On locally compact shift-continuous topologies on the $\alpha$-bicyclic monoid

A topology $\tau$ on a monoid $S$ is called {\em shift-continuous} if for every $a,b\in S$ the two-sided shift $S\to S$, $x\mapsto axb$, is continuous. For every ordinal $\alpha\le \omega$, we describe all shift-continuous locally compact Hausdorff topologies on the $\alpha$-bicyclic monoid $\mathcal{B}_{\alpha}$. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies on $\mathcal{B}_{\alpha}$ is anti-isomorphic to the segment of $[1,\alpha]$ of ordinals, endowed with the natural well-order. Also we prove that for each ordinal $\alpha$ the $\alpha+1$-bicyclic monoid $\mathcal{B}_{\alpha+1}$ is isomorphic to the Bruck extension of the $\alpha$-bicyclic monoid $\mathcal{B}_{\alpha}$.

In this paper all topological spaces are assumed to be Hausdorff.We shall follow the terminology of [9,11,22,25].By N we denote the set of all positive integers and by ω = N ∪ {0} the set of all finite ordinals (= non-negative integer numbers).
A semigroup S is called inverse if for every x ∈ S there exists a unique element x −1 ∈ S (called the inverse of x) such that xx −1 x = x and x −1 xx −1 = x −1 .The map (•) −1 : S → S which assigns to every x ∈ S its inverse element x −1 is called the inversion.A topological (inverse) semigroup is a topological space together with a continuous semigroup operation (and continuous inversion).Obviously, the inversion defined on a topological inverse semigroup is a homeomorphism.If S is an (inverse) semigroup and τ is a topology on S such that (S, τ ) is a topological (inverse) semigroup, then we shall call τ an (inverse) semigroup topology on S. A semitopological semigroup is a topological space together with a separately continuous semigroup operation.
Let α be an arbitrary ordinal and < be the usual order on α (defined by a < b iff a ∈ b).For every a, b ∈ α we write a ≤ b iff either a = b or a ∈ b.Clearly, ≤ is a partial order on α.By + we will denote the operation of addition of ordinals.For two ordinals a, b with a > b let a − b be a unique ordinal c such that a = b + c.For more information on ordinals, see [21], [27] or [28].
By [28,Theorem 17] each ordinal x can be uniquely written in its Cantor's normal form as where n i are positive integers and t 1 , t 2 , ..., t p is a decreasing sequence of ordinals.
In this paper for us will be convenient to use the following modified Cantor's normal form for ordinals x < ω α+1 : where n 1 ∈ ω, n 2 , . . ., n p ∈ N and α, t 2 , t 3 , . . ., t p is a decreasing sequence of ordinals.Obviously that each ordinal x < ω α+1 has a unique modified Cantor's normal form.Later on we denote the tail n 2 ω t 2 + • • • + n p ω tp of the modified Cantor's normal form of the ordinal x by x * .If x = nω α for some non-negative integer n, then x * = 0.The bicyclic monoid B(p, q) is the semigroup with the identity 1 generated by two elements p and q connected by the relation pq = 1.The bicyclic semigroup is isomorphic to the set N × N endowed with the binary operation: It is well known that the bicyclic monoid B(p, q) is a bisimple (and hence simple) combinatorial Eunitary inverse semigroup and every non-trivial congruence on B(p, q) is a group congruence [9].Also, a classic Andersen Theorem states that a simple semigroup S with an idempotent is completely simple if and only if S does not contain an isomorphic copy of the bicyclic semigroup (see [1] and [9,Theorem 2.54]).The bicyclic semigroup does not admit non-discrete semigroup topologies and if a topological semigroup S contains the bicyclic semigroup as a dense subsemigroup, then S \ B(p, q) is a closed ideal in S (see [10]).In [7] this result was extended to semitopological bicyclic semigroups.In [12] it was proved that a Hausdorff locally compact semitopological bicyclic semigroup with adjoined zero is either compact or discrete.
Let (S, ⋆) be an arbitrary semigroup.The Bruck extension B(S) of the semigroup S (see [8]) is the set N × S × N endowed with the following semigroup operation: For numbers n, m ∈ ω by [n, m] we denote the set {(n, s, m) : s ∈ S} ⊂ B(S), called a box.By B 0 (S) we denote the Bruck extension of the semigroup S with adjoined zero.
Possible topologizations of the Bruck extension of a topological semigroup were investigated in [14] and [15].In [16] a topologization of the Bruck-Reilly extension of a semitopological semigroup was studied.A locally compact Bruck extension of a compact topological group was investigated in [13].
Among other important generalizations of bicyclic semigroup let us mention polycyclic monoids and α-bicyclic monoids.Polycyclic monoids were introduced in [24].Algebraic and topological properties of the polycyclic monoids were investigated in [19,20,5,6].The paper [23] is devoted to topological properties of the graph inverse semigroups which are the generalizations of polycyclic monoids.In that paper it was shown that for every finite graph E every locally compact semigroup topology on the graph inverse semigroup over E is discrete, which implies that for every positive integer n every locally compact semigroup topology on the n-polycyclic monoid is discrete.In [4] this result was extended up to a characterization of graph inverse semigroups admitting only discrete locally compact semigroup topology.In [3] it was proved that for each cardinal λ a locally compact semitopological λ-polycyclic monoid is either compact or discrete.
In this paper we are mostly concerned with α-bicyclic monoids.These monoids were introduced in [18] and play a crucial role in the structure of bisimple semigroups with well ordered set of idempotents.
For an ordinal α the α-bicyclic monoid B α is the set ω α × ω α endowed with the binary operation Observe that B 0 is a singleton and B 1 is isomorphic to the bicyclic semigroup B(p, q).In [26] a non-discrete inverse semigroup topology on B 2 was constructed.Inverse semigroup topologies on B α were investigated in [17].In [2] it was proved that α-bicyclic monoid B α is algebraically isomorphic to the semigroup of all order isomorphisms between the principal filters of the ordinal ω α .Also in [2] it was proved that for every ordinal α ≤ ω there exists only discrete locally compact semigroup topology on B α , and an example of a non-discrete locally compact inverse semigroup topology on B ω+1 was constructed.
A topology τ on a monoid S is called shift-continuous if for every a, b ∈ S the two-sided shift S → S, x → axb, is continuous.Observe that a topology τ on a monoid S is shift-continuous if and only if the semigroup operation S × S → S, (x, y) → xy, is separately continuous if and only if (S, τ ) is a semitopological semigroup.We recall that all topologies considered in this paper are Hausdorff.
In this paper for every ordinal α ≤ ω we describe all shift-continuous locally compact topologies on the α-bicyclic monoid B α for α ≤ ω.In particular, we prove that the lattice of shift-continuous locally compact topologies on B α is anti-isomorphic to the segment of ordinals [1, α].We start with the following theorem.
Proof.Define an isomorphism f : B α+1 → B(B α ) by the formula: Since each ordinal a < ω α+1 has a unique modified Cantor's normal form, the map f is a bijection.Now we are going to show that f is a homomorphism.Let There are four cases to consider: Hence the map f is an isomorphism between B α+1 and B(B α ).

Lemma 2. Let α be an ordinal. For any element
The routine calculations show that V ⊂ V (a,b) .
Lemma 6. Assume that 0 * is non-isolated in a locally compact shift-continuous topology τ on B 0 (B k ).
Then for each neighborhood U ∈ τ of 0 * there exists a number n ∈ ω such that the set U ∩ [n, m] is non-empty for infinitely many numbers m ∈ ω.
Proof.To derive a contradiction, assume that for some neighborhood U ∈ τ of 0 * and some n ∈ ω the set A n = {m ∈ ω : U ∩ [n, m] = ∅} is finite.Since the topology τ is locally compact, we can additionally assume that the neighborhood U has compact closure Ū .We claim that each neighborhood V ∈ τ of 0 * intersects infinitely many boxes [n, m].Indeed, suppose that some neighborhood V ∈ τ of 0 * intersects only finitely many boxes, i.e., V ⊂ p i=1 [n i , m i ] ∪ {0 * } for some p ∈ N and some numbers n i , m i ∈ ω.By Lemma 4, for every non-negative integers n, m the set [n, m] ∪ {(n + 1, (0, 0), m + 1} is closed in (B 0 (B k ), τ ) and hence the set F := p i=1 ([n i , m i ] ∪ {(n i + 1, (0, 0), m i + 1}) is closed as well.Then the neighborhood V \ F of 0 * is a singleton, which is not possible as 0 * is non-isolated.
Corollary 9. Let α ≤ ω and τ be a locally compact shift-continuous topology on the monoid B α .If for some n ∈ ω the point (ω n , ω n ) is isolated in the topology τ , then for each non-negative i < α − n the point (ω n+i , ω n+i ) is isolated in (B α , τ ).
Proof.The statement is trivial for i = 0. Assume that for some i ∈ ω we have proved that (ω n+i , ω m+i ) is an isolated point in (B α , τ ).
Proposition 13.Let τ be a locally compact shift-continuous topology on B 0 (B k ).If 0 * is not isolated in the topology τ , then each neighborhood U ∈ τ of 0 * contains all but finitely many boxes [n, m] with n, m ∈ ω.
Proof.To derive a contradiction, assume that for some neighborhood U ∈ τ of 0 * the set The following Theorem generalizes Theorem 1 of Gutik [12].Theorem 14.A locally compact shift-continuous topology τ on the monoid B 0 (B k ) is compact if the point 0 * is not isolated in the topology τ .
To see that the topology τ is compact, take any open cover U ⊂ τ of B 0 (B k ) and find a set U ∈ U containing 0 * .By Proposition 13, the set A = {(n, m) ∈ ω × ω : [n, m] ⊂ U} is finite.For every (n, m) ∈ A choose a finite subcover U n,m ⊂ U of the compact set [n, m] ∪ {(n + 1, (0, 0), m + 1)} and observe that {U} ∪ (n,m)∈A U (n,m) ⊂ U is a finite cover of B 0 (B k ), witnessing that the topology τ is compact.
According to Lemma 2 and Proposition 3, to define a shift-continuous topology on the monoid B α for α ≤ ω, it is sufficient to define open neighborhood bases of the points (ω n , ω n ) for every positive integer n < α.
For every ordinals α ≤ ω and 0 < i ≤ α consider the locally compact Hausdorff shift-continuous topology τ i,α on the α-bicyclic monoid B α uniquely determined by the following two conditions: (i) for every integer j with i ≤ j < α the point (ω j , ω j ) is isolated in the topology τ i,α ; (ii) for every positive integer j with j < i a neighborhood base of the topology τ i,α at (ω j , ω j )consists of the sets U n = {(a, b) : nω j−1 < a < ω j or nω j−1 < b < ω j } ∪ {(ω j , ω j )}, n ∈ N.
Theorem 15.Let α ≤ ω.Each locally compact shift-continuous topology τ on the monoid B α coincides with the topology τ i,α for some non-zero ordinal i ≤ α.
then by [2, Lemma 5] and [2, Proposition 7] the point (a, b) is isolated in the semitopological monoid (B α , τ ) and there is noting to prove.Suppose that n m = k t = 0.By [2, Lemma 3] the set

Lemma 5 .
Assume that 0 * is non-isolated in a locally compact shift-continuous topology τ on B 0 (B k ).Then for any open neighborhoods U, V ∈ τ of 0 * with compact closures U , V the following condition holds:U ∩ [n, m] = V ∩ [n, m] for all but finitely many boxes [n, m], where n, m ∈ ω.