On pseudocompact topological Brandt $\lambda^0$-extensions of semitopological monoids

In the paper we investigate topological properties of a topological Brandt $\lambda^0$-extension $B^0_{\lambda}(S)$ of a semitopological monoid $S$ with zero. In particular we prove that for every Tychonoff pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) semitopological monoid $S$ with zero there exists a unique semiregular pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) extension $B^0_{\lambda}(S)$ of $S$ and establish theirs Stone-\v{C}ech and Bohr compactifications. We also describe a category whose objects are ingredients in the constructions of pseudocompact (resp., countably compact, sequentially compact, compact) topological Brandt $\lambda^0$-extensions of pseudocompact (resp., countably compact, sequentially compact, compact) semitopological monoids with zeros.


Introduction, preliminaries and definitions
In this article we shall follow the terminology of [4,5,8,17,19,21]. All topological spaces are assumed to be Hausdorff. By ω we shall denote the first infinite ordinal and by |A| the cardinality of the set A. All cardinals we shall identify with their corresponding initial ordinals. If Y is a subspace of a topological space X and A ⊆ Y , then by cl Y (A) and int Y (A) we shall denote the topological closure and the interior of the subset A in Y , respectively.
A semigroup is a non-empty 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 the inverse of x and denoted by x −1 . The map defined on an inverse semigroup S which maps every element x of S to its inverse x −1 is called the inversion.
We shall denote the semigroup of λ × λ-matrix units by B λ and the subsemigroup of λ × λmatrix units of the Brandt λ 0 -extension of a monoid S with zero by B 0 λ (1). We always consider the Brandt λ 0 -extension only of a monoid with zero. Obviously, for any monoid S with zero we have B 0 1 (S) = S. Note that every Brandt λ-extension of a group G is isomorphic to the Brandt λ 0 -extension of the group G 0 with adjoined zero. The Brandt λ 0 -extension of the group with adjoined zero is called a Brandt semigroup [5,19]. A semigroup S is a Brandt semigroup if and only if S is a completely 0-simple inverse semigroup [3,18] (cf. also [19,Theorem II.3.5]). We also observe that the trivial semigroup I is isomorphic to the Brandt λ 0 -extension of I for every cardinal λ 1. We shall say that the Brandt λ 0 -extension B 0 λ (S) of a semigroup S is finite if the cardinal λ is finite.
For a topological space X, a family {A s | s ∈ S } of subsets of X is called locally finite if for every point x ∈ X there exists an open neighbourhood U of x in X such that the set {s ∈ S | U ∩ A s = ∅} is finite. A set A of a topological space X is called regular open if A = int X (cl X (A)).
We recall that a topological space X is said to be • semiregular if X has a base with regular open subsets; • compact if each open cover of X has a finite subcover; • sequentially compact if each sequence {x i } i∈ω of X has a convergent subsequence in X; • countably compact if each open countable cover of X has a finite subcover; • countably compact at a subset A ⊆ X if every infinite subset B ⊆ A has an accumulation point x in X; • countably pracompact if there exists a dense subset A in X such that X is countably compact at A; • pseudocompact if each locally finite open cover of X is finite. According to Theorem 3.10.22 of [8], a Tychonoff topological space X is pseudocompact if and only if each continuous real-valued function on X is bounded. Also, a Hausdorff topological space X is pseudocompact if and only if every locally finite family of non-empty open subsets of X is finite. Every compact space and every sequentially compact space are countably compact, every countably compact space is countably pracompact, and every countably pracompact space is pseudocompact (see [2] and [8]).
We recall that the Stone-Čech compactification of a Tychonoff space X is a compact Hausdorff space βX containing X as a dense subspace so that each continuous map f : X → Y to a compact Hausdorff space Y extends to a continuous map f : βX → Y .
A semitopological (resp., topological ) semigroup is a topological space together with a separately (resp., jointly) continuous semigroup operation. Definition 1.1 ( [12]). Let S be some class of semitopological semigroups with zero. Let λ be any cardinal 1, and (S, τ ) ∈ S . Let τ B be a topology on B 0 λ (S) such that: a) (B 0 λ (S), τ B ) ∈ S ; and b) τ B | Sα,α = τ for some α ∈ λ. Then (B 0 λ (S), τ B ) is called the topological Brandt λ 0 -extension of (S, τ ) in S . The notion of the topological Brandt λ 0 -extension of topological semigroup was introduced in the paper [12] on purpose of constructing an example of an absolutely H-closed semigroup S with an absolutely H-closed ideal I such that S/I is not a topological semigroup. In [16] Gutik and Repovš described compact topological Brandt λ 0 -extensions in the class of topological semigroups and countably compact topological Brandt λ 0 -extensions in the class of topological inverse semigroups, and correspondent categories. Also, in [13] countably compact topological Brandt λ 0 -extensions in the class of topological semigroups were described. This paper is motivated by the result obtained in [11] which states that every Hausdorff pseudocompact topology τ on the infinite semigroup of matrix units B λ such that (B λ , τ ) is a semitopological semigroup, is compact.
In this paper we investigate topological properties of a topological Brandt λ 0 -extension B 0 λ (S) of a semitopological monoid S with zero. In particular we prove that for every Tychonoff pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) semitopological monoid S with zero there exists a unique semiregular pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) extension B 0 λ (S) of S and establish their Stone-Čech and Bohr compactifications. We also describe a category whose objects are ingredients in the constructions of pseudocompact (resp., countably compact, sequentially compact, compact) topological Brandt λ 0 -extensions of pseudocompact (resp., countably compact, sequentially compact, compact) semitopological monoids with zeros.

Pseudocompact Brandt λ 0 -extensions of semitopological monoids
The following three lemmas describe the general structure of topological Brandt λ 0 -extensions of semitopological semigroups in the class of semitopological semigroups.
, respectively, are continuous maps and, moreover, the restrictions ϕ : S α,α → S β,β is an isomorphism of subsemigroups S α,α and S β,β of B 0 λ (S). This completes the proof of the lemma.
. Then the continuity of the map ϕ (α,α) and since S α,β = B 0 λ (S) \ (γ,δ)∈(λ×λ)\(α,β) S * γ,δ we have that S α,β is a closed subset of the topological space (B 0 λ (S), τ * B ). Proposition 2.3. Let S be a semitopological monoid with zero, λ 1 be any cardinal, and B 0 λ (S) be a topological Brandt λ 0 -extension of S in the class of semitopological semigroup. Then every non-trivial continuous homomorphic image of B 0 λ (S) in a semitopological semigroup W is a topological Brandt λ 0 -extension of some semitopological monoid with zero. Moreover, if T is the continuous image of B 0 λ (S) under a homomorphism h, then T is topologically isomorphic to a topological Brandt λ 0 -extension of the continuous homomorphic image of the semitopological monoid S α,α under the homomorphism h for any α ∈ λ. Also, the subspaces h(A αβ ) and h(A γδ ) of W are homeomorphic for all α, β, γ, δ ∈ λ, and every nonempty A ⊆ S.
Proof. Let h : B 0 λ (S) → W be a continuous homomorphism of a topological Brandt λ 0 -extension of a semitopological monoid S into a semitopological semigroup W . The algebraic part of the proof follows from Proposition 3.2 of [16]. If h is an annihilating homomorphism, then the statement of the lemma is trivial.
We fix arbitrary α, β, γ, δ ∈ λ and define the maps ϕ for all α, β, γ, δ ∈ λ, x ∈ S, and hence the map ϕ is a homeomorphism and hence the last statement of the proposition holds.
Results of Section 2 of [12] imply that for any infinite cardinal λ and every non-trivial topological semigroup (and hence for a semitopological semigroup) S, there are many topological Brandt λ 0extensions of S in the class of semitopological semigroups. Moreover, for any infinite cardinal λ on the Brandt λ 0 -extension of two-element monoid with zero (i.e., on the infinite semigroup of λ × λunits) there exist many topologies which turns B λ into a topological (and hence a semitopological) semigroup (cf. [11]).
The following proposition describes the properties of finite topological Brandt λ 0 -extensions of semitopological semigroups in the class of semitopological semigroups. Proposition 2.4. Let λ be any finite non-zero cardinal. Let (S, τ ) be a semitopological semigroup and τ B a topology on B 0 λ (S) such that (B 0 λ (S), τ B ) is a semitopological semigroup and τ B | Sα,α = τ for some α ∈ λ. Then the following assertions hold: x is a non-zero element of S and B x is a base of the topology τ at x, then the family is continuous for any α, γ, δ ∈ λ we conclude that is an open subset in (B 0 λ (S), τ B ). Then since the cardinal λ is finite, we have that α,β∈λ A γ,δ and this implies statement (ii).
where 0 is the zero of B 0 λ (S), and B S (s) is a base of the topology τ at the point s ∈ S.
Proof. Let X be a countably pracompact topological space and f : X → Y be a continuous map from X into a Hausdorff topological space Y . Without loss of generality we may assume that f (X) = Y . The countable pracompactness of X implies that there exists a dense subset A in X such that X is countably pracompact at A. Then we have that f (A) is a dense subset of Y (see the proof of Theorem 1.4.10 from [8]). We fix an arbitrary infinite subset B ⊆ f (A). For every b ∈ B we choose an arbitrary point a ∈ A such that f (a) = b and denote by A * the set of all points chosen in such way. Then A * is infinite subset of A and hence A * has an accumulation point Lemma 2.8. Let λ be any cardinal 1. If a topological Brandt λ 0 -extension (B 0 λ (S), τ * B ) of a semitopological monoid (S, τ ) with zero in the class of Hausdorff semitopological semigroups is pseudocompact (resp., countably pracompact, countably compact, sequentially compact, compact), then the topological space (S, τ ) is pseudocompact (resp., countably pracompact, countably compact, sequentially compact, compact).
Next proposition will prove useful for constructing topology at zero of Brandt λ 0 -extension for an infinite cardinal λ in the class of countably compact semitopological semigroups.
for infinitely many pairs of indices (α, β). Then for every such S α,β we fix a unique point x α,β ∈ S α,β \ U(0) and put A = {x α,β }. Then A is infinite and Lemma 2.2 implies that the set A has no accumulation point in (B 0 λ (S), τ * B ). This contradicts Theorem 3.10.3 of [8]. The obtained contradiction implies the statement of the proposition. Proposition 2.9 implies the following corollary: there exist finitely many pairs of indices (α, β) such that S α,β U(0). The following theorem describes the structure of Hausdorff countably compact topological Brandt λ 0 -extensions of semitopological monoids with zero in the class of semitopological semigroups.
Theorem 2.11. For any Hausdorff countably compact semitopological monoid (S, τ ) with zero and for any cardinal λ 1 there exists a unique Hausdorff countably compact topological Brandt λ 0 -extension B 0 λ (S), τ S B of (S, τ ) in the class of semitopological semigroups, and the topology τ S B is generated by the base , and B S (s) is a base of the topology τ at the point s ∈ S.
Proof. In the case when λ is a finite cardinal, Theorem 2.6 and the definition of a countably compact space imply the statements of the theorem. Also, in this case the proof of the separate continuity of the semigroup operation in B 0 λ (S), τ S B is trivial, and hence we omit it. Suppose that λ is an infinite cardinal. Then statement (i) follows from Remark 2.5 and Proposition 2.9 implies that B B (0) is a base of the topology τ S B at the zero 0 ∈ B 0 λ (S). The proof is completed by showing that the semigroup operation in B 0 λ (S), τ S B is separately continuous. We consider only the cases 0 · (α, s, β) and (α, s, β) · 0, because in other cases the proof of the separate continuity of the semigroup operation in B 0 λ (S), τ S B is trivial. Let K be an arbitrary finite subset in λ × λ and let U(0 S ) be an arbitrary open neighbourhood of the zero 0 S in (S, τ ). Then there exists an open neighbourhood This completes the proof of the theorem.
Theorem 2.11 implies the following corollary: Corollary 2.12. For any Hausdorff sequentially compact (resp., compact) semitopological monoid (S, τ ) with zero and for any cardinal λ 1 there exists a unique Hausdorff sequentially compact (resp., compact) topological Brandt λ 0 -extension B 0 λ (S), τ S B of (S, τ ) in the class of semitopological semigroups, and the topology τ S B is generated by the base We proceed to show that the statement analogous to Proposition 2.9 holds in case B 0 λ (S) is a semiregular pseudocompact space.
is a regular open set we conclude that S α,β ⊆ U(0). This completes the proof of the proposition.
The following theorem describes the structure of semiregular pseudocompact topological Brandt λ 0 -extensions of semitopological monoids with zero in the class of semitopological semigroups.
Theorem 2.14. For any semiregular pseudocompact semitopological monoid (S, τ ) with zero and for any cardinal λ 1 there exists a unique semiregular pseudocompact topological Brandt λ 0extension B 0 λ (S), τ S B of (S, τ ) in the class of semitopological semigroups, and the topology τ S B is generated by the base , and B S (s) is a base of the topology τ at the point s ∈ S.
The proof of Theorem 2.14 is similar to the proof of Theorem 2.11 and is based on Proposition 2.13.
Theorem 2.14 implies the following corollary.
Corollary 2.15. For any semiregular countably pracompact semitopological monoid (S, τ ) with zero and for any cardinal λ 1 there exists a unique semiregular countably pracompact topological Brandt λ 0 -extension B 0 λ (S), τ S B of (S, τ ) in the class of semitopological semigroups, and the topology τ S B is generated by the base , and B S (s) is a base of the topology τ at the point s ∈ S.
The following example shows that the statements of Theorem 2.14, Corollaries 2.15 and 2.16 are not true in the case when the pseudocompact (resp., countably pracompact) space B 0 λ (S), τ S B is not semiregular.
and U(s · t) of s, t and s · t in S, respectively, such that Here and subsequently τ S B denotes the the topology on the Brandt λ 0 -extension B 0 λ (S) of a semitopological semigroup S which is defined in Theorems 2.11 and 2.14.
The following proposition is proved by detailed verifications.  (⇐) Suppose that the space S is regular. We consider the case when cardinal λ is infinite. In the case λ < ω the proof is similar and follows from Theorem 2.6.
Fix an arbitrary point x ∈ B 0 λ (S). Suppose that x ∈ S * α,β for some α, β ∈ λ. Then the regularity of S, Remark 2.5 and the definition of the topology τ S B on B 0 λ (S) imply that there exist open neighbourhoods V (x) and U(x) of x in B 0 λ (S), τ S B such that cl B 0 λ (S) (V (x)) ⊆ U(x) ⊆ S * α,β . We now turn to the case x = 0. Let U A (0) be a basic open neighbourhood of 0 in B 0 λ (S), τ S B . Now, the regularity of the space S implies that there exists an open neighbourhood V (0 S ) of 0 S in S such that cl S (V (0 S )) ⊆ U(0 S ). Next, by Remark 2.5 the set (S \ cl S (V (0 S ))) α,β is open in B 0 λ (S), τ S B for any α, β ∈ λ, and hence Remark 2.5 implies the following inclusion . This completes the proof of the statement.
(⇐) Suppose that the space S is Tychonoff. We consider the case when cardinal λ is infinite. In the case λ < ω the proof is similar applying Theorem 2.6.
Fix an arbitrary point x ∈ B 0 λ (S). Suppose that x ∈ S * α,β for some α, β ∈ λ. Since by Lemma 2.2 the subspace S α,β is homeomorphic to S, we identify S α,β with S. Next we shall prove statement (iii).
(⇒) Suppose that B 0 λ (S), τ S B is a normal space. By Lemma 2.2 we have that S α,α is a closed subset of B 0 λ (S), τ S B for every α ∈ λ. Then by Theorem 2.1.6 from [8], S α,α is a normal subspace of B 0 λ (S), τ S B and hence Definition 1.1 and Lemma 2.1 imply that S is a normal space. (⇐) Suppose that S is a normal space. Let F 1 and F 2 be arbitrary closed disjoint subsets of B 0 λ (S), τ S B . Then only one of the following cases holds: 1) 0 ∈ F 1 and 0 / ∈ F 2 ; 2) 0 / ∈ F 1 and 0 / ∈ F 2 .
Further we shall need the following lemma. Proof. On the contrary, suppose that 0 S is not zero in T . Then there exists t ∈ T such that either t · 0 S = 0 S or 0 S · t = 0 S . If t · 0 S = x = 0 S then the Hausdorffness of T implies that there exist open neighbourhoods U(0 S ) and U(x) of the points 0 S and x in T , respectively, such that U(0 S ) ∩ U(x) = ∅. Also, the separate continuity of the semigroup operation in T implies that there exists an open neighbourhood V (t) of the point t in T such that V (t) · 0 S ⊆ U(x). Since S is a dense subsemigroup of T we conclude that V (t) ∩ S = ∅ and hence we have that 0 S ∈ (V (t) ∩ S) · 0 S ⊆ V (t) · 0 S ⊆ U(x), a contradiction. The obtained contradiction implies that t · 0 S = 0 S . The proof of the equality 0 S · t = 0 S is similar.
In the same manner we can see that next lemma holds. Lemma 2.23. Let S be a semigroup with the unit 1 S . If a Hausdorff semitopological semigroup T contains S as a dense subsemigroup then 1 S is unit in T .

Stone-Čech and Bohr compactifications of pseudocompact Brandt λ 0 -extensions of semitopological monoids
We shall say that a Tychonoff space X has the Grothendieck property if for any Tychonoff space Z every separately continuous map f : X × X → Z can be extended to separately continuous map f : βX×βX → βZ. We observe that if X, Y and Z are Tychonoff pseudocompact spaces then every continuous map f : X × X → Z can be extended to separately continuous map f : βX × βX → βZ [20]. Also [1,20], a Tychonoff space X has the Grothendieck property provides that X is pseudocompact and one of the following conditions holds: (i) X is countably compact; (ii) X has the countable tightness (i.e., t(X) = κ, where t(X) is the minimal infinite cardinal κ such that if x ∈ X, A ⊂ X and x ∈ cl X (A) then there exists B ⊆ A such that |B| κ and x ∈ cl X (B)); (iii) X is separable; (iv) X is a k-space (i.e., X is a Hausdorff space which can be represented as a quotient space of a locally compact space).
By Proposition 1.11 of [20] we have that if a Tychonoff space X has the Grothendieck property then every separately continuous map f : X × X → X extends to a separately continuous map f : βX × βX → βX, and hence the following proposition holds: Proposition 3.1. Let S be a Tychonoff semitopological semigroup with the Grothendieck property. Then βS is a compact semitopological semigroup. Moreover, if 1 S is the unit of S (resp., 0 S is the zero of S) then 1 S is the unit of βS (resp., 0 S is the zero of βS).
We observe that the last statement of Proposition 3.1 follows from Lemmas 2.22 and 2.23. If X is a topological space, then by d(X) we denote the density of X, i.e., Fix an arbitrary subset . We consider two cases: (1) x ∈ S * i,j for some i, j ∈ λ; (2) x is zero of the semigroup B 0 λ (S). Suppose that case (1) holds. By Lemma 2.2, S * i,j is an open subspace of B 0 λ (S), τ S B and since t(S * i,j ) t(S) we conclude that there exists a subset B ⊆ A ∩ S * i,j ⊆ A such that x ∈ cl B 0 λ (S) (B) and |B| κ.
Suppose that x is zero of the semigroup B 0 λ (S) and x ∈ cl B 0 λ (S) (A) for some A ⊆ B 0 λ (S). If the set A intersects infinitely many sets S * i,j , i, j ∈ λ, then there exists an infinite countable subset B of B 0 λ (S) such that the following conditions holds: Then the definition of the topology τ S B on B 0 λ (S) implies that x ∈ cl B 0 λ (S) (B). In the other case there exist finitely many subsets S * (iv) If B 0 λ (S), τ S B is a k-space then the definition of the topology τ S B implies that S i,j is a closed subset of B 0 λ (S), τ S B for all i, j ∈ λ, and hence by Theorem 3.3.25 from [8] and Lemma 2.1 we have that S is a k-space.
Suppose that S is a k-space and A (λ) is the one-point Alexandroff compactification of the infinite discrete space λ with the remainder a. Then by Theorem 3.3.27 from [8] we have that A (λ) × S with the product topology is a k-space. We put Then the definition of the space A (λ) × S implies that J is a closed subset of A (λ) × S.
Simple verifications show that the relation Ω J = (J ×J ) ∪ ∆ A (λ)×S , where ∆ A (λ)×S is the diagonal of A (λ) × S, is an equivalence relation on A (λ) × S.
Next we shall show that Ω J is a closed relation on A (λ) × S. Fix an arbitrary ((i, s), (j, t)) ∈ ((A (λ) × S) × (A (λ) × S)) \ Ω J . Since Ω J is symmetric we need only to consider the following cases: (1) i = a, j = a, and t = 0 S ; In case (2) the Hausdorffness of S implies that there exist an open neighbourhoods U(s) and U(t) of the points s = 0 S and t in S, respectively, such that U(s) ∩ U(t) = ∅. Since i = j = a is an isolated point of the space A (λ) we have that {i} × U(s) and {j} × U(t) are open disjoint neighbourhoods of the points (i, s) and (j, t) in A (λ) × S, respectively, and hence ((i, s), (j, t)) is an interior point of the set ((A (λ) × S) × (A (λ) × S)) \ Ω J .
Therefore we have that Ω J is a closed equivalence relation on the topological space A (λ)×S and hence the quotient space (A (λ) × S) /Ω J (with the quotient topology) is a Hausdorff space. Thus, the natural map π Ω J : A (λ) × S → (A (λ) × S) /Ω J is a quotient map and by Theorem 3.3.23 from [8] we get that (A (λ) × S) /Ω J is a k-space for every infinite cardinal λ. Also, we observe that for every finite subset K fin ⊂ A (λ) such that a ∈ K fin we have that π Ω J (K fin × S) is a k-space by Theorem 3.3.23 of [8], because the definition of the space We observe that if λ is an infinite cardinal then |λ × λ| = λ and simple verifications show that the map f : is a homeomorphism in the case when we identify cardinal λ with its square λ × λ. This completes the proof of statement (iv).
Recall [7] that a Bohr compactification of a semitopological semigroup S is a pair (b, B(S)) such that B(S) is a compact semitopological semigroup, b : S → B(S) is a continuous homomorphism, and if g : S → T is a continuous homomorphism of S into a compact semitopological semigroup T , then there exists a unique continuous homomorphism f : B(S) → T such that the diagram commutes. In the sequel, similar as in the general topology by the Bohr compactification of a semitopological semigroup S we shall mean not only pair (b, B(S)) but also the compact semitopological semigroup B(S).
The definition of the Stone-Čech compactification implies that for every Tychonoff semitopological monoid S with the Grothendieck property any continuous homomorphism h : S → T into a compact semitopological semigroup T has the unique extended continuous homomorphism βh : βS → T . This implies that the Bohr compactification of S in this case coincides with its Stone-Čech compactification βS. Hence Proposition 3.4 implies the following two theorems.
where 0 1 and 0 2 are zeros of B 0 λ 1 (S) and B 0 λ 2 (T ), respectively. Moreover, if for the semigroup T the following conditions hold: (i) every idempotent of T lies in the center of T ; (ii) T has B * λ 1 -property, then every non-trivial continuous homomorphism from B 0 λ 1 (S) into B 0 λ 2 (T ) can be constructed in this manner.
Proof. The algebraic part of the proof follows from Theorem 3.10 of [16].
Since the homomorphism h is continuous, I h = h −1 (0 T ) is a closed ideal of the semitopological semigroup S.
Since left and right translations in the topological semigroup B 0 λ 2 (T ) are continuous and any restriction of a continuous map is continuous, we conclude that the continuity of the homomorphism σ : B 0 λ 1 (S) → B 0 λ 2 (T ) implies the continuity of h. Next we define a category of pairs of semiregular pseudocompact semitopological monoids with zero and sets, and a category of semiregular pseudocompact semitopological semigroups.
Let S and T be semiregular pseudocompact semitopological monoids with zeros. Let CHom 0 (S, T ) be a set of all continuous homomorphisms σ : S → T such that σ(0 S ) = 0 T . We put where the map [u, ϕ, h ′ , u ′ ] : λ → H(e) is defined by the formula Straightforward verification shows that T PCB is the category with the identity morphism ε (S,λ) = (Id S , u 0 , Id λ ) for any (S, λ) ∈ Ob(T PCB), where Id S : S → S and Id λ : λ → λ are identity maps and u 0 (α) = 1 S for all α ∈ λ.
A functor F from a category C into a category K is called full if for any a, b ∈ Ob(C ) and for any K -morphism α : Fa → Fb there exists a C -morphism β : a → b such that Fβ = α, and F called representative if for any a ∈ Ob(K ) there exists b ∈ Ob(C ) such that a and Fb are isomorphic [19].
Comfort and Ross [6] proved that the Stone-Čech compactification of a pseudocompact topological group is a topological group. Therefore the functor of the Stone-Čech compactification β from the category of pseudocompact topological groups back into itself determines a monad. Gutik and Repovš in [15] proved the similar result for countably compact 0-simple topological inverse semigroups: every countably compact 0-simple topological inverse semigroup is topologically isomorphic to the topological Brandt λ 0 -extension of a countably compact topological group with adjoined isolated zero for some finite non-zero cardinal λ and the Stone-Čech compactification of a countably compact 0-simple topological inverse semigroup is a compact 0-simple topological inverse semigroup (see Theorems 2 and 3 in [15]). Hence the functor of the Stone-Čech compactification β : B * (C CT G ) → B * (C CT G ) from the category of Hausdorff countably compact Brandt topological semigroups B * (C CT G ) into itself determines a monad [16]. Also, in [14] were obtained similar results to [15] for pseudocompact completely 0-simple topological inverse semigroups. There was proved that every countably compact completely 0-simple topological inverse semigroup is topologically isomorphic to the topological Brandt λ 0 -extension of a pseudocompact topological group with adjoined isolated zero for some finite non-zero cardinal λ and the Stone-Cech compactification of a pseudocompact completely 0-simple topological inverse semigroup is a compact 0-simple topological inverse semigroup (see Theorems 1 and 2 in [14]).
By B * (PCT G ) we denote the category of completely 0-simple Hausdorff pseudocompact topological inverse semigroup, i.e., the objects of B * (PCT G ) are Hausdorff pseudocompact Brandt topological inverse semigroups and morphisms in B * (PCT G ) are continuous homomorphisms between such topological semigroups. Then Theorems 2 from [14] implies the following: