#### Abstract

We study (countably) compact and (absolutely) -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 is called *inverse* if for any there exists a unique such that and . Such an element in is called *inverse* to and denoted by . The map defined on an inverse semigroup which maps to any element of its inverse 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 on a (inverse) semigroup is called (*inverse*) *semigroup* if is a topological (inverse) semigroup.

Further we shall follow the terminology of [2β8]. If is a semigroup, then by we denote the band (the subset of idempotents) of , and by [] we denote the semigroup with the adjoined unit [] (see [7, page 2]). Also if a semigroup has zero , then for any we denote . If is a subspace of a topological space and , then by we denote the topological closure of in . The set of positive integers is denoted by .

If is a semilattice, then the semilattice operation on determines the partial order on :

This order is called * natural*. An element of a partially ordered set is called * minimal* if implies for . An idempotent of a semigroup without zero (with zero) is called *primitive* if is a minimal element in (in ).

Let be a semigroup with zero and let be a set of cardinality . On the set we define the semigroup operation as follows:

and for all and . If , then the semigroup is called the * Brandt **-extension of the semigroup * [9]. Obviously, is the zero of is an ideal of . We put and we shall call the * Brandt **-extension of the semigroup ** with zero* [10]. Further, if , then we shall denote if does not contain zero, and if , for . If is a trivial semigroup (i.e., contains only one element), then by we denote the semigroup with the adjoined zero. Obviously, for any the Brandt -extension of the semigroup is isomorphic to the semigroup of -matrix units and any Brandt -extension of a semigroup with zero contains the semigroup of -matrix units. Further by we shall denote the semigroup of -matrix units and by the subsemigroup of -matrix units of the Brandt -extension of a monoid with zero. A completely -simple inverse semigroup is called a *Brandt semigroup* [8]. A semigroup is a Brandt semigroup if and only if is isomorphic to a Brandt -extension of some group [8, Theorem II.3.5].

A nontrivial inverse semigroup is called a *primitive inverse semigroup* if all its nonzero idempotents are primitive [8]. A semigroup is a primitive inverse semigroup if and only if is an orthogonal sum of Brandt semigroups [8, Theorem II.4.3].

Greenβs relations , , and on a semigroup are defined by

(i) if and only if ; (ii) if and only if ; (iii)for . For details about Green's relations, see [4, Section ] or [11]. We observe that two nonzero elements and of a Brandt semigroup , , , are -equivalent if and only if and (see [8, page 93]).

By we denote some class of topological semigroups.

*Definition 1 (see [9, 12]). *A semigroup is called *-closed in *, if is a closed subsemigroup of any topological semigroup which contains as a subsemigroup. If coincides with the class of all topological semigroups, then the semigroup is called *-closed*.

*Definition 2 (see [13, 14]). *A topological semigroup is called * absolutely **-closed in the class * if any continuous homomorphic image of into is -closed in . If coincides with the class of all topological semigroups, then the semigroup is called * absolutely **-closed*.

A semigroup is called * algebraically closed in * if with any semigroup topology is -closed in and [9]. If coincides with the class of all topological semigroups, then the semigroup is called * algebraically closed*. A semigroup is called * algebraically **-closed in * if with the discrete topology is absolutely -closed in and . If coincides with the class of all topological semigroups, then the semigroup is called * algebraically **-closed*.

Absolutely -closed semigroups and algebraically -closed semigroups were introduced by Stepp in [14]. There, they were called * absolutely maximal* and * algebraic maximal*, respectively.

*Definition 3 (see [9]). *Let be a cardinal and . Let be a topology on such that

(i); (ii) for some .

Then is called a * topological Brandt **-extension of ** in *. If coincides with the class of all topological semigroups, then is called a * topological Brandt **-extension of *.

*Definition 4 (see [10]). *Let be some class of topological semigroups with zero. Let be a cardinal and . Let be a topology on such that

(a); (b) for some .

Then is called a * topological Brandt **-extension of ** in **.* If coincides with the class of all topological semigroups, then is called a * topological Brandt **-extension of *.

Gutik and Pavlyk in [9] proved that the following conditions for a topological semigroup are equivalent:

(i) is an -closed semigroup in the class of topological inverse semigroups; (ii)there exists a cardinal such that any topological Brandt -extension of is -closed in the class of topological inverse semigroups; (iii)for any cardinal every topological Brandt -extension of is -closed in the class of topological inverse semigroups.In [13] they showed that the similar statement holds for absolutely -closed topological semigroups in the class of topological inverse semigroups.

In [10], Gutik and Pavlyk proved the following.

Theorem 5. *Let be a topological inverse monoid with zero. Then the following conditions are equivalent: *

(i)* is an (absolutely) -closed semigroup in the class of topological inverse semigroups; *
(ii)* there exists a cardinal such that any topological Brandt -extension of the semigroup is (absolutely) -closed in the class of topological inverse semigroups; *
(iii)*for each cardinal , every topological Brandt -extension of the semigroup is (absolutely) -closed in the class of topological inverse semigroups. *

Also, an example of an absolutely -closed topological semilattice with zero and a topological Brandt -extension of with the following properties was constructed in [10]:

(i) is an absolutely -closed semigroup for any infinite cardinal ; (ii) is a -compact inverse topological semigroup for any countable cardinal ; (iii) contains an absolutely -closed ideal such that the Rees quotient semigroup is not a topological semigroup.We observe that for any topological Brandt -extension of a topological semigroup there exist a topological monoid with zero and a topological Brandt -extension of , such that the semigroups and are topologically isomorphic. Algebraic properties of Brandt -extensions of monoids with zero and nontrivial homomorphisms between Brandt -extensions of monoids with zero and a category whose objects are ingredients of the construction of Brandt -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 -extensions of topological monoids with zeros.

In [9, 17] for every infinite cardinal , semigroup topologies on Brandt -extensions which preserve an -closedness and an absolute -closedness were constructed. An example of a non-closed topological inverse semigroup in the class of topological inverse semigroups such that for any cardinal there exists an absolute -closed topological Brandt -extension of the semigroup in the class of topological semigroups was constructed in [17].

In this paper we study (countably) compact and (absolutely) -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 be a topological semilattice with zero such that every nonzero idempotent of is primitive. Then every nonzero element of is an isolated point in .*

*Proof. *Let . Since is a Hausdorff topological semilattice, for every open neighbourhood of the point there exists an open neighbourhood of such that . If is not an isolated point of , then which contradicts to the choice of . This implies the assertion of the lemma.

Lemma 7. *Let be a primitive inverse topological semigroup and let be an orthogonal sum of the family of topological Brandt semigroups with zeros, that is, . Let be a nonzero element of . Then *

(i)* there exists an open neighbourhood of such that ;*
(ii)* every nonzero idempotent of is an isolated point in . *

*Proof. *(i) Suppose to the contrary that for any open neighbourhood of the point . Since is a Hausdorff space, there exists an open neighbourhood of the point such that . The continuity of the semigroup operation in implies that there exists an open neighbourhood of the point such that . Since , we have that , a contradiction.

Statement (ii) follows from Lemma 6.

Lemma 7 implies the following.

Corollary 8. *Every nonzero -class of a primitive inverse topological semigroup is an open subset in .*

Lemma 9. *If is a primitive topological inverse semigroup, then every nonzero -class of is a clopen subset in .*

*Proof. *Let be a nonzero -class in for , that is,

Since is a topological inverse semigroup, the maps and defined by the formulae and are continuous. By Lemma 6, and are isolated points in . Then the continuity of the maps and implies the statement of the lemma.

The following example shows that the statement of Lemma 9 does not hold for primitive inverse locally compact -closed topological semigroups.

*Example 10. *Let be the discrete additive group of integers. We extend the semigroup operation from onto as follows:

We observe that is the group with adjoined zero . We determine a semigroup topology on as follows:

(i)every nonzero element of is an isolated point;
(ii)the family is a base of the topology at the point .

A simple verification shows that is a primitive inverse locally compact topological semigroup.

Proposition 11. * is an -closed topological semigroup.*

*Proof. *Suppose that is embedded into a topological semigroup . If is a net in for which converges in to , then the equation implies that for every βwhich is impossible. So is closed in .

Proposition 12. *Every completely -simple topological inverse semigroup is topologically isomorphic to a topological Brandt -extension of some topological group and cardinal in the class of topological inverse semigroups. Furthermore one has the following: *

(i)*any nonzero subgroup of is topologically isomorphic to and every nonzero -class of is homeomorphic to and is a clopen subset in ; *
(ii)*the family , where is a base of the topology at the unity of , is a base of the topology at the nonzero element . *

*Proof. *Let be a nonzero subgroup of . Then by Theorem 3.9 of [4, 5] the semigroup is isomorphic to the Brandt -extension of the subgroup for some cardinal . Since is a topological inverse semigroup, we have that is a topological group.

(i) Let be the unity of . We fix arbitrary and define the maps and by the formulae and . We observe that and for all , , and hence the restrictions and are mutually invertible. Since the maps and are continuous on , the map is a homeomorphism and the map is a topological isomorphism. We observe that the subset of is an -class of and is a subgroup of for all . 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 -simple inverse topological semigroups. Definition 3 implies that is a topological Brandt -extension of the topological group .

Gutik and RepovΕ‘, in [19], studied the structure of -simple countably compact topological inverse semigroups. They proved that any -simple countably compact topological inverse semigroup is topologically isomorphic to a topological Brandt -extension of a countably compact topological group in the class of topological inverse semigroups for some finite cardinal . This implies Pavlyk's Theorem (see [20]) on the structure of -simple compact topological inverse semigroups: *every **-simple compact topological inverse semigroup is topologically isomorphic to a topological Brandt **-extension ** of a compact topological group ** in the class of topological inverse semigroups for some finite cardinal *.

The following theorem describes the structure of primitive countably compact topological inverse semigroups.

Theorem 13. *Every primitive countably compact topological inverse semigroup is topologically isomorphic to an orthogonal sum of topological Brandt -extensions of countably compact topological groups in the class of topological inverse semigroups for some finite cardinals . Moreover the family **
determines a base of the topology at zero of .*

*Proof. *By Theorem II.4.3 of [8] the semigroup is an orthogonal sum of Brandt semigroups and hence is an orthogonal sum of Brandt -extensions of groups . We fix any . Since is a topological inverse semigroup, Proposition II.2 [1] implies that is a topological inverse semigroup. By Proposition 12, is a closed subsemigroup of and hence by Theorem 3.10.4 [6], is a countably compact -simple topological inverse semigroup. Then, by Theorem 2 of [19], the semigroup is a topological Brandt -extension of countably compact topological group in the class of topological inverse semigroups for some finite cardinal . This completes the proof of the first assertion of the theorem.

Suppose on the contrary that is not a base at zero of . Then, there exists an open neighbourhood of zero such that for finitely many indexes . Therefore there exists an infinitely family of nonzero disjoint -classes such that for all . Let be an infinite countable subfamily of . We put . Lemma 9 implies that the family is an open countable cover of . Simple observation shows that the cover does not contain a finite subcover. This contradicts to the countable compactness of . The obtained contradiction implies the last assertion of the theorem.

Since any maximal subgroup of a compact topological semigroup is a compact subset in (see [2, Vol. 1, Theorem ] ), Theorem 13 implies the following.

Corollary 14. *Every primitive compact topological inverse semigroup is topologically isomorphic to an orthogonal sum of topological Brandt -extensions of compact topological groups in the class of topological inverse semigroups for some finite cardinals and the family **
determines a base of the topology at zero of .*

Theorem 15. *Every primitive countably compact topological inverse semigroup is a dense subsemigroup of a primitive compact topological inverse semigroup.*

*Proof. *By Theorem 13 the topological semigroup is topologically isomorphic to an orthogonal sum of topological Brandt -extensions of countably compact topological groups in the class of topological inverse semigroups for some finite cardinals . Since any countably compact topological group is pseudocompact, the Comfort-Ross Theorem (see [21, Theorem 4.1]) implies that the Stone-Δech compactification is a compact topological group and the inclusion mapping of into is a topological isomorphism for all . On the orthogonal sum of Brandt -extensions , , we determine a topology as follows:

(a)the family is a base of the topology at the nonzero element , where is a base of the topology at the unity of the compact topological group ;(b) the family

determines a base of the topology at zero of .

By Theorem II.4.3 of [8], is a primitive inverse semigroup and simple verifications show that with the topology is a compact topological inverse semigroup.

We define a map as follows:

Simple verifications show that is a continuous homomorphism. Since is a topological isomorphism, we have that is a topological isomorphism too.

Gutik and RepovΕ‘ in [19] showed that the Stone-Δech compactification of a -simple countably compact topological inverse semigroup is a -simple compact topological inverse semigroup. In this context the following question arises naturally.

*Question 1. *Is the Stone-Δech compactification of a primitive countably compact topological inverse semigroup a topological semigroup (a primitive topological inverse semigroup)?

Theorem 16. *Let be a topological inverse semigroup such that *

(i)* is an -closed (resp., absolutely -closed) semigroup in the class of topological inverse semigroups for any ; *
(ii)*there exists an -closed (resp., absolutely -closed) subsemigroup of in the class of topological inverse semigroups such that for all , . ** Then is an -closed (resp., absolutely -closed) semigroup in the class of topological inverse semigroups.*

*Proof. *We consider the case of absolute -closedness only.

Suppose on the contrary that there exist a topological inverse semigroup and a continuous homomorphism such that is not closed subsemigroup in . Without loss of generality we can assume that . Thus, by Proposition II.2 of [1], is a topological inverse semigroup.

Then, . Let . Since and are topological inverse semigroups we have that is an inverse subsemigroup in and hence . The semigroup which is an absolutely -closed semigroup in the class of topological inverse semigroups implies that there exists an open neighbourhood of the point in such that . Since is a topological inverse semigroup there exist open neighbourhoods and of the points and in , respectively, such that . But and since is the family of absolutely -closed semigroups in the class of topological inverse semigroups, each of the neighbourhoods and intersects infinitely many subsemigroups in , . Hence, . This contradicts the assumption that . The obtained contradiction implies that is an absolutely -closed semigroup in the class of topological inverse semigroups.

The proof in the case of -closeness is similar to the previous one.

Theorem 16 implies the following.

Corollary 17. *Let be an inverse semigroup such that *

(i)* is an algebraically closed (resp., algebraically -closed) semigroup in the class of topological inverse semigroups for any ; *
(ii)*there exists an algebraically closed (resp., algebraically -closed) sub-semigroup of in the class of topological inverse semigroups such that for all , . ** Then is an algebraically closed (resp., algebraically -closed) semigroup in the class of topological inverse semigroups.*

Theorem 16 implies the following.

Theorem 18. *Let a topological inverse semigroup be an orthogonal sum of the family of -closed (resp., absolutely -closed) topological inverse semigroups with zeros in the class of topological inverse semigroups. Then is an -closed (resp., absolutely -closed) topological inverse semigroup in the class of topological inverse semigroups.*

Corollary 17 implies the following.

Corollary 19. *Let an inverse semigroup be an orthogonal sum of the family of algebraically closed (resp., algebraically -closed) inverse semigroups with zeros in the class of topological inverse semigroups. Then is an algebraically closed (resp., algebraically -closed) inverse semigroup in the class of topological inverse semigroups.*

Recall in [22], that a topological group is called *absolutely closed* if is a closed subgroup of any topological group which contains as a subgroup. In our terminology such topological groups are called -closed in the class of topological groups. In [23] Raikov proved that a topological group is absolutely closed if and only if it is Raikov complete, that is, is complete with respect to the two sided uniformity.

A topological group is called *-complete* if for every continuous homomorphism into a topological group the subgroup of is closed [24]. The -completeness is preserved under taking products and closed central subgroups [24].

Gutik and Pavlyk in [13] showed that a topological group is -closed (resp., absolutely -closed) in the class of topological inverse semigroups if and only if is absolutely closed (resp., -complete).

Theorem 20. *For a primitive topological inverse semigroup the following assertions are equivalent: *

(i)*every maximal subgroup of is absolutely closed; *
(ii)*the semigroup with every inverse semigroup topology is -closed in the class of topological inverse semigroups. *

*Proof. *(i)(ii) Suppose that a primitive topological inverse semigroup is an orthogonal sum of topological Brandt -extensions of topological groups in the class of topological inverse semigroups and every topological group is absolutely closed. Then, by Theorem 3 of [9] any topological Brandt -extension of topological group is -closed in the class of topological inverse semigroups. Theorem 18 implies that is an -closed topological inverse semigroup in the class of topological inverse semigroups.

(ii)(i) Let be any maximal nonzero subgroup of . Since is a primitive topological inverse semigroup, we have that is an orthogonal sum of Brandt -extensions of topological groups and hence there exists a topological Brandt -extension , , such that contains the maximal subgroup and is a subsemigroup of .

Suppose on the contrary that the topological group is not absolutely closed. Then there exists a topological group which contains as a dense proper subgroup. For every we put

On the orthogonal sum of Brandt -extensions , , we determine a topology as follows:

(a)the family is a base of the topology at the nonzero element , where is a base of the topology at the unity of the topological group ;
(b)the zero is an isolated point in .

By Theorem II.4.3 of [8], is a primitive inverse semigroup and simple verifications show that with the topology is a topological inverse semigroup. Also we observe that the semigroup which is induced from topology is a topological inverse semigroup which is a dense proper inverse sub-semigroup of . The obtained contradiction completes the statement of the theorem.

Theorem 20 implies the following.

Corollary 21. *For a primitive inverse semigroup the following assertions are equivalent: *

(i)*every maximal subgroup of is algebraically closed in the class of topological inverse semigroups; *
(ii)*the semigroup is algebraically closed in the class of topological inverse semigroups. *

Theorem 22. *For a primitive topological inverse semigroup the following assertions are equivalent: *

(i)*every maximal subgroup of is -complete; *
(ii)*the semigroup with every inverse semigroup topology is absolutely -closed in the class of topological inverse semigroups. *

*Proof. *(i)(ii) Suppose that a primitive topological inverse semigroup is an orthogonal sum of topological Brandt -extensions of topological groups in the class of topological inverse semigroups and every topological group is -complete. Then by Theorem 14 of [13] any topological Brandt -extension of topological group is absolutely -closed in the class of topological inverse semigroups. Theorem 18 implies that is an absolutely -closed topological inverse semigroup in the class of topological inverse semigroups.

(ii)(i) Let be any maximal nonzero subgroup of . Since is a primitive topological inverse semigroup, is an orthogonal sum of Brandt -extensions of topological groups . Hence there exists a topological Brandt -extension , , such that contains the maximal subgroup and is a subsemigroup of .

Suppose on the contrary that the topological group is not -completed. Then there exist a topological group and continuous homomorphism such that is a dense proper subgroup of . On the Brandt -extension , we determine a topology as follows:

(a)the family is a base of the topology at the nonzero element , where is a base of the topology at the unity of the topological group ;
(b)the zero is an isolated point in .

Then is an inverse semigroup and simple verifications show that with the topology is a topological inverse semigroup.

On the orthogonal sum of Brandt -extensions , , we determine a topology as follows:

(a)the family is a base of the topology at the nonzero element , where is a base of the topology at the unity of the topological group ;
(b)the zero is an isolated point in .

By Theorem II.4.3 of [8], is a primitive inverse semigroup and simple verifications show that with the topology is a topological inverse semigroup.

We define the map as follows:

where is zero of . Evidently the defined map is a continuous homomorphism. Then is a dense proper inverse subsemigroup of the topological inverse semigroup . The obtained contradiction completes the statement of the theorem.

Theorem 22 implies the following.

Corollary 23. *For a primitive inverse semigroup the following assertions are equivalent: *

(i)*every maximal subgroup of is algebraically -closed in the class of topological inverse semigroups; *
(ii)*the semigroup is algebraically -closed in the class of topological inverse semigroups. *

#### Acknowledgment

The authors are grateful to the referee for several comments and suggestions which have considerably improved the original version of the manuscript.