Research Article | Open Access

Volume 2010 |Article ID 671401 | https://doi.org/10.1155/2010/671401

Tetyana Berezovski, Oleg Gutik, Kateryna Pavlyk, "Brandt Extensions and Primitive Topological Inverse Semigroups", International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 671401, 13 pages, 2010. https://doi.org/10.1155/2010/671401

# Brandt Extensions and Primitive Topological Inverse Semigroups

Academic Editor: Volker Runde
Received20 Jul 2009
Accepted01 Feb 2010
Published07 Mar 2010

#### 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 . 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 . Obviously, is the zero of is an ideal of . We put and we shall call the Brandt -extension of the semigroup with zero . 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 . 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 . A semigroup is a primitive inverse semigroup if and only if is an orthogonal sum of Brandt semigroups [8, Theorem II.4.3].

Green’s 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 . 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 . 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 . There, they were called absolutely maximal and algebraic maximal, respectively.

Definition 3 (see ). 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 ). 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  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  they showed that the similar statement holds for absolutely -closed topological semigroups in the class of topological inverse semigroups.

In , 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 :

(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 . 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 .

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 —which 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 .

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 Repovš, in , 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 ) 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  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  implies that is a topological inverse semigroup. By Proposition 12, is a closed subsemigroup of and hence by Theorem 3.10.4 , is a countably compact -simple topological inverse semigroup. Then, by Theorem 2 of , 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-Čech 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 , 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 Repovš in  showed that the Stone-Čech 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-Čech 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 , 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 , 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  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 . The -completeness is preserved under taking products and closed central subgroups .

Gutik and Pavlyk in  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  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 , 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  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 , 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.

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