{ } of B λ (S). ON TOPOLOGICAL BRANDT SEMIGROUPS. O. V. Gutik, 1 K. P. Pavlyk, 2,3 and A. R. Reiter 1 UDC Introduction

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1 Journal of Mathematical Sciences, Vol. 184, No. 1, July, 2012 ON TOPOLOGICAL BRANDT SEMIGROUPS O. V. Gutik, 1 K. P. Pavlyk, 2,3 and A. R. Reiter 1 UDC We describe the structure of pseudocompact completely 0 -simple topological inverse semigroups. We also give sufficient conditions under which the topological Brandt λ 0 -extensions of an (absolutely) H - closed semigroup are (absolutely) H -closed semigroups. 1. Introduction In the present work, we use the same terminology, definitions, and notation as in [4, 9, 10]. All topological spaces are assumed to be Hausdorff spaces. A semigroup is a set with a binary associative operation defined on it. If S is a semigroup, then by S 0 [S 1 ] we denote S with adjoined zero (adjoined unit) and by E(S) the subset of idempotents in S. We also denote by 1 S and 0 S the unit and the zero of the semigroup S, respectively. Recall [10] that a semigroup S is called inverse if, for an arbitrary element x in S, there exists a unique element x 1 (called the inverse of x ) such that xx 1 x = x and x 1 xx 1 = x 1. A semigroup S is called 0 - simple if it does not contain any proper nonzero two-sided ideals. Further, a semigroup S is called completely 0 -simple if S is 0 -simple and every nonzero idempotent in S is primitive. A topological (inverse) semigroup is a Hausdorff topological space with continuous semigroup operation (and inversion, respectively). Let S be a semigroup with identity and let I λ be a set of cardinality λ 1. On the set B λ (S) = (I λ S 1 I λ ) {0}, we define the semigroup operation as follows: (i, s, j) (k,t, l) = (i, st, l), j = k, 0, j k, and (i, s, j) 0 = 0 (i, s, j) = 0 0 = 0 for arbitrary i, j, k, l I λ, s,t S 1. The semigroup B λ (S) is called the Brandt λ-extension of the semigroup S [1, 2]. Obviously, B λ (S) is the matrix Rees semigroup M 0 [S 1 ; I λ, I λ, M ], where M is the unit (I λ I λ )-matrix. If the semigroup S contains only one element, and λ 2, then B λ (S) is the semigroup of (I λ I λ )-matrix units. For arbitrary i, j I λ, we introduce the { } of B λ (S). subset S i, j = (i,s, j) s S 1 Definition 1 [2]. Suppose that λ 1, Ω is a class of topological semigroups, and (S,τ) Ω. Let τ B be a topology on B λ (S) such that the following conditions are satisfied: 1 Franko National University of Lviv, Lviv, Ukraine. 2 Institute of Mathematics, University of Tartu, Tartu, Estonia. 3 Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 54, No. 2, pp. 7 16, April June, Original article submitted December 12, /12/ Springer Science+Business Media, Inc. 1

2 2 O. V. GUTIK, K. P. PAVLYK, AND A. R. REITER (i) (B λ (S),τ B ) Ω, (ii) τ B S i,i =τ for some i I λ. Then (B λ (S),τ B ) is called a topological Brandt λ-extension of the semigroup (S,τ) in the class Ω. If Ω coincides with the class of all topological semigroups, then the topological semigroup (B λ (S),τ B ) is called a topological Brandt λ-extension of the semigroup (S,τ). Let S be a semigroup and let I λ be a set of cardinality λ 1. On the set B λ (S) = (I λ S 0 I λ ) {0}, we define the semigroup operation as follows: (i, s, j) (k,t, l) = (i, st, l), j = k, 0, j k, and (i, s, j) 0 = 0 (i, s, j) = 0 0 = 0 for arbitrary i, j, k, l I λ, s,t S 1. If 0 S is zero of the semigroup S, then the set I 0 = {(i,0 S, j) i, j I λ } {0} is an ideal of B λ (S). A Rees quotient semigroup B 0 λ (S) = B λ (S)/I 0 is called the Brandt λ 0 -extension of the semigroup S [17]. For arbitrary i, j I λ, we introduce the subsets A i, j = {(i, s, j) s A}, 0 A, {(i, s, j) s A \ {0 S }} {0}, 0 A, and A i, j = A i, j \ {0} in B λ 0 (S). Definition 2 [17]. Suppose that λ 1, Ω is a class of topological semigroups, and (S,τ) Ω. Let τ B be a topology on B λ 0 (S) such that the following conditions are satisfied: (i) (B λ 0 (S),τ B ) Ω, (ii) τ B S i,i =τ for some i I λ. Then (B λ 0 (S),τ B ) is called a topological Brandt λ 0 -extension of the semigroup (S,τ) in the class Ω. If Ω coincides with the class of all topological semigroups, then the topological semigroup (B λ 0 (S),τ B ) is called a topological Brandt λ 0 -extension of the semigroup (S,τ). Note that every (topological) Brandt λ-extension B λ (S) of the (topological) semigroup S is the (topological) Brandt λ 0 -extension B λ 0 (T ) of a certain (topological) monoid T with zero, and, in addition, Brandt λ 0 -extensions are preserved by homomorphisms, unlike Brandt λ -extensions [20]. Let Ω be a class of topological semigroups.

3 ON TOPOLOGICAL BRANDT SEMIGROUPS 3 Definition 3 [15, 23]. A topological semigroup S Ω is called H -closed in the class Ω if it is a closed subsemigroup of any semigroup T from the class Ω, which contains S as a subsemigroup. A topological semigroup S Ω is called H -closed if the class Ω coincides with the class of all topological semigroups. Definition 4 [15, 24]. A topological semigroup S Ω is called absolutely H -closed in the class Ω if every continuous homomorphic image of the semigroup S into the semigroup T Ω is an H -closed semigroup in the class Ω. A topological semigroup S is called absolutely H -closed if Ω coincides with the class of all topological semigroups. The topological Brandt λ- and λ 0 -extensions of semigroups were introduced in [1], where their algebraic and topological properties were also studied for the first time. The preservation of H -closedness and absolute H -closedness in the class of topological inverse semigroups by Brandt λ-extensions was investigated in [2] and [15]. In particular, it was proved that a topological inverse semigroup is (absolutely) H -closed in the class of topological inverse semigroups if and only if any topological Brandt λ -extension of it is an (absolutely) H - closed semigroup in the class of topological inverse semigroups. Similar results were obtained in [17] for topological Brandt λ 0 -extensions in the class of topological inverse semigroups. The preservation of H - closedness and absolute H -closedness in the class of topological semigroups by topological Brandt λ- extensions was studied in [2] and [3]. With the use of this technique, the structure of countably compact 0- simple topological inverse semigroups was described in [19], and the structure of compact and countably compact primitive topological inverse semigroups was described in [8]. The categorical properties of Brandt λ 0 - extensions were studied in [20]. The semigroup compactifications of topological semigroups that are semitopological or topological semigroups were investigated in [7]. It was proved there that if the space of a topological semigroup S is openly factorizable, then the semigroup operation of a topological semigroup S extends to a continuous semigroup operation on the Stone Čech compactification βs of the topological space of S. The main properties, structure, and open problems in the theories of pseudocompact and countably compact paratopological groups as well as pseudocompact (countably compact) semitopological and topological semigroups are presented in the monograph by Arhangel skii and Tkachenko [6]. The continuity of inversions in pseudocompact and countably compact paratopological groups was investigated in [5] and [22]. In the present paper, we describe the structure of pseudocompact completely 0 -simple topological inverse semigroups and give sufficient conditions under which the topological Brandt λ 0 -extension of an (absolutely) H -closed semigroup is an (absolutely) H -closed semigroup. 2. Pseudocompact Topological Brandt Semigroups Recall [4] that a topological space X is called pseudocompact if every locally finite family of open subsets of X is finite. Every continuous real-valued function defined on a pseudocompact space is bounded [4]. Theorem 1. Every completely 0 -simple pseudocompact topological inverse semigroup S is the topological Brandt λ-extension B λ (G) of a pseudocompact topological group G for some finite cardinal λ in the class of topological inverse semigroups. Proof. According to Theorem 3.9 in [10], the semigroup S is algebraically isomorphic to the Brandt λ- extension B λ (G) of a certain group G. The continuity of the inversion in the semigroup S implies that the

4 4 O. V. GUTIK, K. P. PAVLYK, AND A. R. REITER mappings ε + : S E(S) and ε : S E(S) defined by the relations ε + (x) = x x 1 and ε (x) = x 1 x, respectively, are continuous. Hence, since the continuous image of a pseudocompact space is pseudocompact, we conclude that E(S) is a pseudocompact space. It follows from Lemma 1 in [18] that the semilattice E(S) is homeomorphic to the one-point Aleksandroff compactification of the discrete space of cardinality E(S) with zero of the semilattice E(S) as a remainder. Thus, an arbitrary nonzero idempotent of the semilattice E(S) is an isolated point in E(S). Since the mappings ε + and ε are continuous, every maximal subgroup H (e) = {x S ε + (x) =ε (x) = e} containing a nonzero idempotent e is an open-and-closed subset of S. Then, according to the Colmex theorem [11] (see also E(d) in [4]), the space of the subgroup H (e) is pseudocompact. Since the semigroup S is 0 -bisimple, it follows from the Green theorem (see Theorem 2.20 in [11]) that an arbitrary subspace H (e, f ) = {x S ε + (x) = e, ε (x) = f } in S, where e, f E(S)\{0}, is homeomorphic to the space of the subgroup H (e). The continuity of the mappings ε + and ε implies that H (e, f ) is a pseudocompact open-and-closed subspace of S. Hence, S \ {0} = H (e, f ). e, f E(S)\{0} Since every nonzero idempotent of the semigroup S is an isolated point in E(S), the set (ε + ) 1 (e) = {H (e, f ) f E(S)\{0}} is open-and-closed in S, and, hence, according to the Colmex theorem [11] (see E(d) in [4]), (ε + ) 1 (e) is a pseudocompact space. It follows from the results obtained above that (ε + ) 1 (e) = H (e, f ), f E(S)\{0} and then, according to Theorem in [4], the semilattice E(S) is finite. Thus, since S is a topological inverse semigroup, we conclude that S is the topological Brandt λ- extension B λ (G) of the pseudocompact topological group G = H (e) for some finite cardinal λ in the class of topological inverse semigroups. Remark 1. It follows from Theorem 1 that every completely 0 -simple pseudocompact topological inverse semigroup S is the finite topological sum of pseudocompact subspaces H (e, f ) and zero of the semigroup S. Let X be a topological space. A pair (Y,c), where Y is a compact Hausdorff space and the mapping c: X Y is a homeomorphic imbedding of the space X into Y such that cl Y (c(x)) = Y, is called a compactification of the space X. We define the partial order on the family C(X) of all compactifications of a space X as follows: c 2 (X) c 1 (X) if and only if there exists a continuous mapping f : c 1 (X) c 2 (X) such that f c 1 = c 2. The greatest element of the family C(X) with respect to the partial order defined above is called the Stone Čech compactification of the topological space X and is denoted by βx [4]. Comfort and

5 ON TOPOLOGICAL BRANDT SEMIGROUPS 5 Ross [12] proved that the Stone Čech compactification of a pseudocompact topological group is a topological group. The theorem below is an analog of the Comfort Ross theorem. Theorem 2. Let S be a completely 0 -simple pseudocompact topological inverse semigroup. Then the Stone Čech compactification βs admits a structure of completely 0 -simple topological inverse semigroup with respect to which the inclusion mapping of the semigroup S into βs is a homeomorphism into. Proof. According to Theorem 1, the semigroup S is topologically isomorphic to a topological Brandt λ- extension B λ (G) of some topological group G in the class of topological inverse semigroups, and, furthermore, λ<ω, and G α,β and G γ,δ are homeomorphic open-and-closed pseudocompact subspaces in B λ (G) for all α,β, γ,δ I λ. Obviously, the topological space B λ (G)\{0} is homeomorphic to the product G I π I π, and, since the set I π I π is finite, it follows from Corollary in [4] that the space B λ (G)\{0} is pseudocompact. Then, by virtue of Theorem 1 in [14], we have β(g I π I π ) =βg βi π βi π =βg I π I π and, hence, β(b λ (G)) = B λ (βg). Theorem 1 yields the following statement: Corollary 1. Every completely 0 -simple pseudocompact topological inverse semigroup is a dense subsemigroup of a 0 -simple compact topological inverse semigroup. Corollary 1 yields the following statement: Corollary 2. The topological space of a completely 0 -simple pseudocompact topological inverse semigroup is completely regular. 3. H -Closed Topological Brandt λ 0 -Extensions Proposition 1. Let the topological Brandt λ 0 -extension B 0 λ (S) of a topological monoid S be a subsemigroup of a topological semigroup T. Then, for every element x = (i, s, j) B 0 λ (S), i, j I λ, there exists an open neighborhood U(x) of the point x in T such that U(x) B 0 λ (S) S i, j. Proof. The continuity of the semigroup operation in T implies that, for an arbitrary open neighborhood W (x) of x in T that does not contain the zero of the semigroup B λ 0 (S), there exists an open neighborhood U(x) of x in T such that (i,1 S,i) U(x) ( j,1 S, j) W (x). Then U(x) B 0 λ (S) S i, j because otherwise we would have 0 (i,1 S,i) U(x) ( j,1 S, j) W (x).

6 6 O. V. GUTIK, K. P. PAVLYK, AND A. R. REITER Proposition 2. Let the topological Brandt λ 0 -extension B λ 0 (S) of a topological monoid S be a dense subsemigroup of a topological semigroup T. Then the following assertions are true: (і) if an element x T \ B λ 0 (S) is an idempotent, then there exists an open neighborhood U(x) of the point x in T such that U(x) B 0 λ (S) S i,i for some i I λ ; (ii) the maximal idempotents of the semigroup B λ 0 (S) are the maximal idempotents of the semigroup T ; (iii) the semigroup T does not contain right (left) identities. Proof. (i) Since the zero of the semigroup B 0 λ (S) is the zero of the semigroup T (see Lemma 1 in [2]), then there exists an open neighborhood W (x) of x in T that does not contain the zero 0 of the semigroup T. It follows from the continuity of the semigroup operation in T that there exists an open neighborhood U(x) of the idempotent x in T such that U(x) U(x) W (x). If the neighborhood U(x) contains either points from two different sets S i,i and S j, j, i j, i, j I λ, or points from the set S i, j, where i j, i, j I λ, then 0 U(x) U(x) W (x), which contradicts the choice of the neighborhood W (x). (ii) Note that every element (i,1 S,i) of the semigroup B 0 λ (S) is the maximal idempotent in B 0 λ (S). Assume that there exists an idempotent e T \ B 0 λ (S) such that (i,1 S,i) < e for some i I λ. Then the continuity of the semigroup operation in T and assertion (і) imply that there exists an open neighborhood U(e) of the idempotent e in T such that U(e) B 0 λ (S) S i,i. However, by virtue of Theorem 1.7 in [9, Vol. 1], the subset (i,1 S,i)T (i,1 S,i) = (i,1 S,i)T T (i,1 S,i) is closed in T, and, in addition, S i,i (i,1 S,i)T (i,1 S,i) and e (i,1 S,i)T (i,1 S,i), which contradicts the statement that e is an accumulation point of the set B 0 λ (S) in T. The contradiction obtained yields assertion (ii). l (iii) Let 1 T be a left identity of the semigroup T. It is clear that 1 l T B 0 λ (S). Then, by virtue of assertion (і), there exists an open neighborhood U(1 l l T ) of the point 1 T in T such that U(1 l T ) B 0 λ (S) S i,i for some i I λ. Let W (i,1 S,i) and W (1 l l T ) be disjoint open neighborhoods of the points (i,1 S,i) and 1 T in T, respectively. It follows from the continuity of the semigroup operation in T that there exist open neighborhoods V (i,1 S,i) and V (1 l l T ) of the elements (i,1 S,i) and 1 T in T such that V (1 T l ) V (i,1 S,i) V (i,1 S,i), V (i,1 S,i) W (i,1 S,i) U(i,1 S,i) and V (1 T l ) W (1 T l ).

7 ON TOPOLOGICAL BRANDT SEMIGROUPS 7 But then (V (1 T l ) (i,1 S,i)) V (1 T l ), which contradicts the choice of the neighborhoods W (i,1 S,i) and W (1 T l ). The contradiction obtained implies that the semigroup T does not contain a left identity. The proof of the statement that T does not contain a right identity is similar. Lemma 1. Let the topological Brandt λ 0 -extension B λ 0 (S) of a topological monoid S be a subsemigroup of a topological semigroup T. If A i, j is a closed subset in T for some i, j I λ, then A k,l is a closed subset in T for all k, l I λ. Proof. We define mappings ϕ:t T and ψ :T T by the formulas ϕ(t) = (k,1 S,i) t ( j,1 S, l) and ψ(t) = (i,1 S, k) t (l,1 S, j). Since the mappings ϕ and ψ are continuous as compositions of shifts in T, we conclude that A =ψ 1 (A i, j ) is a closed subset in T. Then, for the mapping f =ϕ ψ, the restriction f A = f A : A A k,l is a retraction, and the set A k,l is a retract of the space A. Hence, A k,l is a closed subset in T. Theorem 3. Let B λ 0 (S) be a topological Brandt λ 0 -extension of a topological monoid S. If the semigroup S is H -closed and the band of the semigroup B λ 0 (S) is compact, then B λ 0 (S) is an H -closed topological semigroup. Proof. Assume to the contrary that the topological semigroup B λ 0 (S) is not H -closed. Then there exists a topological semigroup T containing B λ 0 (S) as a nonclosed subsemigroup. Since the closure of a subsemigroup in a topological semigroup is a subsemigroup (see [9, Vol. 1, p. 9]), we can assume without loss of generality that B 0 λ (S) is a dense subsemigroup in T, and T \ B 0 λ (S). According to Lemma 1 in [2], the zero 0 of the semigroup B 0 λ (S) is the zero of the semigroup T. Let x T \ B 0 λ (S). Then x 0 = 0 x = 0. It follows from Lemma 1 that the subset S i, j is closed in T for all i, j I λ. Hence, every open neighborhood of the point x intersects infinitely many sets of the type S i, j, i, j I λ. Let U(x) and U(0) be open neighborhoods of the point x and zero 0 in T, respectively, such that U(x) U(0) =. Since x 0 = 0 x = 0, and the semigroup operation in T is continuous, we conclude that there exist open neighborhoods V (x) and V (0) of the point x and zero 0 in T, respectively, such that the following conditions are satisfied: V (x) V (0) U(0), V (0) V (x) U(0), V (0) U(0) and V (x) U(x). It follows from the compactness of the band E(B 0 λ (S)) that, for an arbitrary neighborhood W (0) of zero 0 in T of the semigroup B 0 λ (S), the set A(W (0)) = {(i,1 s,i) (i,1 s,i) W (0)}

8 8 O. V. GUTIK, K. P. PAVLYK, AND A. R. REITER is finite. Thus, since the neighborhood V (x) intersects infinitely many sets S i,i, i I λ, we conclude that the following condition is satisfied: (V (x) V (0)) V (x) or (V (0) V (x)) V (x). This contradicts the choice of the neighborhoods U(x) and U(0). The contradiction obtained yields the statement of the theorem. Lemma 2. Let B λ 0 (S) be a topological Brandt λ 0 -extension of a topological monoid S, let T be a topological semigroup, and let h: B λ 0 (S) T be a continuous homomorphism. Then h(b λ 0 (S)) is the topological Brandt λ 0 -extension of some topological monoid M. Proof. According to Proposition 3.2 in [20], the subsemigroup h(b λ 0 (S)) in T is the Brandt λ 0 - extension of a subsemigroup M = h(s i,i ) in T for some i I λ. Since the subsemigroup M in T is a topological monoid, it follows from Definition 2 that h(b λ 0 (S)) is the topological Brandt λ 0 -extension of the topological monoid M. Lemma 2, Proposition ІІ.2 in [13], and Lemma ІІ.1.10 in [21] yield the following statement: Proposition 3. Let B 0 λ (S) be the topological Brandt λ 0 -extension of a topological inverse monoid S in the class of topological inverse semigroups, let T be a topological inverse semigroup, and let h: B 0 λ (S) T be a continuous homomorphism. Then h(b 0 λ (S)) is the topological Brandt λ 0 -extension of some topological inverse monoid M in the class of topological inverse semigroups. Proposition 4. Let B λ 0 (S) be the topological Brandt λ 0 -extension of a topological monoid S, let T be a topological semigroup, and let h: B λ 0 (S) T be a continuous homomorphism. If A is a nonempty subset in S such that h(a i, j ) is a closed subset in T for some i, j I λ, then h(a k,l ) is a closed subset in T for all k, l I λ. Proof. By virtue of Lemma 2, h(b λ 0 (S)) is the topological Brandt λ 0 -extension of a topological monoid M = h(s i,i ) for some i I λ. We define mappings ϕ:t T and ψ :T T by the formulas ϕ(t) = h((k,1 S,i)) t h(( j,1 S, l)) and ψ(t) = h((i,1 S, k)) t h((l,1 S, j)). Since the mappings ϕ and ψ are continuous as compositions of shifts, we conclude that A =ψ 1 (h(a i, j )) is a closed subset in T. Then, for the mapping f =ϕ ψ, the restriction f A = f A : A h(a k,l ) is a retraction, and the set h(a k,l ) is a retract of the space A. Hence, h(a k,l ) is a closed subset in T. Theorem 4. Let B λ 0 (S) be the topological Brandt λ 0 -extension of a topological monoid S. If the semigroup S is absolutely H -closed and the band of the semigroup B λ 0 (S) is compact, then B λ 0 (S) is an absolutely H -closed topological semigroup.

9 ON TOPOLOGICAL BRANDT SEMIGROUPS 9 Proof. Let T be an arbitrary topological semigroup and let h: B λ 0 (S) T be a continuous homomorphism. Then, by virtue of Lemma 2, the subsemigroup h(b λ 0 (S)) is the topological Brandt λ 0 -extension of a certain topological monoid M, namely of a topological subsemigroup h(s i,i ) in T for some i I λ. Since the semigroup S is absolutely H -closed, we conclude that h(s i,i ) is a closed subset in T, and, according to Theorem 3 and Lemma 2, the subsemigroup h(b λ 0 (S)) is closed in T. Theorem 4 yields the following statement: Corollary 3. Let B 0 λ (S) be the topological Brandt λ 0 -extension of a compact topological monoid S such that the band of the semigroup B 0 λ (S) is compact. Then B 0 λ (S) is an absolutely H -closed topological semigroup. Theorem 5. Let B 0 λ (S) be the topological Brandt λ 0 -extension of a topological inverse monoid S with compact band E(S) in the class of topological inverse semigroups. If E(B 0 λ (S)) is regular and B 0 λ (S) is an H -closed semigroup, then the band E(B 0 λ (S)) is compact. Z k=n k Proof. Assume to the contrary that the band E(B 0 λ (S)) is not a compact subset in B 0 λ (S). Then, since the band E(S) of the semigroup S is compact, it follows from Proposition 1 that there exists an open neighborhood U(0) of the zero 0 of the semigroup B 0 λ (S) in E(B 0 λ (S)) such that the set E(B 0 λ (S)) \ U(0) intersects infinitely many sets of the type E(S) i,i, i I λ. Since the space E(B 0 λ (S)) is regular, there exists an open neighborhood V (0) of zero in E(B 0 λ (S)) such that V (0) U(0). Without loss of generality, we can assume that there are countably many sets of this type and enumerate them by natural numbers: E(S) i,i, i = 1, 2,. Let A i = E(S) i,i \ V (0). Then A i is an open subset in E(B 0 λ (S)) and A i U(0) = for all i = 1, 2,. Further, we define mappings π 1 : B 0 λ (S) E(B 0 λ (S)) and π 2 : B 0 λ (S) E(B 0 λ (S)) by the formulas π 1 (s) = s s 1 and π 2 (s) = s 1 s. Since B 0 λ (S) is a topological inverse semigroup, the mappings π 1 and π 2 are continuous. For any natural n, we denote Z n =π 1 1 (A 2n 1 ) π 1 2 (A 2n ) and P n =. It follows from the continuity of π 1 and π 2 that π 1 1 (V (0)) π 2 1 (V (0)) and P n are open subsets in B λ 0 (S) for any natural n, and then, obviously, π 1 1 (V (0)) π 2 1 (V (0)) P n =. Let x B λ 0 (S). We extend the semigroup operation from B λ 0 (S) to the set T = B λ 0 (S) {x} as follows: x x = s x = x s = 0 for all s B λ 0 (S). It is obvious that the binary operation on T thus defined is associative. Let τ B be a topology on B λ 0 (S). We define a topology τ T on T as follows: that S i, j (i) for every s B λ 0 (S), the bases of the topologies τ B and τ T coincide; (ii) the family I(x) = {U n (x) = {x} P n n = 1, 2, } is the base of the topology τ T at the point x T. Since π 1 1 (V (0)) π 1 2 (V (0)) P n = and, for any subset S i, j in B λ 0 (S), there exists a natural m such P m =, we conclude that T is a Hausdorff space. In addition, for any open neighborhood of zero

10 10 O. V. GUTIK, K. P. PAVLYK, AND A. R. REITER W (0) V (0), we have W (0) U n (x) = U n (x) W (0) = U n (x) U n (x) = {0} W (0). Since, for any subset S i, j in B 0 λ (S), there exists a natural k such that S i, j P k = P k S i, j = {0} W (0), we conclude that T is a topological semigroup containing B λ 0 (S) as a dense subsemigroup. The contradiction obtained implies that E(B λ 0 (S)) is a compact subset in B λ 0 (S). Remark 2. Since every topological Brandt λ-extension B λ (S) of a topological semigroup S is the topological Brandt λ 0 -extension B λ 0 (T ) of some topological monoid T with zero, the statements of Theorems 3 5 hold for the topological Brandt λ-extensions of topological semigroups. Theorem 6. A topological inverse Brandt semigroup B λ (G) with H -closed maximal subgroup in the class of topological semigroups is an H -closed topological semigroup if and only if the band E(B λ (G)) of the semigroup B λ (G) is compact. Proof. The sufficiency follows from Theorem 3 and Remark 2. Let us prove the necessity. Since, according to Lemma 4 in [16], every nonzero idempotent of the band E(B λ (G)) is an isolated point in E(B λ (G)), the topological space E(B λ (G)) is 0 -dimensional and, hence, regular. Thus, all conditions of Theorem 5 are satisfied. Theorem 7. A topological inverse Brandt semigroup B λ (G) with absolutely H -closed maximal subgroup in the class of topological semigroups is an absolutely H -closed topological semigroup if and only if the band E(B λ (G)) of the semigroup B λ (G) is compact. Proof. The necessity follows from Theorem 6, and the sufficiency is a consequence of Theorem 4 and Remark 2. The research of the second author was financially supported by the Estonian Science Foundation and cofunded by the Marie Curie Actions, Postdoctoral Research Grant No. ERMOS36. REFERENCES 1. O. V. Gutik, On Howie semigroup, Mat. Metody Fiz.-Mekh. Polya, 42, No. 4, (1999). 2. O. V. Gutik and K. P. Pavlyk, H -closed topological semigroups and Brandt λ -extensions, Mat. Metody Fiz.-Mekh. Polya, 44, No. 3, (2001). 3. K. P. Pavlyk, Absolutely H -closed topological semigroups and Brandt λ -extensions, Prykl. Probl. Mekh. Mat., No. 2, (2004). 4. R. Engelking, General Topology, PWN, Warsaw (1977). 5. O. T. Alas and M. Sanchis, Countably compact paratopological groups, Semigroup Forum, 74, No. 3, (2007). 6. A. Arhangel skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, Amsterdam (2008). 7. T. Banakh and S. Dimitrova, Openly factorizable spaces and compact extensions of topological semigroups, Comment. Math. Univ. Carolin., 51, No. 1, (2010). 8. T. Berezovski, O. Gutik, and K. Pavlyk, Brandt extensions and primitive topological inverse semigroups, Int. J. Math. Math. Sci., Article ID , doi: /2010/ (2010).

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