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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