^{1}

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We study (countably) compact and (absolutely)

In this paper all spaces are Hausdorff. A semigroup is a nonempty set with a binary associative operation. A semigroup

A

Further we shall follow the terminology of [

If

Let

A nontrivial inverse semigroup is called a

Green’s relations

for

By

A semigroup

A topological semigroup

A semigroup

Absolutely

Let

Then

Let

Then

Gutik and Pavlyk in [

there exists a cardinal

for any cardinal

In [

In [

Let

there exists a cardinal

for each cardinal

Also, an example of an absolutely

We observe that for any topological Brandt

In [

In this paper we study (countably) compact and (absolutely)

Let

Let

Let

there exists an open neighbourhood

every nonzero idempotent of

(i) Suppose to the contrary that

Statement (ii) follows from Lemma

Lemma

Every nonzero

If

Let

The following example shows that the statement of Lemma

Let

every nonzero element of

the family

A simple verification shows that

Suppose that

Every completely

any nonzero subgroup of

the family

Let

(i) Let

(ii) The statement follows from assertion (i) and Theorem 4.3 of [

We observe that Example

Gutik and Repovš, in [

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

Every primitive countably compact topological inverse semigroup

By Theorem II.4.3 of [

Suppose on the contrary that

Since any maximal subgroup of a compact topological semigroup

Every primitive compact topological inverse semigroup

Every primitive countably compact topological inverse semigroup

By Theorem

the family

the family

determines a base of the topology at zero

By Theorem II.4.3 of [

We define a map

Gutik and Repovš in [

Is the Stone-Čech compactification

Let

there exists an

Then

We consider the case of absolute

Suppose on the contrary that there exist a topological inverse semigroup

Then,

The proof in the case of

Theorem

Let

there exists an algebraically closed (resp., algebraically

Then

Theorem

Let a topological inverse semigroup

Corollary

Let an inverse semigroup

Recall in [

A topological group

Gutik and Pavlyk in [

For a primitive topological inverse semigroup

every maximal subgroup of

the semigroup

(i)

(ii)

Suppose on the contrary that the topological group

the family

the zero

By Theorem II.4.3 of [

Theorem

For a primitive inverse semigroup

every maximal subgroup of

the semigroup

For a primitive topological inverse semigroup

every maximal subgroup of

the semigroup

(i)

(ii)

Suppose on the contrary that the topological group

the family

the zero

Then

On the orthogonal sum

the family

the zero

By Theorem II.4.3 of [

We define the map

Theorem

For a primitive inverse semigroup

every maximal subgroup of

the semigroup

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