Author:
Banakh Taras,Bardyla Serhii,Gutik Oleg
Abstract
AbstractFor a Hausdorff topologized semilattice X its Lawson number$$\bar{\Lambda }(X)$$
Λ
¯
(
X
)
is the smallest cardinal $$\kappa $$
κ
such that for any distinct points $$x,y\in X$$
x
,
y
∈
X
there exists a family $$\mathcal U$$
U
of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$
|
U
|
≤
κ
and $$\bigcap \mathcal U$$
⋂
U
is a subsemilattice of X that does not contain y. It follows that $$\bar{\Lambda }(X)\le \bar{\psi }(X)$$
Λ
¯
(
X
)
≤
ψ
¯
(
X
)
, where $$\bar{\psi }(X)$$
ψ
¯
(
X
)
is the smallest cardinal $$\kappa $$
κ
such that for any point $$x\in X$$
x
∈
X
there exists a family $$\mathcal U$$
U
of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$
|
U
|
≤
κ
and $$\bigcap \mathcal U=\{x\}$$
⋂
U
=
{
x
}
. We prove that a compact Hausdorff semitopological semilattice X is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if $$\bar{\Lambda }(X)=1$$
Λ
¯
(
X
)
=
1
. Each Hausdorff topological semilattice X has Lawson number $$\bar{\Lambda }(X)\le \omega $$
Λ
¯
(
X
)
≤
ω
. On the other hand, for any infinite cardinal $$\lambda $$
λ
we construct a Hausdorff zero-dimensional semitopological semilattice X such that $$|X|=\lambda $$
|
X
|
=
λ
and $$\bar{\Lambda }(X)=\bar{\psi }(X)=\mathrm {cf}(\lambda )$$
Λ
¯
(
X
)
=
ψ
¯
(
X
)
=
cf
(
λ
)
. A topologized semilattice X is called (i) $$\omega $$
ω
-Lawson if $$\bar{\Lambda }(X)\le \omega $$
Λ
¯
(
X
)
≤
ω
; (ii) complete if each non-empty chain $$C\subseteq X$$
C
⊆
X
has $$\inf C\in {\overline{C}}$$
inf
C
∈
C
¯
and $$\sup C\in {\overline{C}}$$
sup
C
∈
C
¯
. We prove that for any complete subsemilattice X of an $$\omega $$
ω
-Lawson semitopological semilattice Y, the partial order $$\le _X=\{(x,y)\in X\times X:xy=x\}$$
≤
X
=
{
(
x
,
y
)
∈
X
×
X
:
x
y
=
x
}
of X is closed in $$Y\times Y$$
Y
×
Y
and hence X is closed in Y. This implies that for any continuous homomorphism $$h:X\rightarrow Y$$
h
:
X
→
Y
from a complete topologized semilattice X to an $$\omega $$
ω
-Lawson semitopological semilattice Y the image h(X) is closed in Y.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference16 articles.
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