Brandt Extensions and Primitive Topological Inverse Semigroups

Abstract

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.

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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 𝑆¹ [𝑆⁰] 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 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:

𝑒⩽𝑓iff𝑒𝑓=𝑓𝑒=𝑒. (1) 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:

(𝛼,𝑎,𝛽)⋅(𝛾,𝑏,𝛿)=(𝛼,𝑎𝑏,𝛿),if𝛽=𝛾,0,if𝛽≠𝛾, (2) 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=𝑆¹, 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 𝐵⁰_𝜆(𝑆)=𝐵_𝜆(𝑆)/𝒥 and we shall call 𝐵⁰_𝜆(𝑆) the Brandt 𝜆⁰-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 𝐴_𝛼,𝛽={(𝛼,𝑠,𝛽)∣𝑠∈𝐴⧵{0}}∪{0} if 0∈𝐴, for 𝛼,𝛽∈𝐼_𝜆. If ℐ is a trivial semigroup (i.e., ℐ contains only one element), then by ℐ⁰ we denote the semigroup ℐ with the adjoined zero. Obviously, for any 𝜆⩾2 the Brandt 𝜆⁰-extension of the semigroup ℐ⁰ is isomorphic to the semigroup of 𝐼_𝜆×𝐼_𝜆-matrix units and any Brandt 𝜆⁰-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 𝐵⁰_𝜆(1) the subsemigroup of 𝐼_𝜆×𝐼_𝜆-matrix units of the Brandt 𝜆⁰-extension of a monoid 𝑆 with zero. A completely 0-simple inverse semigroup is called a Brandt semigroup [8]. A semigroup 𝑆 is a Brandt semigroup if and only if 𝑆 is isomorphic to a Brandt 𝜆-extension 𝐵_𝜆(𝐺) of some group 𝐺 [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 𝑆 is a primitive inverse semigroup if and only if 𝑆 is an orthogonal sum of Brandt semigroups [8, Theorem II.4.3].

Green’s relations ℒ, ℛ, and ℋ on a semigroup 𝑆 are defined by

(i)𝑎ℒ𝑏 if and only if 𝑎∪𝑆𝑎=𝑏∪𝑆𝑏; (ii)𝑎ℛ𝑏 if and only if 𝑎∪𝑎𝑆=𝑏∪𝑏𝑆; (iii)ℋ=ℒ∩ℛ

for 𝑎,𝑏∈𝑆. For details about Green's relations, see [4, Section 2.1] or [11]. We observe that two nonzero elements (𝛼₁,𝑠,𝛽₁) and (𝛼₂,𝑡,𝛽₂) of a Brandt semigroup 𝐵_𝜆(𝐺), 𝑠,𝑡∈𝐺, 𝛼₁,𝛼₂,𝛽₁,𝛽₂∈𝐼_𝜆, are ℋ-equivalent if and only if 𝛼₁=𝛼₂ and 𝛽₁=𝛽₂ (see [8, page 93]).

By 𝒮 we denote some class of topological semigroups.

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

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

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 ℎ-closed.

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,𝑆¹_,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 𝒮₀ be some class of topological semigroups with zero. Let 𝜆 be a cardinal ⩾1 and (𝑆,𝜏)∈𝒮₀. Let 𝜏_𝐵 be a topology on 𝐵⁰_𝜆(𝑆) such that
(a)(𝐵⁰_𝜆(𝑆),𝜏_𝐵)∈𝒮₀; (b)𝜏_𝐵∣_{(𝛼,𝑆,𝛼)∪{0}}=𝜏 for some 𝛼∈𝐼_𝜆.
Then (𝐵⁰_𝜆(𝑆),𝜏_𝐵) is called a topological Brandt 𝜆⁰-extension of (𝑆,𝜏) in 𝒮₀. If 𝒮₀ coincides with the class of all topological semigroups, then (𝐵⁰_𝜆(𝑆),𝜏_𝐵) is called a topological Brandt 𝜆⁰-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.

Theorem 5. Let 𝑆 be a topological inverse monoid with zero. Then the following conditions are equivalent:
(i)𝑆 is an (absolutely) 𝐻-closed semigroup in the class of topological inverse semigroups; (ii) there exists a cardinal 𝜆⩾1 such that any topological Brandt 𝜆⁰-extension 𝐵⁰_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 𝜆⁰-extension 𝐵⁰_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 𝜆⁰-extension 𝐵⁰_𝜆(𝒩) of 𝒩 with the following properties was constructed in [10]:

(i)𝐵⁰_𝜆(𝒩) is an absolutely 𝐻-closed semigroup for any infinite cardinal 𝜆; (ii)𝐵⁰_𝜆(𝒩) is a 𝜎-compact inverse topological semigroup for any countable cardinal 𝜆; (iii)𝐵⁰_𝜆(𝒩) contains an absolutely 𝐻-closed ideal 𝐽 such that the Rees quotient semigroup 𝐵⁰_𝜆(𝒩)/𝐽 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 𝜆⁰-extension 𝐵⁰_𝜆(𝑇) of 𝑇, such that the semigroups 𝐵_𝜆(𝑆) and 𝐵⁰_𝜆(𝑇) are topologically isomorphic. Algebraic properties of Brandt 𝜆⁰-extensions of monoids with zero and nontrivial homomorphisms between Brandt 𝜆⁰-extensions of monoids with zero and a category whose objects are ingredients of the construction of Brandt 𝜆⁰-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 𝜆⁰-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.

Proof. Let 𝑥∈𝐸^∗. Since 𝐸 is a Hausdorff topological semilattice, for every open neighbourhood 𝑈(𝑥)∋0̸ of the point 𝑥 there exists an open neighbourhood 𝑉(𝑥) of 𝑥 such that 𝑉(𝑥)⋅𝑉(𝑥)⊆𝑈(𝑥). If 𝑥 is not an isolated point of 𝐸, then 𝑉(𝑥)⋅𝑉(𝑥)∋0 which contradicts to the choice of 𝑈(𝑥). This implies the assertion of the lemma.

Lemma 7. Let 𝑆 be a primitive inverse topological semigroup and let 𝑆 be an orthogonal sum of the family {𝐵_𝜆𝑖(𝐺_𝑖)}_𝑖∈𝒜 of topological Brandt semigroups with zeros, that is, 𝑆=∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐺_𝑖). Let (𝛼_𝑖,𝑔_𝑖,𝛽_𝑖)∈𝐵_𝜆𝑖(𝐺_𝑖) be a nonzero element of 𝑆. Then
(i) there exists an open neighbourhood 𝑈 of (𝛼_𝑖,𝑔_𝑖,𝛽_𝑖) such that 𝑈⊆𝑆^∗_{𝛼𝑖,𝛽𝑖}⊆𝐵_𝜆𝑖(𝐺_𝑖); (ii) every nonzero idempotent of 𝑆 is an isolated point in 𝐸(𝑆).

Proof. (i) Suppose to the contrary that 𝑈⊆𝑆̸^∗_{𝛼𝑖,𝛽𝑖}⊆𝐵_𝜆𝑖(𝐺_𝑖) for any open neighbourhood 𝑈 of the point (𝛼_𝑖,𝑔_𝑖,𝛽_𝑖). Since 𝑆 is a Hausdorff space, there exists an open neighbourhood 𝑉 of the point (𝛼_𝑖,𝑔_𝑖,𝛽_𝑖) such that 0∉𝑉. The continuity of the semigroup operation in 𝑆 implies that there exists an open neighbourhood 𝑊 of the point (𝛼_𝑖,𝑔_𝑖,𝛽_𝑖) such that (𝛼_𝑖,1_𝑖,𝛼_𝑖)⋅𝑊⋅(𝛽_𝑖,1_𝑖,𝛽_𝑖)⊆𝑉. Since 𝑊⊆𝑆̸^∗_{𝛼𝑖,𝛽𝑖}, we have that 0∈𝑉, a contradiction.
Statement (ii) follows from Lemma 6.

Lemma 7 implies the following.

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

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

Proof. Let 𝐻(𝑒,𝑓) be a nonzero ℋ-class in 𝑆 for 𝑒,𝑓∈(𝐸(𝑆))^∗, that is,
𝐻(𝑒,𝑓)={𝑥∈𝑆∣𝑥⋅𝑥⁻¹=𝑒,𝑥⁻¹⋅𝑥=𝑓}. (3) 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 ℤ⁰=ℤ∪{∞} as follows:
𝑥⋅∞=∞⋅𝑥=∞⋅∞=∞,∀𝑥∈ℤ. (4) We observe that ℤ⁰ is the group with adjoined zero ∞. We determine a semigroup topology 𝜏 on ℤ⁰ as follows:
(i)every nonzero element of ℤ⁰ is an isolated point; (ii)the family ℬ(∞)={𝑈_𝑛={∞}∪{𝑥∈ℤ∣𝑥⩾𝑛}∣𝑛isapositiveinteger} is a base of the topology 𝜏 at the point ∞.
A simple verification shows that (ℤ⁰,𝜏) is a primitive inverse locally compact topological semigroup.

Proposition 11. (ℤ⁰,𝜏) is an 𝐻-closed topological semigroup.

Proof. Suppose that ℤ⁰ is embedded into a topological semigroup 𝑇. If {𝑛_𝜄} is a net in ℕ for which {−𝑛_𝜄} converges in 𝑇 to 𝑡∈𝑇⧵ℤ⁰, then the equation −𝑛_𝜄+(𝑛_𝜄+𝑘)=𝑘 implies that 𝑡⋅∞=𝑘 for every 𝑘∈ℕ—which is impossible. So ℤ⁰ is closed in 𝑇.

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^𝛾𝛿_𝛼𝛽∶𝐵_𝜆(𝐺)→𝐵_𝜆(𝐺) and 𝜑^𝛼𝛽_𝛾𝛿∶𝐵_𝜆(𝐺)→𝐵_𝜆(𝐺) by the formulae 𝜑^𝛾𝛿_𝛼𝛽(𝑠)=(𝛾,𝑒,𝛼)⋅𝑠⋅(𝛽,𝑒,𝛿) and 𝜑^𝛼𝛽_𝛾𝛿(𝑠)=(𝛼,𝑒,𝛾)⋅𝑠⋅(𝛿,𝑒,𝛽),𝑠∈𝐵_𝜆(𝐺). We observe that 𝜑^𝛼𝛽_𝛾𝛿(𝜑^𝛾𝛿_𝛼𝛽((𝛼,𝑥,𝛽)))=(𝛼,𝑥,𝛽) and 𝜑^𝛾𝛿_𝛼𝛽(𝜑^𝛼𝛽_𝛾𝛿((𝛾,𝑥,𝛿)))=(𝛾,𝑥,𝛿) for all 𝛼,𝛽,𝛾,𝛿∈𝐼_𝜆, 𝑥∈𝐺, and hence the restrictions 𝜑^𝛾𝛿_𝛼𝛽∣_{(𝛼,𝐺,𝛽)} and 𝜑^𝛼𝛽_𝛾𝛿|_{(𝛾,𝐺,𝛿)} are mutually invertible. Since the maps 𝜑^𝛾𝛿_𝛼𝛽 and 𝜑^𝛼𝛽_𝛾𝛿 are continuous on 𝐵_𝜆(𝐺), the map 𝜑^𝛾𝛿_𝛼𝛽∣_{(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∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of topological Brandt 𝜆_𝑖-extensions 𝐵_𝜆𝑖(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
ℬ(0)={𝑆⧵(𝐵_𝜆𝑖1(𝐺_𝑖1)∪𝐵_𝜆𝑖2(𝐺_𝑖2)∪⋯∪𝐵_𝜆𝑖𝑛(𝐺_𝑖𝑛))^∗∣𝑖₁,𝑖₂,…,𝑖_𝑛∈𝒜,𝑛∈ℕ} (5) 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∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of Brandt 𝜆_𝑖-extensions 𝐵_𝜆𝑖(𝐺_𝑖) of groups 𝐺_𝑖. We fix any 𝑖₀∈𝒜. Since 𝑆 is a topological inverse semigroup, Proposition II.2 [1] implies that 𝐵_𝜆𝑖0(𝐺_𝑖0) is a topological inverse semigroup. By Proposition 12, 𝐵_𝜆𝑖0(𝐺_𝑖0) is a closed subsemigroup of 𝑆 and hence by Theorem 3.10.4 [6], 𝐵_𝜆𝑖0(𝐺_𝑖0) is a countably compact 0-simple topological inverse semigroup. Then, by Theorem 2 of [19], the semigroup 𝐵_𝜆𝑖0(𝐺_𝑖0) is a topological Brandt 𝜆_𝑖-extension of countably compact topological group 𝐺_𝑖0 in the class of topological inverse semigroups for some finite cardinal 𝜆_𝑖0⩾1. This completes the proof of the first assertion of the theorem.
Suppose on the contrary that ℬ(0) is not a base at zero 0 of 𝑆. Then, there exists an open neighbourhood 𝑈(0) of zero 0 such that 𝑈(0)⋃(𝐵_𝜆𝑖1(𝐺_𝑖1)∪𝐵_𝜆𝑖2(𝐺_𝑖2)∪⋯∪𝐵_𝜆𝑖𝑛(𝐺_𝑖𝑛))^∗≠𝑆 for finitely many indexes 𝑖₁,𝑖₂,…,𝑖_𝑛∈𝒜. Therefore there exists an infinitely family ℱ of nonzero disjoint ℋ-classes such that 𝐻⊆𝑈̸(0) for all 𝐻∈ℱ. Let ℱ₀ be an infinite countable subfamily of ℱ. We put 𝑊=⋃{𝐻∣𝐻∈ℱ⧵ℱ₀}. Lemma 9 implies that the family 𝒞={𝑈(0),𝑊}∪ℱ₀ 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∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of topological Brandt 𝜆_𝑖-extensions 𝐵_𝜆𝑖(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
ℬ(0)={𝑆⧵(𝐵_𝜆𝑖1(𝐺_𝑖1)∪𝐵_𝜆𝑖2(𝐺_𝑖2)∪⋯∪𝐵_𝜆𝑖𝑛(𝐺_𝑖𝑛))^∗∣𝑖₁,𝑖₂,…,𝑖_𝑛∈𝒜,𝑛∈ℕ} (6) 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∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of topological Brandt 𝜆_𝑖-extensions 𝐵_𝜆𝑖(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 a topological isomorphism for all 𝑖∈𝒜. On the orthogonal sum ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of Brandt 𝜆-extensions 𝐵_𝜆𝑖(𝛽(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_{𝛽(𝐺𝑖)}(𝑒_𝑖)} is a base of the topology at the nonzero element (𝛼_𝑖,𝑔_𝑖,𝛽_𝑖)∈𝐵_𝜆𝑖(𝛽(𝐺_𝑖)), where ℬ_{𝛽(𝐺𝑖)}(𝑒_𝑖) is a base of the topology at the unity 𝑒_𝑖 of the compact topological group 𝛽(𝐺_𝑖);(b) the family
ℬ(0)={𝑆⧵(𝐵_𝜆𝑖1(𝛽(𝐺_𝑖1))∪𝐵_𝜆𝑖2(𝛽(𝐺_𝑖2))∪⋯∪𝐵_𝜆𝑖𝑛(𝛽(𝐺_𝑖𝑛)))^∗∣𝑖₁,𝑖₂,…,𝑖_𝑛∈𝒜,𝑛∈ℕ} (7)
determines a base of the topology at zero 0 of ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐺_𝑖).
By Theorem II.4.3 of [8], ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝛽(𝐺_𝑖)) is a primitive inverse semigroup and simple verifications show that ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝛽(𝐺_𝑖)) with the topology 𝜏 is a compact topological inverse semigroup.
We define a map 𝑓∶∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐺_𝑖)→∑_𝑖∈𝒜𝐵_𝜆𝑖(𝛽(𝐺_𝑖)) as follows:
𝑓(0)=0,𝑓((𝛼_𝑖,𝑔_𝑖,𝛽_𝑖))=(𝛼_𝑖,𝑓_𝑖(𝑔_𝑖),𝛽_𝑖)∈𝐵_𝜆𝑖(𝛽(𝐺_𝑖))for(𝛼_𝑖,𝑔_𝑖,𝛽_𝑖)∈𝐵_𝜆𝑖(𝐺_𝑖). (8) Simple verifications show that 𝑓 is a continuous homomorphism. Since 𝑓_𝑖∶𝐺_𝑖→𝛽(𝐺_𝑖) is a topological isomorphism, we have that 𝑓∶∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐺_𝑖)→∑_𝑖∈𝒜𝐵_𝜆𝑖(𝛽(𝐺_𝑖)) is a topological isomorphism too.

Gutik and Repovš in [19] showed that the Stone-Čech compactification 𝛽(𝑇) of a 0-simple countably compact topological inverse semigroup 𝑇 is a 0-simple compact topological inverse semigroup. In this context the following question arises naturally.

Question 1. Is the Stone-Čech compactification 𝛽(𝑇) of a primitive countably compact topological inverse semigroup 𝑇 a topological semigroup (a primitive topological inverse semigroup)?

Theorem 16. Let 𝑆=⋃_𝛼∈𝒜𝑆_𝛼 be a topological inverse semigroup such that
(i)𝑆_𝛼 is an 𝐻-closed (resp., absolutely 𝐻-closed) semigroup in the class of topological inverse semigroups for any 𝛼∈𝒜; (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, (𝑉(𝑥)⋅𝑉(𝑥⁻¹)⋅𝑉(𝑥))∩ℎ(𝑇)≠∅. This contradicts the assumption that 𝑈(𝑥)∩ℎ(𝑇)=∅. 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∈𝒜𝑆_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.

Theorem 18. Let a topological inverse semigroup 𝑆 be an orthogonal sum of the family {𝑆_𝛼}_𝛼∈𝒜 of 𝐻-closed (resp., absolutely 𝐻-closed) topological inverse semigroups with zeros in the class of topological inverse semigroups. Then 𝑆 is an 𝐻-closed (resp., absolutely 𝐻-closed) topological inverse semigroup in the class of topological inverse semigroups.

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 ℎ-completeness 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).

Theorem 20. For a primitive topological inverse semigroup 𝑆 the following assertions are equivalent:
(i)every maximal subgroup of 𝑆 is absolutely closed; (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∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of topological Brandt 𝜆_𝑖-extensions 𝐵_𝜆𝑖(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_𝜆𝑖(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∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of Brandt 𝜆-extensions 𝐵_𝜆𝑖(𝐺_𝑖) of topological groups 𝐺_𝑖 and hence there exists a topological Brandt 𝜆_𝑖0-extension 𝐵_𝜆𝑖0(𝐺_𝑖0), 𝑖∈𝒜, such that 𝐵_𝜆𝑖0(𝐺_𝑖0) contains the maximal subgroup 𝐺 and 𝐵_𝜆𝑖0(𝐺_𝑖0) is a subsemigroup of 𝑆.
Suppose on the contrary that the topological group 𝐺=𝐺_𝑖0 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
𝐻_𝑖=𝐺_𝑖,if𝑖≠𝑖₀,𝐻,if𝑖=𝑖₀. (9) On the orthogonal sum ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐻_𝑖) of Brandt 𝜆-extensions 𝐵_𝜆𝑖(𝐻_𝑖), 𝑖∈𝒜, we determine a topology 𝜏₀ as follows:
(a)the family ℬ(𝛼_𝑖,𝑔_𝑖,𝛽_𝑖)={(𝛼_𝑖,𝑔_𝑖⋅𝑈,𝛽_𝑖)|𝑈∈ℬ_𝐻𝑖(𝑒_𝑖)} is a base of the topology at the nonzero element (𝛼_𝑖,𝑔_𝑖,𝛽_𝑖)∈𝐵_𝜆𝑖(𝐻_𝑖), where ℬ_𝐻𝑖(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∈𝒜𝐵_𝜆𝑖(𝐻_𝑖),𝜏₀).
By Theorem II.4.3 of [8], ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐻_𝑖) is a primitive inverse semigroup and simple verifications show that ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐻_𝑖) with the topology 𝜏₀ is a topological inverse semigroup. Also we observe that the semigroup ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) which is induced from (∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐻_𝑖),𝜏₀) topology is a topological inverse semigroup which is a dense proper inverse sub-semigroup of (∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐻_𝑖),𝜏₀). 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∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of topological Brandt 𝜆_𝑖-extensions 𝐵_𝜆𝑖(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_𝜆𝑖(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∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of Brandt 𝜆-extensions 𝐵_𝜆𝑖(𝐺_𝑖) of topological groups 𝐺_𝑖. Hence there exists a topological Brandt 𝜆_𝑖0-extension 𝐵_𝜆𝑖0(𝐺_𝑖0), 𝑖∈𝒜, such that 𝐵_𝜆𝑖0(𝐺_𝑖0) contains the maximal subgroup 𝐺 and 𝐵_𝜆𝑖0(𝐺_𝑖0) is a subsemigroup of 𝑆.
Suppose on the contrary that the topological group 𝐺=𝐺_𝑖0 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_𝜆𝑖0(𝐻), we determine a topology 𝜏_𝐻 as follows:
(a)the family ℬ(𝛼_𝑖0,𝑔_𝑖0,𝛽_𝑖0)={(𝛼_𝑖0,𝑔_𝑖⋅𝑈,𝛽_𝑖0)∣𝑈∈ℬ_𝐻(𝑒)} is a base of the topology at the nonzero element (𝛼_𝑖0,𝑔_𝑖,𝛽_𝑖0)∈𝐵_𝜆𝑖(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_𝜆𝑖0(𝐻),𝜏_𝐻).
Then 𝐵_𝜆𝑖0(𝐻) is an inverse semigroup and simple verifications show that 𝐵_𝜆𝑖0(𝐻) with the topology 𝜏_𝐻 is a topological inverse semigroup.
On the orthogonal sum ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) of Brandt 𝜆-extensions 𝐵_𝜆𝑖(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_𝐺𝑖(𝑒_𝑖)} is a base of the topology at the nonzero element (𝛼_𝑖,𝑔_𝑖,𝛽_𝑖)∈𝐵_𝜆𝑖(𝐺_𝑖), where ℬ_𝐺𝑖(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∈𝒜𝐵_𝜆𝑖(𝐺_𝑖),𝜏_⋆).
By Theorem II.4.3 of [8], ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) is a primitive inverse semigroup and simple verifications show that ∑_𝑖∈𝒜𝐵_𝜆𝑖(𝐺_𝑖) with the topology 𝜏_⋆ is a topological inverse semigroup.
We define the map 𝑓∶𝑆→𝐵_𝜆𝑖0(𝐻) as follows:
𝑓(𝑥)=ℎ(𝑥),if𝑥∈𝐵_𝜆𝑖0(𝐺_𝑖0),0,if𝑥∉𝐵_𝜆𝑖0(𝐺_𝑖0), (10) where 0 is zero of 𝑆. Evidently the defined map 𝑓 is a continuous homomorphism. Then 𝑓(𝑆)=𝐵_𝜆𝑖0(ℎ(𝐺_𝑖0)) is a dense proper inverse subsemigroup of the topological inverse semigroup (𝐵_𝜆𝑖0(𝐻),𝜏_𝐻). The obtained contradiction completes the statement of the theorem.

Theorem 22 implies the following.

Corollary 23. For a primitive inverse semigroup 𝑆 the following assertions are equivalent:
(i)every maximal subgroup of 𝑆 is algebraically ℎ-closed in the class of topological inverse semigroups; (ii)the semigroup 𝑆 is algebraically ℎ-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.