Download Topological Extensions of Linearly Ordered Groups

Topological Extensions of Linearly
Ordered Groups
Kateryna Pavlyk and Oleg Gutik
Institute for Applied Problems of Mechanics and Mathematics,
National Academy of Sciences of Ukraine
and
L’viv National University
AAA80 Workshop on General Algebra,
Będlewo, June 1-10, 2010
Linearly ordered group is a group (G,⋅) equipped with a partial order
"≤" that is translation-invariant; in other words, "≤" has the property
that,
∀ a,b,c∈G , if a ≤b then ac≤bc and ca≤cb .
Let e be the unity of a group G . We put G+ ={x | e≤ x}.
On the set G×G we define a semigroup operation as follows:
(a,b)(c,d ) =(ac (b∧c)−1, bd (b∧c)−1) ,
where b∧c =inf{b,c}, for a,b,c,d ∈G . We denote such semigroup BG .
+ = G+×G+ is a subsemigroup of B and
We observe that the set BG
G
call it bicyclic-like extension of G+ .
Proposition.
• BG is a simple semigroup;
• BG is an inverse semigroup and (a,b)−1=(b,a) ;
• an element (a,b) of BG is an idempotent iff a =b ;
• E(BG ) is linearly ordered;
• the map h:G+ → B+ , defined by the formula h(s) =(s,e) is isomorphic
G
+;
embedding of group G+ into the semigroup BG
• Let (a,b),(c,d )∈ BG . Then:
• (a,b)L (c,d ) iff b = d ;
• (a,b)R (c,d ) iff a = c ;
• (a,b)H (c,d ) iff a = c and b = d , hence every H -class of BG is
singleton;
• BG is a bisimple semigroup;
A topological group G is a topological space with continuous
group operation and inversion.
A topological semigroup is a Hausdorff topological space with
continuous semigroup operation.
A topological inverse semigroup is an inverse topological
semigroup with continuous inversion.
A linearly ordered topological group G is a topological group
with closed invariant linear order on G .
A topological space X is called locally compact if for every
element x∈ X there exists open neighbourhood U ( x) such that the
closure U ( x) is a compact subset of X .
Proposition. Let G be a locally compact linearly ordered
+ with product topology is a topological
topological group. Then BG
inverse semigroup.
Moreover, the map h:G → BG , defined by the formula h(s) =(s,e) is
a topological isomorphism “into”.
A topological space X is called Baire if for each sequence
∞
A1, A2,…, Ai,… of nowhere dense subsets of X the union ∪ Ai is a coi=1
dense subset of X .
Theorem. Let G be a countable linearly ordered commutative
Baire topological group. Then every semigroup inverse topology τB+
+ such that the map i :G ∋ x
on BG
isomorphism is discrete.
G
( x, x)∈ BG is a topological order
Corollary. Let G be a countable linearly ordered commutative
locally compact topological group. Then every semigroup inverse
+ such that the map i :G ∋ x ( x, x)∈ B is a
topology τB+ on BG
G
G
topological order isomorphism is discrete.
A topological space X is called countably compact if any
countable open cover of X contains a finite subcover.
A topological semigroup S is called Γ-compact if for every x∈ S ,
Γ(x ) = {x,x 2,x 3,…,x n,…} is a compact subsemigroup in S .
A family {As }s∈S of subsets of a topological space X is
called locally finite if for every element x∈ X there exists open
neighbourhood U ( x) of x such that the set {s ∈ S :U ∩ As ≠ ∅}
is finite.
A topological space X is called pseudocompact if every locally
finite family of non-open subsets of X is finite.
Corollary. Let G be a linearly ordered group. Then the following
classes of topological semigroups:
1. compact topological semigroups;
2. Γ -compact topological semigroups;
3. countably compact topological inverse semigroups;
4. Hausdorff topological semigroups with the pseudocompact square
+ , and hence do not contain B .
do not contain BG
G