Brandt extensions and primitive topological inverse semigroups

In the paper we study (countably) compact and (absolutely) $H$-closed primitive topological inverse semigroups. We describe the structure of compact and countably compact primitive topological inverse semigroups and show that any countably compact primitive topological inverse semigroup embeds into a compact primitive topological inverse semigroup.

In this paper all spaces are Hausdorff. A semigroup is a nonempty set with a binary associative operation. A semigroup S is called inverse if for any x ∈ S there exists a unique y ∈ S such that x · y · x x and y · x · y y. Such an element y in S is called inverse to x and denoted by x −1 . The map defined on an inverse semigroup S which maps to any element x of S its inverse x −1 is called the inversion.
A topological semigroup is a Hausdorff topological space with a jointly continuous semigroup operation. A topological semigroup which is an inverse semigroup is called an inverse topological semigroup. A topological inverse semigroup is an inverse topological semigroup with continuous inversion. A topological group is a topological space with a continuous group operation and an inversion. We observe that the inversion on a topological inverse semigroup is a homeomorphism see 1, Proposition II.1 . A Hausdorff topology τ on a inverse semigroup S is called inverse semigroup if S, τ is a topological inverse semigroup.
Further we shall follow the terminology of 2-8 . If S is a semigroup, then by E S we denote the band the subset of idempotents of S, and by S 1 S 0 we denote the semigroup S with the adjoined unit zero see 7, page 2 . Also if a semigroup S has zero 0 S , then for any A ⊆ S we denote A * A \ {0 S }. If Y is a subspace of a topological space X and A ⊆ Y , then 2 International Journal of Mathematics and Mathematical Sciences by cl Y A we denote the topological closure of A in Y . The set of positive integers is denoted by N.
If E is a semilattice, then the semilattice operation on E determines the partial order on E: This order is called natural. An element e of a partially ordered set X is called minimal if f e implies f e for f ∈ X. An idempotent e of a semigroup S without zero with zero is called primitive if e is a minimal element in E S in E S * . Let S be a semigroup with zero and let I λ be a set of cardinality λ 1. On the set B λ S I λ × S × I λ ∪ {0} we define the semigroup operation as follows: and α, a, β · 0 0 · α, a, β 0 · 0 0, for all α, β, γ, δ ∈ I λ and a, b ∈ S. If S S 1 , then the semigroup B λ S is called the Brandt λ-extension of the semigroup S 9 . Obviously, J {0} ∪ { α, O, β | O is the zero of S} is an ideal of B λ S . We put B 0 λ S B λ S /J and we shall call B 0 λ S the Brandt λ 0 -extension of the semigroup S with zero 10 . Further, if A ⊆ S, then we shall denote A α,β { α, s, β | s ∈ A} if A does not contain zero, and If I is a trivial semigroup i.e., I contains only one element , then by I 0 we denote the semigroup I with the adjoined zero. Obviously, for any λ 2 the Brandt λ 0 -extension of the semigroup I 0 is isomorphic to the semigroup of I λ × I λmatrix units and any Brandt λ 0 -extension of a semigroup with zero contains the semigroup of I λ ×I λ -matrix units. Further by B λ we shall denote the semigroup of I λ ×I λ -matrix units and by A semigroup S is called algebraically closed in S if S with any semigroup topology τ is H-closed in S and S, τ ∈ S 9 . If S coincides with the class of all topological semigroups, then the semigroup S is called algebraically closed. A semigroup S is called algebraically h-closed in S if S with the discrete topology d is absolutely H-closed in S and S, d ∈ S. If S coincides with the class of all topological semigroups, then the semigroup S is called algebraically hclosed.
Absolutely H-closed semigroups and algebraically h-closed semigroups were introduced by Stepp in 14 . There, they were called absolutely maximal and algebraic maximal, respectively.
Definition 3 see 9 . Let λ be a cardinal 1 and S, τ ∈ S. Let τ B be a topology on B λ S such that i B λ S , τ B ∈ S; ii τ B | α,S 1 ,α τ for some α ∈ I λ .
Then B λ S , τ B is called a topological Brandt λ-extension of S, τ in S. If S coincides with the class of all topological semigroups, then B λ S , τ B is called a topological Brandt λ-extension of S, τ .
Definition 4 see 10 . Let S 0 be some class of topological semigroups with zero. Let λ be a cardinal 1 and S, τ ∈ S 0 . Let τ B be a topology on B 0 λ S such that a B 0 λ S , τ B ∈ S 0 ; b τ B | α,S,α ∪{0} τ for some α ∈ I λ .
Then B 0 λ S , τ B is called a topological Brandt λ 0 -extension of S, τ in S 0 . If S 0 coincides with the class of all topological semigroups, then B 0 λ S , τ B is called a topological Brandt λ 0 -extension of S, τ .
Gutik and Pavlyk in 9 proved that the following conditions for a topological semigroup S are equivalent: i S is an H-closed semigroup in the class of topological inverse semigroups; ii there exists a cardinal λ 1 such that any topological Brandt λ-extension of S is H-closed in the class of topological inverse semigroups; iii for any cardinal λ 1 every topological Brandt λ-extension of S is H-closed in the class of topological inverse semigroups.
In 13 they showed that the similar statement holds for absolutely H-closed topological semigroups in the class of topological inverse semigroups. In 10 , Gutik and Pavlyk proved the following. ii there exists a cardinal λ 1 such that any topological Brandt λ 0 -extension B 0 λ S of the semigroup S is (absolutely) H-closed in the class of topological inverse semigroups; iii for each cardinal λ 1, every topological Brandt λ 0 -extension B 0 λ S of the semigroup S is (absolutely) H-closed in the class of topological inverse semigroups.
Also, an example of an absolutely H-closed topological semilattice N with zero and a topological Brandt λ 0 -extension B 0 λ N of N with the following properties was constructed in 10 : is not a topological semigroup. We observe that for any topological Brandt λ-extension B λ S of a topological semigroup S there exist a topological monoid T with zero and a topological Brandt λ 0 -extension B 0 λ T of T , such that the semigroups B λ S and B 0 λ T are topologically isomorphic. Algebraic properties of Brandt λ 0 -extensions of monoids with zero and nontrivial homomorphisms between Brandt λ 0 -extensions of monoids with zero and a category whose objects are ingredients of the construction of Brandt λ 0 -extensions of monoids with zeros were described in 15 . Also, in 15, 16 was described a category whose objects are ingredients in the constructions of finite compact, countably compact topological Brandt λ 0 -extensions of topological monoids with zeros.
In 9, 17 for every infinite cardinal λ, semigroup topologies on Brandt λ-extensions which preserve an H-closedness and an absolute H-closedness were constructed. An example of a nonH-closed topological inverse semigroup S in the class of topological inverse semigroups such that for any cardinal λ 1 there exists an absolute H-closed topological Brandt λ-extension of the semigroup S in the class of topological semigroups was constructed in 17 .
In this paper we study countably compact and absolutely H-closed primitive topological inverse semigroups. We describe the structure of compact and countably compact primitive topological inverse semigroups and show that any countably compact primitive topological inverse semigroup embeds into a compact primitive topological inverse semigroup.

Lemma 6. Let E be a topological semilattice with zero 0 such that every nonzero idempotent of E is primitive. Then every nonzero element of E is an isolated point in E.
0 which contradicts to the choice of U x . This implies the assertion of the lemma.

International Journal of Mathematics and Mathematical Sciences
5 Lemma 7 implies the following.

Proof.
Let H e, f be a nonzero H-class in S for e, f ∈ E S * , that is, Since S is a topological inverse semigroup, the maps ϕ : S → E S and ψ : S → E S defined by the formulae ϕ x x · x −1 and ψ x x −1 · x are continuous. By Lemma 6, e and f are isolated points in E S . Then the continuity of the maps ϕ and ψ implies the statement of the lemma.
The following example shows that the statement of Lemma 9 does not hold for primitive inverse locally compact H-closed topological semigroups.
Example 10. Let Z be the discrete additive group of integers. We extend the semigroup operation from Z onto Z 0 Z ∪ {∞} as follows: We observe that Z 0 is the group with adjoined zero ∞. We determine a semigroup topology τ on Z 0 as follows: i every nonzero element of Z 0 is an isolated point; ii the family B ∞ {U n {∞} ∪ {x ∈ Z | x n} | n is a positive integer} is a base of the topology τ at the point ∞.
A simple verification shows that Z 0 , τ is a primitive inverse locally compact topological semigroup. 6 International Journal of Mathematics and Mathematical Sciences

Proposition 12.
Every completely 0-simple topological inverse semigroup S is topologically isomorphic to a topological Brandt λ-extension B λ G of some topological group G and cardinal λ 1 in the class of topological inverse semigroups. Furthermore one has the following: i any nonzero subgroup of S is topologically isomorphic to G and every nonzero H-class of S is homeomorphic to G and is a clopen subset in S; ii the family B α, g, β { α, g · U, β | U ∈ B G e }, where B G e is a base of the topology at the unity e of G, is a base of the topology at the nonzero element α, g, β ∈ B λ G .
Proof. Let G be a nonzero subgroup of S. Then by Theorem 3.9 of 4, 5 the semigroup S is isomorphic to the Brandt λ-extension of the subgroup G for some cardinal λ 1. Since S is a topological inverse semigroup, we have that G is a topological group.
i Let e be the unity of G. We fix arbitrary α, β, γ, δ ∈ I λ and define the maps ϕ | α,G,β : α, G, β → γ, G, δ is a homeomorphism and the map ϕ γγ αα | α,G,α : α, G, α → γ, G, γ is a topological isomorphism. We observe that the subset α, G, β of B λ G is an H-class of B λ G and α, G, α is a subgroup of B λ G for all α, β ∈ I λ . This completes the proof of assertion i .
ii The statement follows from assertion i and Theorem 4.3 of 18 .
We observe that Example 10 implies that the statements of Proposition 12 are not true for completely 0-simple inverse topological semigroups. Definition 3 implies that S is a topological Brandt λ-extension B λ G of the topological group G.
Gutik and Repovš, in 19 , studied the structure of 0-simple countably compact topological inverse semigroups. They proved that any 0-simple countably compact topological inverse semigroup is topologically isomorphic to a topological Brandt λ-extension B λ H of a countably compact topological group H in the class of topological inverse semigroups for some finite cardinal λ 1. This implies Pavlyk's Theorem see 20 on the structure of 0-simple compact topological inverse semigroups: every 0-simple compact topological inverse semigroup is topologically isomorphic to a topological Brandt λ-extension B λ H of a compact topological group H in the class of topological inverse semigroups for some finite cardinal λ 1.
The following theorem describes the structure of primitive countably compact topological inverse semigroups.

Theorem 13. Every primitive countably compact topological inverse semigroup S is topologically isomorphic to an orthogonal sum i∈A B λ i G i of topological Brandt λ i -extensions B λ i G i of countably compact topological groups G i in the class of topological inverse semigroups for some finite cardinals λ i 1. Moreover the family
determines a base of the topology at zero 0 of S.
Proof. By Theorem II.4.3 of 8 the semigroup S is an orthogonal sum of Brandt semigroups and hence S is an orthogonal sum i∈A B λ i G i of Brandt λ i -extensions B λ i G i of groups G i . We fix any i 0 ∈ A. Since S is a topological inverse semigroup, Proposition II.
for finitely many indexes i 1 , i 2 , . . . , i n ∈ A. Therefore there exists an infinitely family F of nonzero disjoint H-classes such that H/ ⊆U 0 for all H ∈ F. Let F 0 be an infinite countable subfamily of F. We put W {H | H ∈ F \ F 0 }. Lemma 9 implies that the family C {U 0 , W} ∪ F 0 is an open countable cover of S. Simple observation shows that the cover C does not contain a finite subcover. This contradicts to the countable compactness of S. The obtained contradiction implies the last assertion of the theorem.
Since any maximal subgroup of a compact topological semigroup T is a compact subset in T see 2, Vol. 1, Theorem 1.11 , Theorem 13 implies the following.

Corollary 14. Every primitive compact topological inverse semigroup S is topologically isomorphic to an orthogonal sum i∈A B λ i G i of topological Brandt λ i -extensions B λ i G i of compact topological groups G i in the class of topological inverse semigroups for some finite cardinals λ i 1 and the family
determines a base of the topology at zero 0 of S.

Theorem 15. Every primitive countably compact topological inverse semigroup S is a dense subsemigroup of a primitive compact topological inverse semigroup.
Proof. By Theorem 13 the topological semigroup S is topologically isomorphic to an orthogonal sum i∈A B λ i G i of topological Brandt λ i -extensions B λ i G i of countably compact topological groups G i in the class of topological inverse semigroups for some finite cardinals λ i 1. Since any countably compact topological group G i is pseudocompact, the Comfort-Ross Theorem see 21, Theorem 4.1 implies that the Stone-Čech compactification β G i is a compact topological group and the inclusion mapping f i of G i into β G i is 8 International Journal of Mathematics and Mathematical Sciences a topological isomorphism for all i ∈ A. On the orthogonal sum i∈A B λ i G i of Brandt λextensions B λ i β G i , i ∈ A, we determine a topology τ as follows: determines a base of the topology at zero 0 of i∈A B λ i G i . By Theorem II.4.3 of 8 , i∈A B λ i β G i is a primitive inverse semigroup and simple verifications show that i∈A B λ i β G i with the topology τ is a compact topological inverse semigroup.
We define a map f : i∈A B λ i G i → i∈A B λ i β G i as follows: Simple verifications show that f is a continuous homomorphism. Since f i : Gutik and Repovš in 19 showed that the Stone-Čech compactification β T of a 0simple countably compact topological inverse semigroup T is a 0-simple compact topological inverse semigroup. In this context the following question arises naturally. ii there exists an H-closed (resp., absolutely H-closed) subsemigroup T of S in the class of topological inverse semigroups such that S α · S β ⊆ T for all α / β, α, β ∈ A.
Then S is an H-closed (resp., absolutely H-closed) semigroup in the class of topological inverse semigroups.
Proof. We consider the case of absolute H-closedness only.
Suppose on the contrary that there exist a topological inverse semigroup G and a continuous homomorphism h : S → G such that h S is not closed subsemigroup in G. Without loss of generality we can assume that cl G h S G. Thus, by Proposition II.2 of 1 , G is a topological inverse semigroup.
Then, G \ h S / ∅. Let x ∈ G \ h S . Since S and G are topological inverse semigroups we have that h S is an inverse subsemigroup in G and hence x −1 ∈ G \ h S . The semigroup T which is an absolutely H-closed semigroup in the class of topological inverse semigroups implies that there exists an open neighbourhood U x of the point x in T such that U x ∩ h T ∅. Since G is a topological inverse semigroup there exist open neighbourhoods V x and V x −1 of the points x and x −1 in G, respectively, such that V x · V x −1 · V x ⊆ U x . But x, x −1 ∈ cl G h S \ h S and since {S α | α ∈ A} is the family of absolutely H-closed semigroups in the class of topological inverse semigroups, each of the neighbourhoods V x and V x −1 intersects infinitely many subsemigroups h S β in G, β ∈ A. Hence, The obtained contradiction implies that S is an absolutely H-closed semigroup in the class of topological inverse semigroups.
The proof in the case of H-closeness is similar to the previous one.
Theorem 16 implies the following.

Corollary 17. Let S α∈A S α be an inverse semigroup such that
i S α is an algebraically closed (resp., algebraically h-closed) semigroup in the class of topological inverse semigroups for any α ∈ A; ii there exists an algebraically closed (resp., algebraically h-closed) sub-semigroup T of S in the class of topological inverse semigroups such that S α · S β ⊆ T for all α / β, α, β ∈ A.
Then S is an algebraically closed (resp., algebraically h-closed) semigroup in the class of topological inverse semigroups.
Theorem 16 implies the following. Corollary 17 implies the following.

Corollary 19.
Let an inverse semigroup S be an orthogonal sum of the family {S α } α∈A of algebraically closed (resp., algebraically h-closed) inverse semigroups with zeros in the class of topological inverse semigroups. Then S is an algebraically closed (resp., algebraically h-closed) inverse semigroup in the class of topological inverse semigroups.
Recall in 22 , that a topological group G is called absolutely closed if G is a closed subgroup of any topological group which contains G as a subgroup. In our terminology such topological groups are called H-closed in the class of topological groups. In 23 Raikov proved that a topological group G is absolutely closed if and only if it is Raikov complete, that is, G is complete with respect to the two sided uniformity.
A topological group G is called h-complete if for every continuous homomorphism f : G → H into a topological group H the subgroup f G of H is closed 24 . The hcompleteness is preserved under taking products and closed central subgroups 24 .
Gutik and Pavlyk in 13 showed that a topological group G is H-closed resp., absolutely H-closed in the class of topological inverse semigroups if and only if G is absolutely closed resp., h-complete . ii the semigroup S with every inverse semigroup topology τ is H-closed in the class of topological inverse semigroups.
Proof. i ⇒ ii Suppose that a primitive topological inverse semigroup S is an orthogonal sum i∈A B λ i G i of topological Brandt λ i -extensions B λ i G i of topological groups G i in the class of topological inverse semigroups and every topological group G i is absolutely closed. Then, by Theorem 3 of 9 any topological Brandt λ i -extension B λ i G i of topological group G i is H-closed in the class of topological inverse semigroups. Theorem 18 implies that S is an H-closed topological inverse semigroup in the class of topological inverse semigroups.
ii ⇒ i Let G be any maximal nonzero subgroup of S. Since S is a primitive topological inverse semigroup, we have that S is an orthogonal sum i∈A B λ i G i of Brandt λ-extensions B λ i G i of topological groups G i and hence there exists a topological Brandt Suppose on the contrary that the topological group G G i 0 is not absolutely closed. Then there exists a topological group H which contains G as a dense proper subgroup. For every i ∈ A we put On the orthogonal sum i∈A B λ i H i of Brandt λ-extensions B λ i H i , i ∈ A, we determine a topology τ 0 as follows: a the family B α i , g i , β i { α i , g i · U, β i |U ∈ B H i e i } is a base of the topology at the nonzero element α i , g i , β i ∈ B λ i H i , where B H i e i is a base of the topology at the unity e i of the topological group H i ; b the zero 0 is an isolated point in i∈A B λ i H i , τ 0 .
By Theorem II.4.3 of 8 , i∈A B λ i H i is a primitive inverse semigroup and simple verifications show that i∈A B λ i H i with the topology τ 0 is a topological inverse semigroup. Also we observe that the semigroup i∈A B λ i G i which is induced from i∈A B λ i H i , τ 0 topology is a topological inverse semigroup which is a dense proper inverse sub-semigroup of i∈A B λ i H i , τ 0 . The obtained contradiction completes the statement of the theorem.
Theorem 20 implies the following.

Corollary 21.
For a primitive inverse semigroup S the following assertions are equivalent: i every maximal subgroup of S is algebraically closed in the class of topological inverse semigroups; ii the semigroup S is algebraically closed in the class of topological inverse semigroups.

Theorem 22.
For a primitive topological inverse semigroup S the following assertions are equivalent: i every maximal subgroup of S is h-complete; ii the semigroup S with every inverse semigroup topology τ is absolutely H-closed in the class of topological inverse semigroups.
Proof. i ⇒ ii Suppose that a primitive topological inverse semigroup S is an orthogonal sum i∈A B λ i G i of topological Brandt λ i -extensions B λ i G i of topological groups G i in the class of topological inverse semigroups and every topological group G i is h-complete. Then by Theorem 14 of 13 any topological Brandt λ i -extension B λ i G i of topological group G i is absolutely H-closed in the class of topological inverse semigroups. Theorem 18 implies that S is an absolutely H-closed topological inverse semigroup in the class of topological inverse semigroups.
ii ⇒ i Let G be any maximal nonzero subgroup of S. Since S is a primitive topological inverse semigroup, S is an orthogonal sum i∈A B λ i G i of Brandt λ-extensions B λ i G i of topological groups G i . Hence there exists a topological Brandt λ i 0 -extension B λ i 0 G i 0 , i ∈ A, such that B λ i 0 G i 0 contains the maximal subgroup G and B λ i 0 G i 0 is a subsemigroup of S.
Suppose on the contrary that the topological group G G i 0 is not h-completed. Then there exist a topological group H and continuous homomorphism h : G → H such that h G is a dense proper subgroup of H. On the Brandt λ-extension B λ i 0 H , we determine a topology τ H as follows: a the family B α i 0 , g i 0 , β i 0 { α i 0 , g i · U, β i 0 | U ∈ B H e } is a base of the topology at the nonzero element α i 0 , g i , β i 0 ∈ B λ i H , where B H e is a base of the topology at the unity e of the topological group H; b the zero 0 is an isolated point in B λ i 0 H , τ H .
Then B λ i 0 H is an inverse semigroup and simple verifications show that B λ i 0 H with the topology τ H is a topological inverse semigroup.
On the orthogonal sum i∈A B λ i G i of Brandt λ-extensions B λ i G i , i ∈ A, we determine a topology τ as follows: a the family B α i , g i , β i { α i , g i · U, β i | U ∈ B G i e i } is a base of the topology at the nonzero element α i , g i , β i ∈ B λ i G i , where B G i e i is a base of the topology at the unity e i of the topological group G i ; b the zero 0 is an isolated point in i∈A B λ i G i , τ .
By Theorem II.4.3 of 8 , i∈A B λ i G i is a primitive inverse semigroup and simple verifications show that i∈A B λ i G i with the topology τ is a topological inverse semigroup.
We define the map f : S → B λ i 0 H as follows: ii the semigroup S is algebraically h-closed in the class of topological inverse semigroups.