(X) // N2(X) // υ(X)
In [1] — [4] we studied the properties of the compact right-topological
semigroup υ(X) and its subsemigroups for groups X. In this paper we
shall study the algebraic structure of the semigroups λ(X), ϕ(X), N2(X),
and υ(X) for semilattices X.
Let us recall that a semilattice is a commutative idempotent semigroup.
Idempotent semigroups are called bands. So, in a band each element x
is an idempotent, which means that xx = x. A semigroup S is linear if
xy ∈ {x, y} for any elements x, y ∈ X. It follows that each linear semigroup
S is a band. Each (linear) semilattice is partially (linearly) ordered by the
relation ≤ defined by x ≤ y iff xy = x.
A semigroup S is cancellative if for each element a ∈ S the left shift
la : S → S, la : x 7→ ax, and the right shift ra : S → S, ra : x 7→ xa,
are injective. A semigroup S is called Clifford (resp. sub-Clifford) if S
is a union of groups (resp. of cancellative semigroups). Observe that
a subsemigroup of a sub-Clifford semigroup is sub-Clifford and a finite
semigroup S is Clifford if and only if it is sub-Clifford. It is easy to see
that a semigroup S is sub-Clifford if and only if for every pair of natural
numbers n,m it is (n,m)-Clifford in the sense that for any element x ∈ S
the equality xn+1 = xm+1 implies xn = xm.
A semigroup S is called a regular semigroup if a ∈ aSa for any a ∈ S.
Such a semigroup S is called an inverse semigroup if ab = ba for any
idempotents a, b ∈ S. Observe that each band is a Clifford semigroup and
every Clifford semigroup is sub-Clifford and regular. An inverse semigroup
with a unique idempotent is a group.
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T. Banakh, V. Gavrylkiv 29
These algebraic properties of semigroups relate as follows:
linear
��
sub-Clifford // (1,2)-Clifford
semilattice //
��
band // Clifford
OO
// regular
commutative inverse // Clifford inverse //
OO
inverse
OO
commutative group //
OO
group
OO
In this paper we shall characterize semigroups X whose extensions
υ(X), λ(X), ϕ(X) or N2(X) are bands, linear semigroups, commutative
semigroups, or semilattices. In Section 5 we shall characterize lattices X
whose extensions υ(X), λ(X), ϕ(X) are lattices. The results obtained in
this paper will be applied in the paper [5] devoted to the superextensions
of inverse semigroups.
1. Semigroups whose extensions are bands
In this section we shall characterize semigroups X whose extensions
υ(X), λ(X) or ϕ(X) are bands. Let us recall that a semigroup S is a
(linear) band if xx = x for all x ∈ X (and xy ∈ {x, y} for all x, y ∈ X).
Let us recall that an element a of a semigroup S is regular in S if
a ∈ aSa. It is clear that each idempotent is a regular element.
Theorem 1.1. For a semigroup X the following conditions are equivalent:
(1) X is linear;
(2) υ(X) is a band;
(3) ϕ(X) is a band;
(4) λ(X) is a band.
Proof. (1) ⇒ (2) Assume that the semigroup X is linear. To show that
υ(X) is a band, we should check that A∗A = A for any upfamily A ∈ υ(X).
Since X is linear, for any A ∈ A we get A = A ∗ A ∈ A ∗ A and hence
A ⊂ A ∗ A.
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30 Algebra in superextensions of semilattices
To show that A ⊃ A ∗ A, fix any basic subset B =
⋃
x∈A
x∗Ax ∈ A ∗ A
where A ∈ A and Ax ∈ A for all x ∈ A.
Now we consider two cases.
(i) There is x ∈ A such that xa = a for all a ∈ Ax. In this case
A ∋ Ax = x∗Ax ⊂ B and thus B ∈ A.
(ii) For every x ∈ A there is a ∈ Ax such that xa 6= a and hence
xa = x (as X is linear). In this case A ∋ A ⊂
⋃
x∈A x ∗Ax = B and hence
B ∈ A.
The implications (2) ⇒ (3, 4) are trivial.
(3) ⇒ (1) Assume that ϕ(X) is a band. Then X, being a subsemigroup
of ϕ(X), also is a band. To show that X is linear, take any two points x, y ∈
X and consider the filter F = 〈{x, y}〉 ∈ ϕ(X). Being an idempotent,
the filter F is a regular element of the semigroup υ(X). Consequently,
we can find an upfamily A ∈ υ(X) such that F ∗ A ∗ F = F . It follows
that there are sets Ax, Ay ∈ A such that (xAx ∪ yAy) · {x, y} ⊂ {x, y}.
In particular, for every ax ∈ Ax we get xaxy ∈ {x, y}. If xaxy = x, then
xy = xaxyy = xaxy = x. If xaxy = y, then xy = xxaxy = xaxy = y,
witnessing that the band X is linear.
(4) ⇒ (1) Assume that λ(X) is a band. Then X, being a subsemigroup
of λ(X), is a band as well. Assuming that the band X is not linear, we
can find two points x, y ∈ X such that xy /∈ {x, y}. We claim that
the maximal linked system L = 〈{x, y}, {x, xy}, {y, xy}〉 ∈ λ(X) is not
an idempotent. We shall prove more: the element L is not regular in
the semigroup υ(X). Assuming the converse, we can 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 sets Bu ∈ A ∗ L,
u ∈ L. The linked property of the 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 ∈ LA 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}, 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 points x, y /∈ {x, y}.
Observe that the proof of Theorem 1.1 yields a bit more, namely:
Proposition 1.2. For a band X the following conditions are equivalent:
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T. Banakh, V. Gavrylkiv 31
(1) X is linear;
(2) each element of ϕ(X) is regular in υ(X);
(3) each element of λ(X) is regular in υ(X).
The linearity of a semilattice X can be also characterized via the
(1, 2)-Clifford property of the semigroups ϕ(X) and λ(X).
Theorem 1.3. For a semilattice X the following conditions are equivalent:
(1) X is linear;
(2) ϕ(X) is (1, 2)-Clifford;
(3) λ(X) is (1, 2)-Clifford.
Proof. The implications (1) ⇒ (2, 3) follow from Theorem 1.1 because
each band is a (1, 2)-Clifford semigroup.
(2, 3) ⇒ (1) Assume that the semilattice X is not linear. Then X
contains two elements x, y ∈ X such that yx = xy /∈ {x, y}.
Consider the filter F = 〈{x, y}〉 and observe that F 6= F ∗ F =
〈{x, xy, y}〉 = F ∗ F ∗ F , which means that the semigroup ϕ(X) is not
(1, 2)-Clifford.
To see that the semigroup λ(X) is not (1, 2)-Clifford, consider the
maximal linked system L = 〈{x, y}, {x, xy}, {y, xy}〉 ∈ λ(X) and observe
that L 6= L ∗ L = 〈{xy}〉 = L ∗ L ∗ L.
Next we characterize semigroups X whose Stone-Čech extension β(X)
is a band. A sequence (xn)n∈ω of points of some set X is called injective
if xn 6= xm for any distinct numbers n,m ∈ ω.
Theorem 1.4. For a band X the semigroup β(X) is a band if and only
if for each injective sequence (xn)n∈ω in X there are numbers n < m such
that xnxm ∈ {xn, xm}.
Proof. To prove the “only if” part, assume that (xn)n∈ω is an injective
sequence in X such that xnxm /∈ {xn, xm} for all n < m. We claim that
there is an infinite subset Ω ⊂ ω such that xnxm 6= xk for any numbers
n,m, k ∈ Ω with n < m. For this we shall apply the famous Ramsey
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32 Algebra in superextensions of semilattices
Theorem. Consider the 4-coloring χ : [ω]3 → 4 = {0, 1, 2, 3} of the set
[ω]3 = {(k, n,m) ∈ ω3 : k < n < m}, defined by
χ(k, n,m) =
1 if xkxn = xm,
2 if xkxm = xn,
3 if xnxm = xk,
0 otherwise.
By the Ramsey Theorem [11, 5.1], there is an infinite set Ω ⊂ ω such that
χ(Ω3 ∩ [ω]3) is a singleton. It follows from the definition of the coloring Ω
that this singleton is {0}, which means that for any numbers k, n,m ∈ Ω
with n < m and k /∈ {n,m} we get xnxm 6= xk. Since xnxm /∈ {xn, xm}
for any numbers n < m, we conclude that xnxm 6= xk for any numbers
k, n,m ∈ Ω with n < m.
Now take any free ultrafilter A that contains the set A = {xn}n∈Ω.
Then for every n ∈ ω the set A>n = {xm : n < m ∈ Ω} belongs to the
ultrafilter A. The choice of the sequence A = {xn}n∈Ω guarantees that
A ∩
⋃
n∈Ω xn ∗ A>n = ∅, which implies that A 6= A ∗ A and hence the
ultrafilter A is not an idempotent in β(X).
To prove the “if” part, assume that β(X) is not a band and find an
ultrafilter F ∈ β(X) with F ∗ F 6= F . In particular, F ∗ F * F . This
implies that for some A ∈ F and {Ax}x∈A ⊂ F the set
⋃
x∈A x∗Ax /∈ F .
Consider the set X↑
F = {x ∈ X : ↑x ∈ F} where ↑x = {y ∈ X :
xy = x}. We claim that X↑
F /∈ F . Assuming that X↑
F ∈ F , we conclude
that A ∩ X↑
F ∈ F . This implies that ↑a ∈ F and ↑a ∩ Aa ∈ F for any
a ∈ A ∩X↑
F . Therefore a ∗ (↑a ∩Aa) = {a} and hence
⋃
x∈A
x ∗Ax ⊃
⋃
x∈A∩X
↑
F
x ∗ (↑x ∩Ax) =
⋃
x∈A∩X
↑
F
{x} = A ∩X↑
F ∈ F .
Thus
⋃
x∈A x∗Ax ∈ F . This contradiction shows that X↑
F /∈ F .
Next, consider the set X↓
F = {x ∈ X : ↓x ∈ F} where ↓x = {y ∈ X :
xy = y}. We claim that X↓
F /∈ F . Assume that X↓
F ∈ F . Then A∩X↓
F ∈ F .
This implies that ↓a ∈ F and ↓a∩Aa ∈ F for any a ∈ A∩X↓
F . Therefore
↓a ∩Aa ⊂ a ∗ (↓a ∩Aa) ⊂ a ∗Aa ⊂
⋃
x∈A
x ∗Ax
and
⋃
x∈A x∗Ax ∈ F . This contradiction shows that X↓
F /∈ F .
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T. Banakh, V. Gavrylkiv 33
Since F is an ultrafilter, X↑
F ∪X↓
F /∈ F and ZF = X \ (X↑
F ∪X↓
F ) ∈ F .
Let x0 ∈ ZF be arbitrary and by induction, for every n ∈ ω choose a point
xn+1 ∈ ZF \
⋃
i≤n(↑xi ∪ ↓xi) ∈ F . Then the injective sequence (xn)n∈ω
has the required property: xnxm /∈ {xn, xm} for n < m (which follows
from xm /∈ ↓xn ∪ ↑xn).
A subset A of a semigroup X is called an antichain if ab /∈ {a, b} for any
distinct points a, b ∈ A. Theorem implies the following characterization:
Corollary 1.5. For a semilattice X the semigroup β(X) is a band if and
only if each antichain in X is finite.
2. Semilattices whose extensions are commutative
In this section we recognize the structure of semilattices X whose
extensions υ(X), N2(X) or λ(X) are commutative.
Commutative semigroups of ultrafilters were characterized in [9, 4.27]
as follows:
Theorem 2.1. The Stone-Čech extension β(X) of a semigroup X is not
commutative if and only if there are sequences (xn)n∈ω and (yn)n∈ω in X
such that {xkyn : k < n} ∩ {ykxn : k < n} = ∅.
This characterization implies the following (well-known) fact:
Corollary 2.2. If the Stone-Čech extension β(X) of a semigroup X is
commutative, then each linear subsemigroup in X in finite.
Proof. Assume conversely that X contains an infinite linear subsemilattice
L. Using Ramsey’s Theorem, we can find an injective sequence (zn)n∈ω
in L such that either znzm = zn for all n < m or else znzm = zm for all
n < m. Put xn = z2n and yn = z2n+1 for n ∈ ω. Applying Theorem 2.1
to the sequences (xn)n∈ω and (yn)n∈ω we conclude that the semigroup
β(L) is not commutative. Then β(X) is not commutative either.
In spite of Theorem 2.1 the following problem seems to be open.
Problem 2.3. Describe the structure of a semilattice X whose Stone-Čech
extension β(X) is commutative.
A similar problem on commutativity of semigroups υ(X) also is open:
Problem 2.4. Characterize semigroups X whose extension υ(X) is com-
mutative.
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34 Algebra in superextensions of semilattices
We shall resolve this problem for bands. First we prove a useful result
on multiplication of upfamilies on linear semigroups.
For a semigroup X denote by υ•(X) the subsemigroup of υ(X) con-
sisting of all upfamilies A ∈ υ(X) such that for each set A ∈ A there is a
finite subset F ∈ A with F ⊂ A.
For a semigroup X and two upfamilies A,B ∈ υ(X) let
A⊗ B = 〈A ∗B : A ∈ A, B ∈ B〉.
It is clear that A⊗B ⊂ A ∗ B. In the following theorem we show that for
finite linear semigroups the converse inclusion also holds.
Theorem 2.5. If X is a linear semilattice, then A ∗ B = A⊗ B for any
upfamilies A ∈ υ•(X) and B ∈ υ(X).
Proof. On the semilattice X consider the linear order ≤ defined by: x ≤ y
iff yx = x. For a subset A ⊂ X and a point x ∈ X we write A ≤ x if a ≤ x
for all a ∈ A. It follows from the definition of the semigroup operation
∗ on υ(X) that A ⊗ B ⊂ A ∗ B. To prove the reverse inclusion, fix any
basic set C =
⋃
a∈A a∗Ba ∈ A ∗ B where A ∈ A and Ba ∈ B for all a ∈ A.
Since A ∈ υ•(X), we can assume that the set A is finite and hence can
be enumerated as A = {a1, . . . , an} where ai ≤ ai+1 for all i < n. Now let
us consider two cases.
1. There is i ≤ n such that Bai ≤ ai, which means that aib = b for all
b ∈ Bai and hence ai∗Bai = Bai . For every j ≥ i the inequality Bai ≤ ai ≤
aj implies aj ∗Bai = Bai . Consequently, A ∗Bai ⊂ {a1, . . . , ai−1} ∪Bai .
We can assume that i is the smallest number such that Bai ≤ ai. In
this case the minimality of i implies that Baj 6≤ aj for all j < i. This means
bj 6≤ aj for some bj ∈ Baj and hence ajbj = aj (as ajbj ∈ {aj , bj} and
ajbj 6= bj). Then aj∗Baj ∋ ajbj = aj and thus A ∗Bai ⊂ {a1, . . . , ai−1} ∪
Bai ⊂
⋃n
j=1 ajBaj , which implies that C ∈ A⊗ B.
2. Bai 6≤ ai for all i ≤ n. In this case ai ∈ ai ∗ Bai for all i. Observe
that for any b ∈ Ban and i ≤ n we get aib ∈ {ai, b} by the linearity of X.
If aib 6= ai, then aib = b and aib = b = anaib = anb ∈ anBan . So,
A⊗ B ∋ A ∗Ban ⊂ {a1, . . . , an} ∪ anBan ⊂
n
⋃
i=1
aiBai = C
and hence C ∈ A⊗ B.
Now we are able to characterize bands X with commutative extensions
υ(X) and N2(X).
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T. Banakh, V. Gavrylkiv 35
Theorem 2.6. For a band X the following conditions are equivalent:
(1) X is a finite linear semilattice;
(2) the semigroup υ(X) is commutative;
(3) the semigroup N2(X) is commutative;
(4) the semigroup λ(X) is commutative and (1, 2)-Clifford.
Proof. The implication (1) ⇒ (2) follows from Theorem 2.5 as A ∗ B =
A⊗ B = B ⊗A = B ∗ A for every A,B ∈ υ•(X) = υ(X).
The implication (2) ⇒ (3) is trivial.
(3) ⇒ (1) Assume that the semigroup N2(X) is commutative. Then
so is the semigroup X. Being a commutative band, the semigroup X
is a semilattice. Assuming that X is not linear, we can find two points
x, y ∈ X with xy /∈ {x, y}. It can be shown that the linked upfamilies
A = 〈{x, y}〉 and B = 〈{x, xy}, {y, xy}〉 ∈ N2(X) do not commute because
{xy} ∈ A∗B\B∗A. Therefore,X is a linear semilattice. Since β(X) ⊂ υ(X)
is commutative, Corollary 2.2 implies that the linear semilattice X is finite.
(1) ⇔ (4) If X is a finite linear semilattice, then λ(X) is commutative
by the implication (1) ⇒ (2) of this theorem and is (1, 2)-Clifford by
Theorem 1.3.
If the semigroup λ(X) is commutative and (1, 2)-Clifford, then the
semigroup X ⊂ λ(X) is commutative and by Theorem 1.3, X is linear.
By Corollary 2.2, the linear semilattice X is finite.
Now we shall characterize semilattices X with commutative superex-
tension λ(X). A semilattice X is called a bush if for any maximal linear
subsemilattices A,B ⊂ X the product A ∗ B is the singleton {minX}
containing the smallest element minX of X. This definition implies that
A ∩ B = A ∗ B = {minX}. By a branch of a bush X we understand a
maximal linear subsemilattice of X.
Theorem 2.7. A semilattice X has commutative superextension λ(X) if
and only if X is a bush with finite branches.
Proof. First assume that X is a bush with finite branches, and take any
two maximal linked systems A,B ∈ λ(X). Since the products A ∗ B and
B∗A are maximal linked upfamilies, the equality A∗B = B∗A will follow
as soon as we check that any two basic sets CAB =
⋃
a∈A a∗Ba ∈ A ∗ B
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36 Algebra in superextensions of semilattices
and CBA =
⋃
b∈B b∗Ab ∈ B ∗A have non-empty intersection. Here A ∈ A,
(Ba)a∈A ∈ BA, B ∈ B, and (Ab)b∈B ∈ AB. Assume conversely that
CAB ∩ CBA = ∅. Then either minX /∈ CAB or minX /∈ CBA.
Without loss of generality, minX /∈ CAB. Then minX /∈ A and for
each a ∈ A the set {a} ∪ Ba lies in a branch of X. Since branches of X
meet only at the point minX, all the sets {a} ∪ Ba, a ∈ A, lie in the
same (finite) branch. Repeating the argument of Theorem 2.5, we can
show that CAB ⊃ AB′ for some set B′ ∈ B. Since B is linked, there is a
point b ∈ B ∩B′. By the same reason, there is a point a ∈ A ∩Ab. Then
ab = ba ∈ AB′ ∩ bAb ⊂ CAB ∩ CBA and we are done.
Now assume that X is a semilattice with commutative superextension
λ(X). Corollary 2.2 implies that all branches of X are finite. We claim
that for every z ∈ X the lower set ↓z = {x ∈ X : xz = x} is linear.
Assuming the converse, find two points x, y ∈ ↓z such that xy /∈ {x, y}.
It follows that the points x, y, z, xy are pairwise distinct. It is easy to
check that the maximal linked upfamilies A = 〈{x, y}, {x, z}, {y, z}〉 and
B = 〈{x, y}, {x, xy}, {y, xy}〉 do not commute because {x, y} ∈ B∗A\A∗B.
Thus ↓z is linear for every z ∈ X, which means that X is a tree.
Assuming that the tree X is not a bush, we can find two points x, y ∈ X
such that xy /∈ {x, y, z} where z = minX. Now consider the maximal
linked systems A = 〈{x, y}, {x, z}, {y, z}〉, B = 〈{x, y}, {x, xy}, {y, xy}〉
and observe that they do not commute as {xy} ∈ A ∗ B misses the set
{x, y, z} ∈ B ∗ A.
3. Semigroups whose extensions are semilattices
In this section we shall characterize semigroups X whose extensions
υ(X), λ(X), ϕ(X), or N2(X) are semilattices.
Theorem 3.1. For a semigroup X the following conditions are equivalent:
(1) X is a finite linear semilattice;
(2) υ(X) is a semilattice;
(3) λ(X) is a semilattice;
(4) ϕ(X) is a semilattice.
Proof. (1) ⇒ (2) If X is a finite linear semilattice, then υ(X) is a semi-
lattice (=commutative band) by Theorems 1.1 and 2.6.
The implications (2) ⇒ (3, 4) are trivial.
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T. Banakh, V. Gavrylkiv 37
The implication (3) ⇒ (1) follows from Theorems 1.1 and 2.7.
(4) ⇒ (1) Assume that ϕ(X) is a semilattice. Then X, being a sub-
semigroup of the commutative semigroup ϕ(X) is commutative. Since
ϕ(X) is a band, X is a linear semigroup by Theorem 1.1. Thus X, being
a commutative linear semigroup, is a linear semilattice. Since the subsemi-
group β(X) ⊂ λ(X) is commutative, the linear semilattice X is finite by
Corollary 2.2.
4. Semigroups whose extensions are linear
In this section we characterize semigroups X whose extensions υ(X),
λ(X) or ϕ(X) are linear semigroups.
A semigroup S is called a semigroup of left (right) zeros if xy = x
(resp. xy = y) for all x, y ∈ X.
Theorem 4.1. For a semigroup X the semigroup υ(X) is linear if and
only if X is either a semigroup of right zeros or a semigroup of left zeros.
Proof. If X is a semigroup of left zeros, then for any upfamilies A,B ∈ υ(X)
and any basic element
⋃
x∈A xBx ∈ A∗B we get
⋃
x∈A xBx =
⋃
x∈A{x} =
A and thus A ∗ B ⊂ A. On the other hand, each A ∈ A belongs to A ∗ B
as A = A ∗B ∈ A ∗ B for any B ∈ B. Thus A ∗ B = A and the semigroup
υ(X) is a semigroup of left zeros.
If X is a semigroup of right zeros, then so is the semigroup υ(X). In
both cases, υ(X) is linear.
Now assume that the semigroup υ(X) is linear. Then X, being a
subsemigroup of υ(X), also is linear. Let x, y be any two distinct elements
of X. First we prove that xy 6= yx. Assume conversely that xy = yx. Then
xy = yx ∈ {x, y} and we lose no generality assuming that xy = x. Now
consider two upfamilies A = 〈{x, y}〉 and B = 〈{x}, {y}〉 and observe that
B ∗ A = 〈{xx, xy}, {yx, yy}〉 = 〈{x}, {x, y}〉 = 〈{x}〉 /∈ {A,B},
so υ(X) is not linear and this is a required contradiction.
Thus xy 6= yx for all distinct points x, y ∈ X. We call a pair (x, y) ∈ X2
left if xy = x and yx = y and right if xy = y and yx = x. Since X is
linear, each pair (x, y) ∈ X2 is either left or right. We claim that either
all pairs (x, y) ∈ X2 are left or else all such pairs are right. Assuming the
opposite, find pairs (x, y), (a, b) ∈ X2 such that (x, y) is not left and (a, b)
is not right. Then x 6= y, a 6= b and the pair (x, y) is right while (a, b) is
left. Consider the filters A = 〈{x, a}〉 and B = 〈{y, b}〉 and observe that
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38 Algebra in superextensions of semilattices
A ∗ B = 〈{xy, xb, ay, ab}〉 = 〈{y, xb, ay, a}〉. Since υ(X) is linear, either
A ∗ B = A or A ∗ B = B. In the first case {x, a} ⊃ {y, xb, ay, a} ⊃ {y, a}
and hence y = a. In the second case, {y, a} ⊂ {y, b} and thus a = y.
Now consider the filters C = 〈{x, b}〉 and D = 〈{a}〉 and observe that
C ∗ D = 〈{xa, ba}〉 = 〈{xy, b}〉 = 〈{y, b}〉 = 〈{a, b}〉 /∈ {C,D}, which
contradicts the linearity of υ(X).
Therefore either each pair (x, y) ∈ X2 is left and then X is a semigroup
of left zeros or else each pair (x, y) ∈ X2 is right and then X is a semigroup
of right zeros.
Theorem 4.2. For a semigroup X the following conditions are equivalent:
(1) the semigroup ϕ(X) is linear;
(2) the semigroup N2(X) is linear;
(3) either X is a semigroup of left zeros or X is a semigroup of right
zeros or else X is a semilattice of order |X| ≤ 2.
Proof. (3) ⇒ (2) If |X| = 1, then N2(X) is a singleton and hence is a
linear semigroup. If X is a semilattice of order |X| = 2, then X = {0, 1}
for some elements 0, 1 with 0 · 1 = 1 · 0 = 0. In this case N2(X) = ϕ(X) is
a 3-element linear semilattice ordered as:
〈{0}〉 ≤ 〈{0, 1}〉 ≤ 〈{1}〉.
If X is a semigroup of left or right zeros, then the semigroup υ(X) is
linear by Theorem 4.1 and so is its subsemigroup N2(X).
(2) ⇒ (1) Is the semigroup N2(X) is linear, then so is its subsemigroup
ϕ(X).
(1) ⇒ (3) Assume that the semigroup ϕ(X) is linear. Then X, being
a subsemigroup of ϕ(X), is linear as well. If |X| ≤ 2, then either X is
a linear semilattice or a semigroup of left or right zeros. So, we assume
that |X| ≥ 3. We claim that distinct elements x, y ∈ X do not commute.
Assume conversely that xy = yx for some distinct elements x, y ∈ X.
Since xy = yx ∈ {x, y} we lose no generality assuming that xy = yx = x.
Fix any element z ∈ X \ {x, y}. Now consider 3 cases:
1. zx = z. In this case we can consider the filters A = 〈{z, y}〉 and
B = 〈{x, y}〉 and observe that
A ∗ B = 〈{zx, yx, zy, yy}〉 = 〈{z, x, zy, y}〉 /∈ {A,B},
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T. Banakh, V. Gavrylkiv 39
which contradicts the linearity of ϕ(X).
2. zx = x and zy = z. In this case we can consider the filters A =
〈{z, y}〉 and B = 〈{x, y}〉 and observe that A ∗ B = 〈{zx, yx, zy, yy}〉 =
〈{x, x, z, y}〉 /∈ {A,B}, which contradicts the linearity of ϕ(X).
3. zx = x and zy = y. In this case we can consider the filters A =
〈{x, z}〉 and B = 〈{y, z}〉 and observe that A ∗ B = 〈{xy, xz, zy, zz}}〉 =
〈{x, xz, y, z}〉 /∈ {A,B}, which again contradicts the linearity of ϕ(X).
Those contradictions show that distinct elements of X do not commute.
Continuing as in the proof of Theorem 4.1, we can show that X is a
semigroup of right or left zeros.
Finally, we characterize commutative semigroups with linear superex-
tensions.
Theorem 4.3. For a commutative semigroup X the semigroup λ(X) is
linear if and only if X is a linear semilattice of order |X| ≤ 3.
Proof. If X is a linear semilattice of order |X| ≤ 2, then the semigroup
λ(X) = X is linear.
If X is a linear semilattice of order |X| = 3, then X can be identified
with the set 3 = {0, 1, 2} endowed with the operation xy = min{x, y}. The
semigroup λ(X) contains 4 elements: 0, 1, 2 and ∆ = {A ⊂ 3 : |A| ≥ 2}.
One can check that λ(3) is a linear semilattice ordered as follows:
0 ≤ ∆ ≤ 1 ≤ 2.
This proves the “if” part of the theorem. To prove the “only if” part,
we first shall analyze the structure of the superextension λ(4) of the
semilattice 4 = {0, 1, 2, 3} endowed with the operation xy = min{x, y}.
By Theorem 3.1, λ(4) is a semilattice. It contains 12 elements:
〈k〉, ∆k = 〈{A ⊂ n : |A| = 2, k /∈ A} and
�k = 〈{n \ {k}, A : A ⊂ n, |A| = 2, k ∈ A}〉 where k ∈ 4.
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40 Algebra in superextensions of semilattices
The order structure of the semilattice λ(4) is described in the following
diagram:
〈3〉
�3
OO
∆1
==
∆2
OO
∆0
aa
�0
OO ==
�2
aa ==
�1
OOaa
〈2〉
OO
∆3
OO
XX FF
〈1〉
OO
〈0〉
OO
Looking at this diagram we see that the semilattice λ(4) is not linear.
Now assume that X is a commutative semigroup whose superextension
λ(X) is linear. Then X is a linear semilattice. If |X| > 3, then λ(X) is
not linear as it contains a subsemigroup isomorphic to the semilattice
λ(4), which is not linear.
5. Lattices whose extensions are lattices
In this section we characterize lattices whose extensions υ(X), λ(X)
or ϕ(X) are lattices.
A lattice is a set X endowed with two semilattice operations ∧,∨ :
X ×X → X such that (x∧ y)∨ y = y and (x∨ y)∧ y = y for all x, y ∈ X.
Both operations ∧ and ∨ of a lattice X can be extended to right-
topological operations ∧ and ∨ on the compact Hausdorff space υ(X). Is
it natural to ask if the triple (υ(X),∧,∨) is a lattice.
A lattice will be called linear if x ∧ y, x ∨ y ∈ {x, y} for all x, y ∈ X.
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T. Banakh, V. Gavrylkiv 41
Theorem 5.1. For a lattice X the following conditions are equivalent:
(1) X is a linear lattice of order |X| ≤ 2.
(2) υ(X) is a lattice;
(3) λ(X) is a lattice;
(4) ϕ(X) is a lattice.
Proof. (1) ⇒ (2) If X is a linear lattice of order |X| = 1, then υ(X) = X is
a trivial lattice. If X is a linear lattice of order 2, then X can be identified
with the lattice 2 = {0, 1} endowed with the operations x∧ y = min{x, y}
and x∨ y = max{x, y}. In this case λ(2) = β(2) coincides with the lattice
2, ϕ(2) = {〈{0}〉, 〈{0, 1}〉, 〈{1}〉} is a 3-element lattice, isomorphic to
the lattice 3 = {0, 1, 2} endowed with the operations min and max, and
υ(2) =
{
〈{0}〉, 〈{0, 1}〉, 〈{0}, {1}〉, 〈{1}〉
}
is a 4-element lattice isomorphic
to the lattice {0, 1}2.
The implications (2) ⇒ (3, 4) are trivial.
(3, 4) ⇒ (1) Assume that λ(X) or ϕ(X) is a lattice. By Theorem 3.1,
the lattice X is finite and linear. We claim that |X| ≤ 2. Assuming the
converse, we conclude that the lattice X contains a sublattice isomorphic
to the lattice (3,min,max).
Consider the maximal linked upfamily ∆ = {A ⊂ 3 : |A| ≥ 2} and
observe that max{∆, 〈1〉} = 〈1〉 = min{∆, 〈1〉}, which implies that λ(3)
is not a lattice and then λ(X) also is not a lattice.
Next, consider the filters A = 〈{0, 1, 2}〉 and B = 〈{0, 2}〉 and observe
that max{A,B} = A = min{A,B} implying that ϕ(3) is not a lattice and
then ϕ(X) also cannot be a lattice.
References
[1] T. Banakh, V. Gavrylkiv, O. Nykyforchyn, Algebra in superextensions of groups,
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[2] T. Banakh, V. Gavrylkiv. Algebra in superextension of groups, II: cancelativity
and centers, Algebra Discrete Math. (2008), No.4, 1–14.
[3] T. Banakh, V. Gavrylkiv. Algebra in superextension of groups: the minimal ideal
of λ(G), Mat. Stud. 31 (2009), 142–148.
[4] T. Banakh, V. Gavrylkiv. Algebra in the superextensions of twinic groups, Dissert.
Math. 473 (2010), 74pp.
[5] T. Banakh, V. Gavrylkiv. Algebra in the superextensions of inverse semigroups,
Algebra Discr. Math. (to appear).
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42 Algebra in superextensions of semilattices
[6] A.H. Clifford, G.B. Preston, The algebraic theory of semigroups. Vol. I., Mathe-
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Contact information
T. Banakh Ivan Franko National 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 19.01.2012.
T. Banakh, V. Gavrylkiv
|
| id | nasplib_isofts_kiev_ua-123456789-152184 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T17:19:46Z |
| publishDate | 2012 |
| publisher | Інститут прикладної математики і механіки НАН України |
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| spelling | Banakh, T. Gavrylkiv, V. 2019-06-08T09:42:17Z 2019-06-08T09:42:17Z 2012 Algebra in superextensions of semilattices / T. Banakh, V. Gavrylkiv // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 26–42. — Бібліогр.: 14 назв. — англ. 1726-3255 2010 Mathematics Subject Classification: 06A12, 20M10. https://nasplib.isofts.kiev.ua/handle/123456789/152184 Given a semilattice X we study the algebraic properties of the semigroup υ(X) of upfamilies on X. The semigroup υ(X) contains the Stone-ˇCech extension β(X), the superextension λ(X), and the space of filters φ(X) on X as closed subsemigroups. We prove that υ(X) is a semilattice iff λ(X) is a semilattice iff φ(X) is a semilattice iff the semilattice X is finite and linearly ordered. We prove that the semigroup β(X) is a band if and only if X has no infinite antichains, and the semigroup λ(X) is commutative if and only if X is a bush with finite branches. The first author has been partially financed by NCN means granted by decision DEC-2011/01/B/ST1/01439. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Algebra in superextensions of semilattices Article published earlier |
| spellingShingle | Algebra in superextensions of semilattices Banakh, T. Gavrylkiv, V. |
| title | Algebra in superextensions of semilattices |
| title_full | Algebra in superextensions of semilattices |
| title_fullStr | Algebra in superextensions of semilattices |
| title_full_unstemmed | Algebra in superextensions of semilattices |
| title_short | Algebra in superextensions of semilattices |
| title_sort | algebra in superextensions of semilattices |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152184 |
| work_keys_str_mv | AT banakht algebrainsuperextensionsofsemilattices AT gavrylkivv algebrainsuperextensionsofsemilattices |