Download Compact topological semilattices

Compact topological semilattices
Oleg Gutik
National University of Lviv
Uniwersytet Jana Kazimierza we Lwowie
Tartu Ülikooli,
14 detsember, 2011
Oleg Gutik
Compact topological semilattices
General Topology
Definition:
A topological space X is called T4 -space, or a normal space, if X is:
T1 -space (i.e., for every pairs of distinct points x1 , x2 ∈ X there exists an
open set U in X such that x1 ∈ U and x2 ∈
/ U ); and
for every pairs of disjoint closed subsets A, B ⊂ X there exist open
disjoint subsets U, V ∈ X such that A ⊆ U and B ⊆ V .
Definition:
A topological space X is called T3 1 -space, or a Tychonoff space, or a
2
completely regular space, if X is:
T1 -space; and
for every x ∈ X and every closed subset F ⊂ X such that x ∈
/ F there
exists a continuous function f : X → I such that f (x) = 0 and f (y) = 1
for all y ∈ F .
Oleg Gutik
Compact topological semilattices
General Topology
Definition:
A topological space X is called T4 -space, or a normal space, if X is:
T1 -space (i.e., for every pairs of distinct points x1 , x2 ∈ X there exists an
open set U in X such that x1 ∈ U and x2 ∈
/ U ); and
for every pairs of disjoint closed subsets A, B ⊂ X there exist open
disjoint subsets U, V ∈ X such that A ⊆ U and B ⊆ V .
Definition:
A topological space X is called T3 1 -space, or a Tychonoff space, or a
2
completely regular space, if X is:
T1 -space; and
for every x ∈ X and every closed subset F ⊂ X such that x ∈
/ F there
exists a continuous function f : X → I such that f (x) = 0 and f (y) = 1
for all y ∈ F .
Oleg Gutik
Compact topological semilattices
General Topology
Urysohn’s Lemma (1925):
For every pairs A and B of disjoint closed subsets of a normal space X there
exists a continuous function f : X → I such that f (x) = 0 for x ∈ A and
f (y) = 1 for y ∈ B.
Definition:
For a topological space X the cardinal number
w(X) = min {|B| : B is a base of X}
is called the weight of X.
Definition:
The Tychonoff
cube of weight m > ℵ0 is the space I m , i.e., the Cartesian
Q
product s∈S Is with Tychonoff topology, where Is = I and |S | = m. The
Tychonoff cube I ℵ0 is called the Hilbert cube.
Tychoff’s Theorem (1930):
The Tychonoff cube I m is universal for all Tychonoff topological spaces of
weight m > ℵ0 .
Oleg Gutik
Compact topological semilattices
General Topology
Urysohn’s Lemma (1925):
For every pairs A and B of disjoint closed subsets of a normal space X there
exists a continuous function f : X → I such that f (x) = 0 for x ∈ A and
f (y) = 1 for y ∈ B.
Definition:
For a topological space X the cardinal number
w(X) = min {|B| : B is a base of X}
is called the weight of X.
Definition:
The Tychonoff
cube of weight m > ℵ0 is the space I m , i.e., the Cartesian
Q
product s∈S Is with Tychonoff topology, where Is = I and |S | = m. The
Tychonoff cube I ℵ0 is called the Hilbert cube.
Tychoff’s Theorem (1930):
The Tychonoff cube I m is universal for all Tychonoff topological spaces of
weight m > ℵ0 .
Oleg Gutik
Compact topological semilattices
General Topology
Urysohn’s Lemma (1925):
For every pairs A and B of disjoint closed subsets of a normal space X there
exists a continuous function f : X → I such that f (x) = 0 for x ∈ A and
f (y) = 1 for y ∈ B.
Definition:
For a topological space X the cardinal number
w(X) = min {|B| : B is a base of X}
is called the weight of X.
Definition:
The Tychonoff
cube of weight m > ℵ0 is the space I m , i.e., the Cartesian
Q
product s∈S Is with Tychonoff topology, where Is = I and |S | = m. The
Tychonoff cube I ℵ0 is called the Hilbert cube.
Tychoff’s Theorem (1930):
The Tychonoff cube I m is universal for all Tychonoff topological spaces of
weight m > ℵ0 .
Oleg Gutik
Compact topological semilattices
General Topology
Urysohn’s Lemma (1925):
For every pairs A and B of disjoint closed subsets of a normal space X there
exists a continuous function f : X → I such that f (x) = 0 for x ∈ A and
f (y) = 1 for y ∈ B.
Definition:
For a topological space X the cardinal number
w(X) = min {|B| : B is a base of X}
is called the weight of X.
Definition:
The Tychonoff
cube of weight m > ℵ0 is the space I m , i.e., the Cartesian
Q
product s∈S Is with Tychonoff topology, where Is = I and |S | = m. The
Tychonoff cube I ℵ0 is called the Hilbert cube.
Tychoff’s Theorem (1930):
The Tychonoff cube I m is universal for all Tychonoff topological spaces of
weight m > ℵ0 .
Oleg Gutik
Compact topological semilattices
Partial Order
Definition:
A binary relation 6 on a non-empty set X is called a partial order if the
following conditions hold:
x 6 x for all x ∈ X;
if x 6 y and y 6 x, then x = y, for x, y ∈ X;
if x 6 y and y 6 z, then x 6 z, for x, y, z ∈ X.
A set X equipped with a partial order 6 is called a partially ordered set and it
will be denoted by (X, 6).
For a partially ordered set (X, 6), x ∈ X and A ⊆ X we denote:
[
[
↓x = {y ∈ X : y 6 x}, ↑x = {y ∈ X : x 6 y}, ↓A =
↓a and ↑A =
↑a.
a∈A
Definition:
A partial order determined on a topological space X is called closed (or
continuous) if the relation 6 is a closed subset of the product X × X.
Oleg Gutik
Compact topological semilattices
a∈A
Partial Order
Definition:
A binary relation 6 on a non-empty set X is called a partial order if the
following conditions hold:
x 6 x for all x ∈ X;
if x 6 y and y 6 x, then x = y, for x, y ∈ X;
if x 6 y and y 6 z, then x 6 z, for x, y, z ∈ X.
A set X equipped with a partial order 6 is called a partially ordered set and it
will be denoted by (X, 6).
For a partially ordered set (X, 6), x ∈ X and A ⊆ X we denote:
[
[
↓x = {y ∈ X : y 6 x}, ↑x = {y ∈ X : x 6 y}, ↓A =
↓a and ↑A =
↑a.
a∈A
Definition:
A partial order determined on a topological space X is called closed (or
continuous) if the relation 6 is a closed subset of the product X × X.
Oleg Gutik
Compact topological semilattices
a∈A
Partial Order
Definition:
A binary relation 6 on a non-empty set X is called a partial order if the
following conditions hold:
x 6 x for all x ∈ X;
if x 6 y and y 6 x, then x = y, for x, y ∈ X;
if x 6 y and y 6 z, then x 6 z, for x, y, z ∈ X.
A set X equipped with a partial order 6 is called a partially ordered set and it
will be denoted by (X, 6).
For a partially ordered set (X, 6), x ∈ X and A ⊆ X we denote:
[
[
↓x = {y ∈ X : y 6 x}, ↑x = {y ∈ X : x 6 y}, ↓A =
↓a and ↑A =
↑a.
a∈A
Definition:
A partial order determined on a topological space X is called closed (or
continuous) if the relation 6 is a closed subset of the product X × X.
Oleg Gutik
Compact topological semilattices
a∈A
Closed Partial Order
Exercise:
A partial order 6 on a topological space X is continuous if and only if for
x, y ∈ X, x y implies there exist open neighbourhoods U (x) and U (y) of x
and y in X such that a b for all a ∈ U (x) and b ∈ U (y).
Definition:
A topological space X equipped with a continuous partial order 6 is called a
partially ordered topological space (or pospace).
Exercise:
Every partially ordered topological space is Hausdorff.
Exercise:
Q
The Cartesian product s∈S Xs of an arbitrary family of partially ordered sets
{(Xs , 6)}s∈S with the product order (i.e., (xs ) 6 (ys ) if and only if xs 6 ys
for all s ∈ S ) is a partially ordered set.
Oleg Gutik
Compact topological semilattices
Closed Partial Order
Exercise:
A partial order 6 on a topological space X is continuous if and only if for
x, y ∈ X, x y implies there exist open neighbourhoods U (x) and U (y) of x
and y in X such that a b for all a ∈ U (x) and b ∈ U (y).
Definition:
A topological space X equipped with a continuous partial order 6 is called a
partially ordered topological space (or pospace).
Exercise:
Every partially ordered topological space is Hausdorff.
Exercise:
Q
The Cartesian product s∈S Xs of an arbitrary family of partially ordered sets
{(Xs , 6)}s∈S with the product order (i.e., (xs ) 6 (ys ) if and only if xs 6 ys
for all s ∈ S ) is a partially ordered set.
Oleg Gutik
Compact topological semilattices
Closed Partial Order
Exercise:
A partial order 6 on a topological space X is continuous if and only if for
x, y ∈ X, x y implies there exist open neighbourhoods U (x) and U (y) of x
and y in X such that a b for all a ∈ U (x) and b ∈ U (y).
Definition:
A topological space X equipped with a continuous partial order 6 is called a
partially ordered topological space (or pospace).
Exercise:
Every partially ordered topological space is Hausdorff.
Exercise:
Q
The Cartesian product s∈S Xs of an arbitrary family of partially ordered sets
{(Xs , 6)}s∈S with the product order (i.e., (xs ) 6 (ys ) if and only if xs 6 ys
for all s ∈ S ) is a partially ordered set.
Oleg Gutik
Compact topological semilattices
Closed Partial Order
Exercise:
A partial order 6 on a topological space X is continuous if and only if for
x, y ∈ X, x y implies there exist open neighbourhoods U (x) and U (y) of x
and y in X such that a b for all a ∈ U (x) and b ∈ U (y).
Definition:
A topological space X equipped with a continuous partial order 6 is called a
partially ordered topological space (or pospace).
Exercise:
Every partially ordered topological space is Hausdorff.
Exercise:
Q
The Cartesian product s∈S Xs of an arbitrary family of partially ordered sets
{(Xs , 6)}s∈S with the product order (i.e., (xs ) 6 (ys ) if and only if xs 6 ys
for all s ∈ S ) is a partially ordered set.
Oleg Gutik
Compact topological semilattices
Normally Ordered Space
Definition:
A map f from a partially ordered set X into a partially ordered set Y is said to
be monotone if x 6 y implies f (x) 6 f (y), for x, y ∈ X.
Definition:
A subset A of a partially ordered set X is called order convex if a 6 b 6 c and
a, b ∈ A imply c ∈ A.
Definition (Nachbin, 1948):
A pospace X said to be normally ordered (or monotone normal), if given two
closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint
subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V .
Notation:
By I we denote the unit interval [0, 1] with the usual topology and the usual
order 6.
Oleg Gutik
Compact topological semilattices
Normally Ordered Space
Definition:
A map f from a partially ordered set X into a partially ordered set Y is said to
be monotone if x 6 y implies f (x) 6 f (y), for x, y ∈ X.
Definition:
A subset A of a partially ordered set X is called order convex if a 6 b 6 c and
a, b ∈ A imply c ∈ A.
Definition (Nachbin, 1948):
A pospace X said to be normally ordered (or monotone normal), if given two
closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint
subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V .
Notation:
By I we denote the unit interval [0, 1] with the usual topology and the usual
order 6.
Oleg Gutik
Compact topological semilattices
Normally Ordered Space
Definition:
A map f from a partially ordered set X into a partially ordered set Y is said to
be monotone if x 6 y implies f (x) 6 f (y), for x, y ∈ X.
Definition:
A subset A of a partially ordered set X is called order convex if a 6 b 6 c and
a, b ∈ A imply c ∈ A.
Definition (Nachbin, 1948):
A pospace X said to be normally ordered (or monotone normal), if given two
closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint
subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V .
Notation:
By I we denote the unit interval [0, 1] with the usual topology and the usual
order 6.
Oleg Gutik
Compact topological semilattices
Normally Ordered Space
Definition:
A map f from a partially ordered set X into a partially ordered set Y is said to
be monotone if x 6 y implies f (x) 6 f (y), for x, y ∈ X.
Definition:
A subset A of a partially ordered set X is called order convex if a 6 b 6 c and
a, b ∈ A imply c ∈ A.
Definition (Nachbin, 1948):
A pospace X said to be normally ordered (or monotone normal), if given two
closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint
subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V .
Notation:
By I we denote the unit interval [0, 1] with the usual topology and the usual
order 6.
Oleg Gutik
Compact topological semilattices
Normally Ordered Space
The Urysohn-Nachbin Lemma (Nachbin, 1948):
Let X be a normally ordered pospace. If A = ↑A and B = ↓B are closed
disjoint subsets in X, then there exists a monotone function f : X → I such
that f (B) = 0 and f (A) = 1.
Theorem (Nachbin, 1948):
Every compact partially ordered space is normally ordered.
Theorem (Nachbin, 1948):
Every compact partially ordered space X is ordered convex (i.e., X has a base
of the topology which consists of open ordered convex subsets).
Oleg Gutik
Compact topological semilattices
Normally Ordered Space
The Urysohn-Nachbin Lemma (Nachbin, 1948):
Let X be a normally ordered pospace. If A = ↑A and B = ↓B are closed
disjoint subsets in X, then there exists a monotone function f : X → I such
that f (B) = 0 and f (A) = 1.
Theorem (Nachbin, 1948):
Every compact partially ordered space is normally ordered.
Theorem (Nachbin, 1948):
Every compact partially ordered space X is ordered convex (i.e., X has a base
of the topology which consists of open ordered convex subsets).
Oleg Gutik
Compact topological semilattices
Normally Ordered Space
The Urysohn-Nachbin Lemma (Nachbin, 1948):
Let X be a normally ordered pospace. If A = ↑A and B = ↓B are closed
disjoint subsets in X, then there exists a monotone function f : X → I such
that f (B) = 0 and f (A) = 1.
Theorem (Nachbin, 1948):
Every compact partially ordered space is normally ordered.
Theorem (Nachbin, 1948):
Every compact partially ordered space X is ordered convex (i.e., X has a base
of the topology which consists of open ordered convex subsets).
Oleg Gutik
Compact topological semilattices
Topological Semilattices
Definition:
A commutative semigroup of idempotents is called a semilattice.
The Natural Partial Order on a Semilattice:
Let E be a semilattice. We put
e6f
if and only if
e · f = f · e = e,
for e, f ∈ E.
Definition:
A Hausdorff topological space E with a continuous semilattice operation is
called a topological semilattice (i.e., for any e, f ∈ E and for every open
neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods
V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )).
Example 1:
Imin = ([0, 1], min).
Example 2:
(exp∞ (X), ∪) with the Vietories topology.
Oleg Gutik
Compact topological semilattices
Topological Semilattices
Definition:
A commutative semigroup of idempotents is called a semilattice.
The Natural Partial Order on a Semilattice:
Let E be a semilattice. We put
e6f
if and only if
e · f = f · e = e,
for e, f ∈ E.
Definition:
A Hausdorff topological space E with a continuous semilattice operation is
called a topological semilattice (i.e., for any e, f ∈ E and for every open
neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods
V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )).
Example 1:
Imin = ([0, 1], min).
Example 2:
(exp∞ (X), ∪) with the Vietories topology.
Oleg Gutik
Compact topological semilattices
Topological Semilattices
Definition:
A commutative semigroup of idempotents is called a semilattice.
The Natural Partial Order on a Semilattice:
Let E be a semilattice. We put
e6f
if and only if
e · f = f · e = e,
for e, f ∈ E.
Definition:
A Hausdorff topological space E with a continuous semilattice operation is
called a topological semilattice (i.e., for any e, f ∈ E and for every open
neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods
V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )).
Example 1:
Imin = ([0, 1], min).
Example 2:
(exp∞ (X), ∪) with the Vietories topology.
Oleg Gutik
Compact topological semilattices
Topological Semilattices
Definition:
A commutative semigroup of idempotents is called a semilattice.
The Natural Partial Order on a Semilattice:
Let E be a semilattice. We put
e6f
if and only if
e · f = f · e = e,
for e, f ∈ E.
Definition:
A Hausdorff topological space E with a continuous semilattice operation is
called a topological semilattice (i.e., for any e, f ∈ E and for every open
neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods
V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )).
Example 1:
Imin = ([0, 1], min).
Example 2:
(exp∞ (X), ∪) with the Vietories topology.
Oleg Gutik
Compact topological semilattices
Topological Semilattices
Definition:
A commutative semigroup of idempotents is called a semilattice.
The Natural Partial Order on a Semilattice:
Let E be a semilattice. We put
e6f
if and only if
e · f = f · e = e,
for e, f ∈ E.
Definition:
A Hausdorff topological space E with a continuous semilattice operation is
called a topological semilattice (i.e., for any e, f ∈ E and for every open
neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods
V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )).
Example 1:
Imin = ([0, 1], min).
Example 2:
(exp∞ (X), ∪) with the Vietories topology.
Oleg Gutik
Compact topological semilattices
Topological Semilattices
Theorem:
The natural partial order on a Hausdorff topological semilattice is closed, and
hence every Hausdorff topological semilattice is a partially ordered topological
space.
Definition:
Let E be a topological semilattice. By Hom(E, Imin ) we denote the set of all
continuous homomorphisms from E into Imin . We shall say that Hom(E, Imin )
separates points of E if for any distinct e, f ∈ E with e 6 f there exists
h ∈ Hom(E, Imin ) such that h(e) = 0 and h(f ) = 1.
Anderson Problem (1953):
Let E be a compact topological semilattice. Does Hom(E, Imin ) separate
points of E?
Oleg Gutik
Compact topological semilattices
Topological Semilattices
Theorem:
The natural partial order on a Hausdorff topological semilattice is closed, and
hence every Hausdorff topological semilattice is a partially ordered topological
space.
Definition:
Let E be a topological semilattice. By Hom(E, Imin ) we denote the set of all
continuous homomorphisms from E into Imin . We shall say that Hom(E, Imin )
separates points of E if for any distinct e, f ∈ E with e 6 f there exists
h ∈ Hom(E, Imin ) such that h(e) = 0 and h(f ) = 1.
Anderson Problem (1953):
Let E be a compact topological semilattice. Does Hom(E, Imin ) separate
points of E?
Oleg Gutik
Compact topological semilattices
Topological Semilattices
Theorem:
The natural partial order on a Hausdorff topological semilattice is closed, and
hence every Hausdorff topological semilattice is a partially ordered topological
space.
Definition:
Let E be a topological semilattice. By Hom(E, Imin ) we denote the set of all
continuous homomorphisms from E into Imin . We shall say that Hom(E, Imin )
separates points of E if for any distinct e, f ∈ E with e 6 f there exists
h ∈ Hom(E, Imin ) such that h(e) = 0 and h(f ) = 1.
Anderson Problem (1953):
Let E be a compact topological semilattice. Does Hom(E, Imin ) separate
points of E?
Oleg Gutik
Compact topological semilattices
Lawson Semilattices
Definition (Lawson, 1969) & (McWaters, 1969):
A topological semilattice S is said to have small semilattices at x ∈ S if the
point x has a basis of neighborhoods which are subsemilattices of S. The
topological semilattice S has small semilattices (or is a Lawson semilattice) iff
it has small semilattices at every point.
Proposition (Lawson, 1969) & (McWaters, 1969):
(i) Let S be a Lawson topological semilattice and let T be a subsemilattice
(equipped with the relative topology). Then T is a Lawson semilattice.
(ii) Let
Q {Sj : j ∈ J } be a collection of Lawson topological semilattices. Then
{Sj : j ∈ J } endowed with coordinatewise operations and the product
topology is a Lawson semilattice.
Oleg Gutik
Compact topological semilattices
Lawson Semilattices
Definition (Lawson, 1969) & (McWaters, 1969):
A topological semilattice S is said to have small semilattices at x ∈ S if the
point x has a basis of neighborhoods which are subsemilattices of S. The
topological semilattice S has small semilattices (or is a Lawson semilattice) iff
it has small semilattices at every point.
Proposition (Lawson, 1969) & (McWaters, 1969):
(i) Let S be a Lawson topological semilattice and let T be a subsemilattice
(equipped with the relative topology). Then T is a Lawson semilattice.
(ii) Let
Q {Sj : j ∈ J } be a collection of Lawson topological semilattices. Then
{Sj : j ∈ J } endowed with coordinatewise operations and the product
topology is a Lawson semilattice.
Oleg Gutik
Compact topological semilattices
Lawson Semilattices
Theorem (Lawson, 1969):
Let S be a locally compact Hausdorff topological semilattice. For x ∈ S the
following statements are equivalent:
(i) S has small semilattices at x;
(ii) the semilattice ↓x has small semilattices at x;
(iii) if V is open in S and x ∈ V , then there exists an element y ∈ V such that
x ∈ Int(↑y).
Theorem (Lawson, 1969):
Let S and T be compact topological semilattices and let f be a continuous
homomorphism from S onto T . If S is Lawson, then T is Lawson, too.
Oleg Gutik
Compact topological semilattices
Lawson Semilattices
Theorem (Lawson, 1969):
Let S be a locally compact Hausdorff topological semilattice. For x ∈ S the
following statements are equivalent:
(i) S has small semilattices at x;
(ii) the semilattice ↓x has small semilattices at x;
(iii) if V is open in S and x ∈ V , then there exists an element y ∈ V such that
x ∈ Int(↑y).
Theorem (Lawson, 1969):
Let S and T be compact topological semilattices and let f be a continuous
homomorphism from S onto T . If S is Lawson, then T is Lawson, too.
Oleg Gutik
Compact topological semilattices
The Fundamental Theorem of Compact Semilattices
The Fundamental Theorem of Compact Semilattices (Lawson, 1969):
Let S be a compact Hausdorff topological semilattice. Then the following
statements are equivalent:
(i) S is Lawson;
(ii) Hom(S, Imin ) separates points of S;
(iii) S is topologically isomorphic to a closed subsemilattice of a product of
copies of Imin .
Examples of Lawson semilattices:
(i) Every linearly ordered topological semilattice.
(ii) Every compact 0-dimensional topological semilattice (A topological space
X is called 0-dimensional if X has a base which consists of
closed-and-open subsets of X).
(iii) (exp∞ (X), ∪) with the Vietories topology: the free Lawson semilattice
over a topological space S.
Oleg Gutik
Compact topological semilattices
The Fundamental Theorem of Compact Semilattices
The Fundamental Theorem of Compact Semilattices (Lawson, 1969):
Let S be a compact Hausdorff topological semilattice. Then the following
statements are equivalent:
(i) S is Lawson;
(ii) Hom(S, Imin ) separates points of S;
(iii) S is topologically isomorphic to a closed subsemilattice of a product of
copies of Imin .
Examples of Lawson semilattices:
(i) Every linearly ordered topological semilattice.
(ii) Every compact 0-dimensional topological semilattice (A topological space
X is called 0-dimensional if X has a base which consists of
closed-and-open subsets of X).
(iii) (exp∞ (X), ∪) with the Vietories topology: the free Lawson semilattice
over a topological space S.
Oleg Gutik
Compact topological semilattices
Examples of non-Lawson Semilattices
Jimmie D. Lawson. Lattices with no interval homomorphisms. Pacific Journal
of Mathematics 32 (1970), 459—465.
Gerhard Gierz. A compact semilattice on the Hilbert cube with no interval
homomorphism. Proceedings of the American Mathematical Society 101
(1987), 592–594.
Oleg Gutik
Compact topological semilattices
Examples of non-Lawson Semilattices
Jimmie D. Lawson. Lattices with no interval homomorphisms. Pacific Journal
of Mathematics 32 (1970), 459—465.
Gerhard Gierz. A compact semilattice on the Hilbert cube with no interval
homomorphism. Proceedings of the American Mathematical Society 101
(1987), 592–594.
Oleg Gutik
Compact topological semilattices
The Fundamental Theorem of Compact 0-dimensional Semilattices
The Fundamental Theorem of Compact 0-dimensional Semilattices (Hofmann,
Mislove, Stralka, 1974):
Let S be a compact Hausdorff topological semilattice. Then the following
statements are equivalent:
(i) S is 0-dimensional;
(ii) Hom(S, {0, 1}min ) separates points of S;
(iii) S is topologically isomorphic to a closed subsemilattice of a product of
copies of {0, 1}min .
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Definition:
A continuous map f : X → Y of topological spaces X and Y is called open if
f (U ) is an open subset in Y for any open subset U ⊆ X.
Definition:
A topological semigroup S is called a semigroup with open translations if all
left λs : S → S : x 7→ s · x and right ρs : S → S : x 7→ x · s translations in S are
open maps, for all s ∈ S.
Guran’s Problem (1991):
To describe compact topological semilattices with open translations.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Definition:
A continuous map f : X → Y of topological spaces X and Y is called open if
f (U ) is an open subset in Y for any open subset U ⊆ X.
Definition:
A topological semigroup S is called a semigroup with open translations if all
left λs : S → S : x 7→ s · x and right ρs : S → S : x 7→ x · s translations in S are
open maps, for all s ∈ S.
Guran’s Problem (1991):
To describe compact topological semilattices with open translations.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Definition:
A continuous map f : X → Y of topological spaces X and Y is called open if
f (U ) is an open subset in Y for any open subset U ⊆ X.
Definition:
A topological semigroup S is called a semigroup with open translations if all
left λs : S → S : x 7→ s · x and right ρs : S → S : x 7→ x · s translations in S are
open maps, for all s ∈ S.
Guran’s Problem (1991):
To describe compact topological semilattices with open translations.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices
Compact Topological Semilattices with Open Translations
Theorem:
Let E be a compact topological semilattice with open translations. Then the
following assertions hold:
(i) the semilattice E is 0-dimensional, and hence is Lawson;
(ii) the topological space E is scattered (i.e., every subset A in E has an
isolated point in itself);
(iii) E contains a dense subset K(E) of isolated points of E;
(iv) K(E) is a subsemilattice of E;
(v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable;
(vi) every maximal antichain (i.e., a subset with point-wise incomparable
elements) in K(E) is finite;
(vii) K(E) is countable;
(viii) E is a first countable space;
(ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the
topology at the point e ∈ E;
(x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and
χ(e, E) = min {|B(e)| : B(e) is a base at e};
(xi) χ(E) 6 ℵ0 , and hence E is countable.
Oleg Gutik
Compact topological semilattices