On locally compact semitopological $0$-bisimple inverse $\omega$-semigroups

TL;DR

This paper characterizes the structure of certain locally compact semitopological inverse semigroups, showing they are either compact or a topological sum of classes, with specific results for Reilly semigroups with monothetic subgroups.

Contribution

It provides a detailed structural description and a dichotomy result for locally compact semitopological $0$-bisimple inverse $ ext{omega}$-semigroups, including Reilly semigroups with monothetic subgroups.

Findings

Such semigroups are either compact or topologically sum of $ ext{H}$-classes.

Reilly semigroups with non-annihilating homomorphism are either compact or discrete.

The paper discusses the closure and remainder of the semigroup in larger semigroups.

Abstract

We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its $\mathscr{H}$-classes. We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroups with a monothetic maximal subgroups. In particular we prove the dichotomy for $T_1$ locally compact semitopological Reilly semigroup $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^0,\tau\right)$ with adjoined zero and with a non-annihilating homomorphism $\theta\colon \mathbb{Z}_{+}\to \mathbb{Z}_{+}$: $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^0,\tau\right)$ is either compact or discrete. At the end we discuss on the remainder under the closure of the discrete Reilly semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^0$ in a semitopological semigroup.

Equations

$$(i,s,j)\cdot (m,t,n)=\left(i+m-\min \{j,m\},{\theta }^{m-\min \{j,m\}}(s){\theta }^{j-\min \{j,m\}}(t),j+n-\min \{j,m\}\right),$$

$$(i,s,j){\eta }^{♮}(m,t,n)if\; and\; only\; ifi=mandj=n$$

$$S_{i,i}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(i,1_S,i)\cdot (m,s,n)=\left(m,s,n\right)and(m,s,n)\cdot (i,1_S,i)=\left(m,s,n\right),for\; all\; integersm,n⩾i,$$

$$\begin{array}{rl}(i,1_S,i)\cdot (m,s,n)=& (i,{\theta }^{i-m}(s),i+n-m)\ne \left(m,s,n\right)and\\ (m,s,n)\cdot (i,1_S,i)=& (i-n+m,{\theta }^{i-n}(s),i)\ne \left(m,s,n\right),for\; all\; non-negative\; integersm,n<i.\end{array}$$

$$V\subseteq \text{BR}(S,\theta )\setminus \left((k+i+1,1_S,k+i+1)\text{BR}(S,\theta )\cup \text{BR}(S,\theta )(i+1,1_S,i+1)\right).$$

$$(m,t,n)\cdot (i,1_S,0)=\left(m+i-n,{\theta }^{i-n}(t),0\right)\in S_{k+i,0},$$

$$(i,s,j){\eta }^{♮}(m,t,n)if\; and\; only\; ifi=mandj=n,and\text{0}{\eta }^{♮}\text{0}$$

$$U^{\ast }=\bigcup \left\{G_{i,j}:G_{i,j}\cap U\ne \varnothing \right\}\cup \{\text{0}\}.$$

$$\left(\text{B}(G,\theta {)}^0,{\tau }_{\text{B}}^0\right)=⨁\left\{G_{i,j}:i,j\in {\mathbb{N}}_0\right\}\oplus \{\text{0}\}.$$

$$\left\{j\in {\mathbb{N}}_0:G_{i_0,j}⊈U_{\text{0}}\right\}and\left\{j\in {\mathbb{N}}_0:G_{j,i_0}⊈U_{\text{0}}\right\}$$

$${\mathfrak{H}}_{i_0}^0(U_{\text{0}})=\{\text{0}\}\cup \bigcup _{j\in {\mathbb{N}}_0}{\left[{\mathrm{cl}}_{\text{B}(G,\theta {)}^0}(U_{\text{0}})\right]}_{i_0,j}$$

$${\left[V_{\text{0}}\right]}_{i_0,n}={\left[U_{\text{0}}\right]}_{i_0,n}={\left[{\mathrm{cl}}_{\text{B}(G,\theta {)}^0}(U_{\text{0}})\right]}_{i_0,n}.$$

$${\tilde{U}}_{\text{0}}=U_{\text{0}}\setminus \left(G_{i_0,0}\cup \dots \cup G_{i_0,j_2-1}\right).$$

$$\big{[}\widetilde{U}_{\textsf{0}}\big{]}_{i_{0},n}\setminus\left[W_{\textsf{0}}\right]_{i_{0},n}\neq\varnothing\qquad\hbox{and}\qquad\left[W_{\textsf{0}}\right]_{i_{0},n}\setminus\big{[}\widetilde{U}_{\textsf{0}}\big{]}_{i_{0},n}\neq\varnothing,$$

$${\mathfrak{H}}_{i_0}^0(U_{\text{0}})=\{\text{0}\}\cup \bigcup _{j\in {\mathbb{N}}_0}{\left[{\mathrm{cl}}_{\text{B}(G,\theta {)}^0}(U_{\text{0}})\right]}_{i_0,j}$$

$${\left[V_{\text{0}}\right]}_{i_n,j_n}\subseteq {\left[U_{\text{0}}\right]}_{i_n,j_n}\ne G_{i_n,j_n}and{\left[U_{\text{0}}\right]}_{i_n,j_n+1}=G_{i_n,j_n+1},$$

$${\mathcal{B}}_{\text{B}}^{\oplus }(i,g,j)=\left\{U_{i,j}:U\in {\mathcal{B}}_G(g)\right\}$$

$${\mathcal{B}}_{\text{B}}^{\mathrm{Ac}}(i,g,j)=\left\{U_{i,j}:U\in {\mathcal{B}}_G(g)\right\};$$

$$U_{(i_1,j_1),\dots ,(i_k,j_k)}=\text{B}(G,\theta {)}^0\setminus \left\{G_{i_1,j_1}\cup \cdots \cup G_{i_k,j_k}\right\}.$$

$${\mathcal{B}}_{\mathbf{cf}}(i,0_{{\mathbb{Z}}_{+}},j)=\left\{{\left(U_F\right)}_{i-1,j-1}^0={\left({\mathbb{Z}}_{+}\setminus F\right)}_{i-1,j-1}^0:Fis\; a\; finite\; subset\; of\mathbb{Z}\right\}$$

$${\left(U_F\right)}_{i,j}^0\cdot (m,g,n)and(m,g,n)\cdot {\left(U_F\right)}_{i,j}^0,$$

$$\left(U_{F}\right)_{i,j}^{0}\cdot(m,g,n)=\left\{\begin{array}[]{cl}\left\{(m+i-j,g,n)\right\},&\hbox{if~{}}j<m;\\ \left(U_{F+g}\right)_{i,n}^{0},&\hbox{if~{}}j=m;\\ \left(U_{F}\right)_{i,j+n-m}^{0},&\hbox{if~{}}j>m,\end{array}\right.$$

$$(i,g,j)\cdot\left(U_{F}\right)_{m,n}^{0}=\left\{\begin{array}[]{cl}\left(U_{F}\right)_{i+m-j,n}^{0},&\hbox{if~{}}j<m;\\ \left(U_{g+F}\right)_{i,n}^{0},&\hbox{if~{}}j=m;\\ \left\{(i,g,j+n-m)\right\},&\hbox{if~{}}j>m,\end{array}\right.$$

$$\mathrm{inv}\left({\left(U_F\right)}_{i,j}^0\right)={\left(U_{-F}\right)}_{j,i}^0,$$

$${\mathcal{B}}_{+}(i,0_{{\mathbb{Z}}_{+}},j)=\left\{{\left(U_F^{+}\right)}_{i-1,j-1}^0={\left(\mathbb{N}\setminus F\right)}_{i-1,j-1}^0:Fis\; a\; finite\; subset\; of\mathbb{N}\right\}$$

$${\left(U_F^{+}\right)}_{i,j}^0\cdot (m,g,n),(m,g,n)\cdot {\left(U_F^{+}\right)}_{i,j}^0,and{\left(U_F^{+}\right)}_{k,l}^0\cdot {\left(U_{F^{\prime }}^{+}\right)}_{i,j}^0,$$

$$\left(U_{{F-g}}^{+}\right)_{i,j}^{0}\cdot(m,g,n)=\left\{\begin{array}[]{cl}\left\{(m+i-j,g,n)\right\},&\hbox{if~{}}j<m;\\ \left(U_{F}^{+}\right)_{i,n}^{0},&\hbox{if~{}}j=m;\\ \left(U_{{F-g}}^{+}\right)_{i,j+n-m}^{0},&\hbox{if~{}}j>m,\end{array}\right.$$

$$(i,g,j)\cdot\left(U_{{F-g}}^{+}\right)_{m,n}^{0}=\left\{\begin{array}[]{cl}\left(U_{{F-g}}^{+}\right)_{i+m-j,n}^{0},&\hbox{if~{}}j<m;\\ \left(U_{F}^{+}\right)_{i,n}^{0},&\hbox{if~{}}j=m;\\ \left\{(i,g,j+n-m)\right\},&\hbox{if~{}}j>m,\end{array}\right.$$

$$\left(U_{F}^{+}\right)_{i,j}^{0}\cdot\left(U_{F}^{+}\right)_{m,n}^{0}=\left\{\begin{array}[]{cl}\left(U_{F}^{+}\right)_{m,n}^{0},&\hbox{if~{}}j<m;\\ \left(U_{F+F}^{+}\right)_{i,n}^{0},&\hbox{if~{}}j=m;\\ \left(U_{F}^{+}\right)_{i,j}^{0},&\hbox{if~{}}j>m,\end{array}\right.$$

Full text

1. Introduction and preliminaries

Further we shall follow the terminology of [10, 13, 17, 43]. By $\mathbb{N}$ (resp. $\mathbb{N}_{0}$) we denote the set of all positive (resp., non-negative) integers and by $\mathbb{Z}_{+}$ we denote the additive group of integers. We shall say that a set $A$ intersects almost all elements of a infinite family $\mathscr{K}=\{K_{i}\}_{i\in\Omega}$ of non-empty sets if $A\cap K_{i}=\varnothing$ only for finitely many elements of $\mathscr{K}$. All topological spaces, considered in this paper, are Hausdorff, if the otherwise is not stated explicitely. If $A$ is a subset of a topological space $X$ then by $\operatorname{cl}_{X}(A)$ and $\operatorname{int}_{X}(A)$ we denote the closure and interior of $A$ in $X$, respectively.

A subset $A$ of a topological space $X$ is called regular open if $A=\operatorname{int}_{X}(\operatorname{cl}_{X}(A))$.

We recall that a topological space $X$ is called

• •

semiregular if $X$ has a base consisting of regular open subsets;

• •

compact if each open cover of $X$ has a finite subcover;

• •

locally compact if each point of $X$ has an open neighbourhood with the compact closure;

• •

Čech-complete if $X$ is Tychonoff and there exists a compactification $cX$ of $X$ such that the remainder $cX\setminus c(X)$ is an $F_{\sigma}$-set in $cX$.

It is well-known (see [17, Section 3.9]) and easy to show that every locally compact space is semiregular and Čech-complete.

Given a semigroup $S$, we shall denote the set of idempotents of $S$ by $E(S)$. A semigroup $S$ with the adjoined zero will be denoted by $S^{0}$ (cf. [13]).

A semigroup $S$ is called inverse if for every $x\in S$ there exists a unique $x^{-1}\in S$ such that $xyx=x$ and $yxy=y$. In the following this element $y$ we shall denote by $x^{-1}$ and call inverse of $x$. A map $\operatorname{\textsf{inv}}\colon S\to S$ which poses every $s\in S$ its inverse is called inversion.

The subset of idempotents of a semigroup $S$ we shall denote by $E(S)$. If $S$ is an inverse semigroup then $E(S)$ is closed under multiplication and we shall refer to $E(S)$ as a band (or the band of $S$). If the band $E(S)$ is non-empty, then the semigroup operation on $S$ determines the following natural partial order $\leqslant$ on $E(S)$: $e\leqslant f$ if and only if $ef=fe=e$. A semilattice is a commutative semigroup of idempotents. A semilattice $E$ is called linearly ordered or a chain if its natural order is a linear order. A chain $L$ is called an $\omega$-chain if $L$ is isomorphic to $\{0,-1,-2,-3,\ldots\}$ with the semigroup operation $x\cdot y=\min\{x,y\}$, thus the natural partial order on $\{0,-1,-2,-3,\ldots\}$ coincides with the usual order $\leqslant$.

A semitopological (topological) semigroup is a topological space with a separately continuous (jointly continuous) semigroup operation. An inverse topological semigroup with continuous inversion is called a topological inverse semigroup. Also, a semitopological (topological) group is a topological space with a separately continuous (jointly continuous) group operation (and inversion).

A topology $\tau$ on a semigroup $S$ is called:

• •

semigroup if $(S,\tau)$ is a topological semigroup;

• •

shift-continuous if $(S,\tau)$ is a semitopological semigroup;   and

• •

inverse semigroup if $(S,\tau)$ is a topological inverse semigroup.

The bicyclic semigroup ${\mathscr{C}}(p,q)$ is the semigroup with the identity $1$ generated by two elements $p$ and $q$ subject only to the condition $pq=1$. The bicyclic monoid is a combinatorial bisimple $F$-inverse semigroup and it plays the important role in the algebraic theory of semigroups and in the theory of topological semigroups. For example well-known Andersen’s Theorem [1] states thata ([math]-)simple semigroup with an idempotent is completely ([math]-)simple if and only if it does not contain the bicyclic semigroup. Eberhart and Selden showed that the bicyclic semigroup admits only the discrete semigroup topology and if a topological semigroup $S$ contains it as a dense subsemigroup then ${\mathscr{C}}(p,q)$ is an open subset of $S$ [15]. Bertman and West in [8] extended these results for semitopological semigroups. In [6, 7, 11, 12, 18, 22, 24, 26, 28, 29, 37] these topologizability results were extended to generalizations of the bicyclic semigroup. Stable and $\Gamma$-compact topological semigroups do not contain the bicyclic semigroup [2, 31]. The problem of an embedding of the bicyclic monoid into compact-like topological semigroups was discussed in [4, 5, 27].

In [15] Eberhart and Selden proved that if the bicyclic monoid ${\mathscr{C}}(p,q)$ is a dense subsemigroup of a topological monoid $S$ and $I=S\setminus{\mathscr{C}}(p,q)\neq\varnothing$ then $I$ is a two-sided ideal of the semigroup $S$. Also, there they described the closure of the bicyclic monoid ${\mathscr{C}}(p,q)$ in a locally compact topological inverse semigroup. The closure of the bicyclic monoid in a countably compact (pseudocompact) topological semigroups was studied in [5].

In the paper [22] was shown the following dichotomy: a locally compact semitopological bicyclic semigroup with an adjoined zero is either compact or discrete. Also there was shown a counterpart of this result for a locally compact semitopological bicyclic semigroup with an adjoined compact ideal. On the other hand, there was constructed a Čech-complete first countable topological inverse bicyclic semigroup with adjoined non-isolated zero which is not compact. The above dichotomy was extended by Bardyla in [6] to locally compact $\lambda$-polycyclic semitopological monoids and to locally compact semitopological interassociates of the bicyclic monoid [23].

Let $S$ be a monoid and $\theta\colon S\to H_{S}(1)$ a homomorphism from $S$ into the group of units $H_{S}(1)$ of $S$. The set $\mathbb{N}_{0}\times S\times\mathbb{N}_{0}$ with the semigroup operation

[TABLE]

where $i,j,m,n\in\mathbb{N}_{0}$, $s,t\in S$ and $\theta^{0}$ is an identity map on $S$, is called the Bruck–Reilly extension of the monoid $S$ and it will be denoted by $\textbf{BR}(S,\theta)$. We observe that if $S$ is a trivial monoid (i.e., $S$ is a singleton set), then $\textbf{BR}(S,\theta)$ is isomorphic to the bicyclic semigroup ${\mathscr{C}}(p,q)$ and in case when $\theta\colon S\to H_{S}(1)$ is the annihilating homomorphism (i.e., $\theta(s)=1_{S}$ for all $s\in S$, where $1_{S}$ is the identity of the monoid $S$), then $\textbf{BR}(S,\theta)$ is called the Bruck semigroup over the monoid $S$ [9].

For arbitrary $i,j\in\mathbb{N}_{0}$ and non-empty subset $A$ of the semigroup $S$ we define subsets $A_{i,j}$ and $A_{i,j}^{0}$ of $\textbf{BR}(S,\theta)$ as follows: $A_{i,j}=\left\{(i,s,j)\colon s\in A\right\}$ and $A_{i,j}^{0}=A_{i,j}\cup\left\{(i+1,1_{S},j+1)\right\}$.

A regular semigroup $S$ is $\omega$-semigroup if $E(S)$ is isomorphic to the $\omega$-chain and $S$ is bisimple if $S$ constitutes a single $\mathscr{D}$-class.

The construction of the Bruck semigroup over a monoid was used in [9] for the proof of the statement that every semigroup embeds into a simple monoid. Also Reilly and Warne prove that every bisimple regular $\omega$-semigroup $S$ is isomorphic to the Bruck–Reilly extension of some group [42, 48], i.e., for every bisimple regular $\omega$-semigroup $S$ there exists a group $G$ and a homomorphism $\theta\colon G\to G$ such that $S$ is isomorphic to the Bruck–Reilly extension $\textbf{BR}(G,\theta)$. A bisimple regular $\omega$-semigroup is called Reilly semigroup [41]. Later similar as in the Petrich book [41], the Reilly semigroup $\textbf{BR}(G,\theta)$ we shall denote by $\textbf{B}(G,\theta)$.

Gutik constructed a topological counterpart of the Bruck construction in [19]. Also in [19, 21] he studied the problem of a topologization of the Bruck semigroup over a topological semigroup. A. Selden described the structure of locally compact bisimple regular topological $\omega$-semigroups and studied the closures of such semigroups in locally compact topological semigroups in [44, 45, 46]. In the paper [25] topologizations of the Bruck–Reilly extension $\textbf{BR}(S,\theta)$ of a semitopological semigroup $S$ which turns $\textbf{BR}(S,\theta)$ into a semitopological semigroup are studied.

An inverse semigroup $S$ is called a [math]-bisimple $\omega$-semigroup if $S$ has two $\mathscr{D}$-classes: $S\setminus\{0\}$ and $\{0\}$, and $E(S)\setminus\{0\}$ is order isomorphic to the $\omega$-chain. The results of the Lallement and Petrich paper [33] imply that every [math]-bisimple inverse $\omega$-semigroup is isomorphic to a Reilly semigroup with adjoined zero $\textbf{B}(G,\theta)^{0}$.

In this paper we describe the structure of locally compact semitopological [math]-bisimple inverse $\omega$-semigroups with a compact maximal subgroup. In particular, we show that a locally compact semitopological [math]-bisimple inverse $\omega$-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its $\mathscr{H}$-classes. We describe the structure of locally compact semitopological [math]-bisimple inverse $\omega$-semigroups with a monothetic maximal subgroups. In particular we prove the dichotomy that a locally compact $T_{1}$-semitopological Reilly semigroup $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ over the additive group of integers $\mathbb{Z}_{+}$, with adjoined zero and with a non-annihilating homomorphism $\theta$: the semigroup $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is either compact or discrete. At the end we discuss on the remainder under the closure of the discrete Reilly semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)$ in a semitopological semigroup.

2. On a topological Bruck–Reilly extension of a semitopological monoid

Later we need the following proposition which is a simple generalization of Lemma 1.2 from [40] and follows from Theorem XI.1.1 of [41].

Proposition 2.1**.**

Let $S$ be an arbitrary monoid and $\theta\colon S\to H_{S}(1)$ a homomorphism from $S$ into the group of units $H_{S}(1)$ of $S$. Then a map $\eta\colon\textbf{BR}(S,\theta)\to\mathscr{C}(p,q)$, $(i,s,j)\mapsto q^{i}p^{j}$ is a homomorphism and hence the relation $\eta^{\natural}$ on $\textbf{BR}(S,\theta)$ defined in the following way

[TABLE]

is a congruence.

Proposition 2.2**.**

Let $\tau_{\textbf{BR}}$ be a topology on $\textbf{BR}(S,\theta)$ such that the set $S_{i,j}$ is open in $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ for all $i,j\in\mathbb{N}_{0}$. Then $\eta^{\natural}$ is a closed congruence on $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$.

Proof.

Fix an arbitrary non-$\eta^{\natural}$-equivalent elements $(i,s,j),(m,t,n)\in\textbf{BR}(S,\theta)$. Then the definition of the relation $\eta^{\natural}$ implies that $S_{i,j}\cap S_{m,n}=\varnothing$. Since $S_{i,j}$ and $S_{m,n}$ are open disjoint neighbourhoods of the points $(i,s,j)$ and $(m,t,n)$ in the topological space $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$, Proposition 2.1 implies that $\eta^{\natural}$ is a closed congruence on $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$. ∎

Definition 2.3**.**

Let $\mathcal{C}$ be a class of semitopological semigroups and $(S,\tau_{S})\in\mathcal{C}$. If $\tau_{\textbf{BR}}$ is a topology on $\textbf{BR}(S,\theta)$ such that $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)\in\mathcal{C}$ and for some $i\in\mathbb{N}_{0}$ the subsemigroup $S_{i,i}$ with the topology restricted from $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ is topologically isomorphic to $(S,\tau_{S})$ under the map $\xi_{i}\colon S_{i,i}\to S$, $(i,s,i)\mapsto s$, then $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ is called a topological Bruck–Reilly extension of $(S,\tau_{S})$ in the class $\mathcal{C}$.

For all non-negative integers $i,j,m,n$ and an arbitrary Bruck–Reilly extension $\textbf{BR}(S,\theta)$ we define a map $\phi^{i,j}_{m,n}\colon\textbf{BR}(S,\theta)\to\textbf{BR}(S,\theta)$, $x\mapsto(m,1_{S},i)\cdot x\cdot(j,1_{S},n)$.

Proposition 2.4**.**

Let $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ be a topological Bruck–Reilly extension of a semitopological semigroup $(S,\tau_{S})$ in the class of semitopological semigroups. Then the following assertions hold:

• $(i)$

for all non-negative integers $i,j,m,n$ the restrictions $\phi^{i,j}_{m,n}|_{S_{i,j}}\colon S_{i,j}\to S_{m,n}$ and $\phi^{m,n}_{i,j}|_{S_{m,n}}\colon S_{m,n}\to S_{i,j}$ are mutually invertible homeomorphisms between subspaces $S_{i,j}$ and $S_{m,n}$ of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$;

• $(ii)$

for all non-negative integers $i,m$ the restrictions $\phi^{i,i}_{m,m}|_{S_{i,i}}\colon S_{i,i}\to S_{m,m}$ and $\phi^{m,m}_{i,i}|_{S_{m,m}}\colon S_{m,m}\to S_{i,i}$ are mutually invertible topological isomorphisms between semitopological subsemigroups $S_{i,i}$ and $S_{m,m}$ of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$;

• $(iii)$

the homomorphism $\theta\colon S\to H_{S}(1)$ is a continuous map;

• $(iv)$

for all non-negative integers $i,j,m,n$ the restrictions $\phi^{i,j}_{m,n}|_{S_{i,j}^{0}}\colon S_{i,j}^{0}\to S_{m,n}^{0}$ and $\phi^{m,n}_{i,j}|_{S_{m,n}^{0}}\colon S_{m,n}^{0}\to S_{i,j}^{0}$ are mutually invertible homeomorphisms between subspaces $S_{i,j}^{0}$ and $S_{m,n}^{0}$ of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$;

• $(v)$

if $\theta\colon S\to H_{S}(1)$ is an annihilating homomorphism then for all non-negative integers $i,m$ the restrictions $\phi^{i,i}_{m,m}|_{S_{i,i}^{0}}\colon S_{i,i}^{0}\to S_{m,m}^{0}$ and $\phi^{m,m}_{i,i}|_{S_{m,m}^{0}}\colon S_{m,m}^{0}\to S_{i,i}^{0}$ are mutually invertible topological isomorphisms between semitopological subsemigroups $S_{i,i}^{0}$ and $S_{m,m}^{0}$ of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$;

• $(vi)$

for every positive integer $i$ the left and right shifts $\rho_{(i,1_{S},i)}\colon\textbf{BR}(S,\theta)\to\textbf{BR}(S,\theta)$, $x\mapsto x\cdot(i,1_{S},i)$ and $\lambda_{(i,1_{S},i)}\colon\textbf{BR}(S,\theta)\to\textbf{BR}(S,\theta)$, $x\mapsto(i,1_{S},i)\cdot x$ are retractions, and hence $(i,1_{S},i)\textbf{BR}(S,\theta)$, $\textbf{BR}(S,\theta)(i,1_{S},i)$ are closed subsets, and hence $S_{0,0}$ is an open subset of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$;

• $(vii)$

for every positive integer $i$ the sets $S_{0,i}$ and $S_{i,0}$ are open in $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$;

• $(viii)$

for any non-negative integer $k$ and for every positive integer $i$ the sets $S_{0,k}\cup S_{1,k+1}\cup\cdots\cup S_{i,k+i}$ and $S_{k,0}\cup S_{k+1,1}\cup\cdots\cup S_{k+i,i}$ are open in $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$.

Proof.

Assertions $(i)$ and $(ii)$ follow from the definition of the semigroup $\textbf{BR}(S,\theta)$ and the separate continuity of the semigroup operation in $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$.

$(iii)$ Since $\left(i,s,i\right)\cdot\left(i+1,1_{S},i+1\right)=\left(i+1,\theta(s),i+1\right)$ for any $s\in S$, assertions $(i)$ and $(ii)$, and the separate continuity of the semigroup operation of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ imply that the homomorphism $\theta\colon S\to H_{S}(1)$ is a continuous map, because the following diagram

[TABLE]

commutes, where $\rho_{(i+1,1_{S},i+1)}$ is the right shift on the semigroup $\textbf{BR}(S,\theta)$ by the element $(i+1,1_{S},i+1)$..

$(iv)$ Since $\left(n,1_{S},i\right)(i+1,1_{S},j+1)(j,1_{S},m)=(n+1,1_{S},m+1)$, assertion $(iv)$ follows from the definition of the semigroup $\textbf{BR}(S,\theta)$, the separate continuity of the semigroup operation in $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ and $(iii)$.

$(v)$ If $\theta\colon S\to H_{S}(1)$ is the annihilating homomorphism then it is obvious that $S_{i,i}^{0}$ is a subsemigroup of $\textbf{BR}(S,\theta)$ for any non-negative integer $i$ such that $(i+1,1_{S},i+1)$ is zero of $S_{i,i}^{0}$, and next we apply item $(iv)$.

$(vi)$ The semigroup operation of $\textbf{BR}(S,\theta)$ implies that

[TABLE]

and

[TABLE]

Then the above arguments, [17, Exercise. 1.5.C], Hausdorffness of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ and the separate continuity of the semigroup operation of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ imply our assertion.

$(vii)$ Item $(vi)$ implies that $S_{0,0}\cup S_{0,1}\cup\cdots\cup S_{0,i}$ and $S_{0,0}\cup S_{1,0}\cup\cdots\cup S_{i,0}$ are open subsets of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$. Then $S_{0,i}=(S_{0,0}\cup S_{0,1}\cup\cdots\cup S_{0,i})\cap\rho_{(i,1_{S},0)}^{-1}(S_{0,0})$ is an open subset of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$, as well. Similar arguments show that $S_{i,0}$ is an open subset of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$, too.

$(viii)$ Fix an arbitrary non-negative integer $k$, an arbitrary positive integer $i$ and any element $(k+i,s,i)\in S_{k+i,i}$. Then $(k+i,s,i)\cdot(i,1_{S},0)=(k+i,s,0)$ and hence the separate continuity of the semigroup operation in $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ and assertion $(vii)$ imply that there exists an open neighbourhood $V$ of the point $(k+i,s,i)$ in $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ such that $V\cdot(i,1_{S},0)\subseteq S_{k+i,0}$. Without loss of generality we may assume that

[TABLE]

Fix an arbitrary $(m,t,n)\in V$. Then we have that $m\leqslant k+i$ and $n\leqslant i$. This implies that

[TABLE]

and hence $m=k+n$. This implies that $S_{k,0}\cup S_{k+1,1}\cup\cdots\cup S_{k+i,i}=\rho_{(i,1_{S},0)}(S_{k+i,0})$ is an open subset of $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$, because all shifts are continuous in $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$. The proof of openness of the set $S_{0,k}\cup S_{1,k+1}\cup\cdots\cup S_{i,k+i}$ in $\left(\textbf{BR}(S,\theta),\tau_{\textbf{BR}}\right)$ is similar. ∎

Proposition 2.4 and Theorem 3.3.8 of [17] imply the following

Corollary 2.5**.**

Every semitopological bisimple inverse $\omega$-semigroup $S$ is topologically isomorphic to a topological Bruck–Reilly extension $\left(\textbf{B}(G,\theta),\tau_{\textbf{B}}\right)$ of a semitopological group $(G,\tau_{G})$ in the class of semitopological semigroups. Moreover, if $S$ is locally compact then $(G,\tau_{G})$ is a locally compact topological group.

We observe that the continuity of the group operation and inversion in a locally compact semitopological group $(G,\tau_{G})$ follows from well-known Ellis’ Theorem (see [16, Theorem 2]).

The following corollary describes the structure of [math]-bisimple inverse $\omega$-semigroups and it follows from Theorem 4.2 of [39] (also see corresponding statements in [33] and [34]).

Corollary 2.6**.**

Every [math]-bisimple inverse $\omega$-semigroup $S$ is isomorphic to a Reilly semigroup with adjoined zero $\textbf{B}(G,\theta)^{0}=\textbf{B}(G,\theta)\sqcup\{\textsf{0}\}$.

Later in this paper by 0 we shall denote zero of a Reilly semigroup with adjoined zero $\textbf{B}(G,\theta)^{0}$.

By Theorem 3.3.8 from [17] every open subspace of a locally compact space is locally compact and hence Corollaries 2.5 and 2.6 imply the following theorem.

Theorem 2.7**.**

Every semitopological [math]-bisimple inverse $\omega$-semigroup $S$ is topologically isomorphic to a topological Bruck–Reilly extension $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ of a semitopological group $(G,\tau_{G})$ with adjoined zero (not necessarily as an isolated point) in the class of semitopological semigroups. Moreover, if $S$ is locally compact then $(G,\tau_{G})$ is a locally compact topological group.

3. On locally compact semitopological [math]-bisimple inverse $\omega$-semigroups with compact maximal subgroups

In this section we describe the structure of locally compact semitopological [math]-bisimple inverse $\omega$-semigroups with compact maximal subgroups.

Lemma 3.1**.**

Let $S$ be a semitopological [math]-bisimple inverse $\omega$-semigroup with compact maximal subgroups. Then every non-zero $\mathscr{H}$-class of $S$ is an open-and-closed subset of $S$.

Proof.

By Theorem 2.7 there exist a compact semitopological group (which by the Ellis Theorem is a topological group [16]) $(G,\tau_{G})$ and a continuous homomorphism $\theta\colon G\to G$ such that the semitopological semigroup $S$ is topologically isomorphic to a topological Bruck–Reilly extension $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ of $(G,\tau_{G})$ with an adjoined zero. It is obvious that every non-zero $\mathscr{H}$-class of the semigroup $\textbf{B}(G,\theta)^{0}$ coincides with $G_{i,j}$ for some $i,j\in\mathbb{N}_{0}$. Then items $(ii)$ and $(vi)$ of Proposition 2.4 imply the assertion of the lemma. ∎

For an arbitrary group $G$ and a homomorphism $\theta\colon G\to G$ we define a map $\eta\colon\textbf{BR}(G,\theta)^{0}\to\mathscr{C}^{0}={\mathscr{C}}(p,q)\cup\{0\}$ by the formulae $\eta(i,g,j)=q^{i}p^{j}$ and $\eta(\textsf{0})=0$, for $g\in G$ and $i,j\in\mathbb{N}_{0}$. Then using Proposition 2.1 we can show that the map $\eta$ is a homomorphism and hence the relation $\eta^{\natural}$ on $\textbf{BR}(G,\theta)$ defined in the following way

[TABLE]

is a congruence on the semigroup $\textbf{BR}(G,\theta)^{0}$.

Lemma 3.2**.**

Let $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ be a semitopological semigroup with a compact group of units. Then $\eta^{\natural}$ is a closed congruence on $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$.

Proof.

Fix arbitrary non-$\eta^{\natural}$-equivalent non-zero elements $(i,s,j)$ and $(m,t,n)$ of the Reilly semigroup with adjoined zero $\textbf{B}(G,\theta)^{0}$. Then $G_{i,j}$ and $G_{m,n}$ are open-and-closed neighbourhoods of the points $(i,s,j)$ and $(m,t,n)$ in the space $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$, respectively, such that $\eta^{\natural}\cap\left(G_{i,j}\times G_{m,n}\right)=\varnothing$. Also, the above arguments imply that $G_{i,j}\times\left(\textbf{B}(G,\theta)^{0}\setminus G_{i,j}\right)$ is an open-and-closed neighbourhood of the ordered pair $\left((i,s,j),\textsf{0}\right)$ in $\textbf{B}(G,\theta)^{0}\times\textbf{B}(G,\theta)^{0}$ with the product topology which does not intersect the congruence $\eta^{\natural}$ of semigroup $\textbf{B}(G,\theta)^{0}$. Hence we get that $\eta^{\natural}$ is a closed congruence on the semitopological semigroup $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$. ∎

Later we shall need the following notions. A continuous map $f\colon X\to Y$ from a topological space $X$ into a topological space $Y$ is called:

• $\bullet$

quotient if the set $f^{-1}(U)$ is open in $X$ if and only if $U$ is open in $Y$ (see [38] and [17, Section 2.4]);

• $\bullet$

hereditarily quotient (or pseudoopen) if for every $B\subset Y$ the restriction $f|_{B}\colon f^{-1}(B)\to B$ of $f$ is a quotient map (see [35, 36, 3] and [17, Section 2.4]);

• $\bullet$

open if $f(U)$ is open in $Y$ for every open subset $U$ in $X$;

• $\bullet$

closed if $f(F)$ is closed in $Y$ for every closed subset $F$ in $X$;

• $\bullet$

perfect if $X$ is Hausdorff, $f$ is a closed map and all fibers $f^{-1}(y)$ are compact subsets of $X$ [47].

Every perfect map is closed, every closed map and every hereditarily quotient map are quotient [17]. Moreover a continuous map $f\colon X\to Y$ from a topological space $X$ onto a topological space $Y$ is hereditarily quotient if and only if for every $y\in Y$ and every open subset $U$ in $X$ which contains $f^{-1}(y)$ we have that $y\in\operatorname{int}_{Y}(f(U))$ (see [17, 2.4.F]).

Lemma 3.3**.**

Let $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ be a semitopological semigroup with a compact group of units. Then the quotient natural homomorphism $\eta\colon\textbf{B}(G,\theta)^{0}\to\mathscr{C}^{0}$ is an open map.

Proof.

If $U$ is an open subset of $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ such that $U\not\ni\textsf{0}$ then $\eta(U)$ is an open subset of $\mathscr{C}^{0}$, because $\mathscr{C}(p,q)$ is a discrete open subset of the space $\mathscr{C}^{0}$ (see [8, Proposition 1]).

Suppose $U\ni\textsf{0}$ is an open subset of $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$. Put $U^{*}=\eta^{-1}\left(\eta(U)\right)$. Then $U^{*}=\eta^{-1}\left(\eta(U^{*})\right)$. Since $\eta\colon\textbf{B}(G,\theta)^{0}\to\mathscr{C}^{0}$ is a natural homomorphism,

[TABLE]

Since every [math]-bisimple inverse $\omega$-semigroup is isomorphic to a Reilly semigroup with adjoined zero, the last equality and Lemma 3.1 imply that $U^{*}$ is an open subset of the space $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$, and since $\eta$ is a quotient map and $U^{*}=\eta^{-1}\left(\eta(U^{*})\right)$, we conclude that $\eta(U)$ is an open subset of the space $\mathscr{C}^{0}$. ∎

The following simple example from the paper [22] shows that the semigroup $\mathscr{C}^{0}$ admits a topology $\tau_{\operatorname{\textsf{Ac}}}$ making it a compact semitopological semigroup.

Example 3.4** ([22]).**

On the semigroup $\mathscr{C}^{0}$ we define a topology $\tau_{\operatorname{\textsf{Ac}}}$ in the following way:

• $(i)$

every element of the bicyclic monoid ${\mathscr{C}}(p,q)$ is an isolated point in the space $(\mathscr{C}^{0},\tau_{\operatorname{\textsf{Ac}}})$;

• $(ii)$

the family $\mathscr{B}(0)=\left\{U\subseteq\mathscr{C}^{0}\colon U\ni 0\hbox{~{}and~{}}{\mathscr{C}}(p,q)\setminus U\hbox{~{}is finite}\right\}$ is a base of the topology $\tau_{\operatorname{\textsf{Ac}}}$ at zero $0\in\mathscr{C}^{0}$,

i.e., $\tau_{\operatorname{\textsf{Ac}}}$ is the topology of the Alexandroff one-point compactification of the discrete space ${\mathscr{C}}(p,q)$ with the remainder $\{0\}$. The semigroup operation in $(\mathscr{C}^{0},\tau_{\operatorname{\textsf{Ac}}})$ is separately continuous, because all elements of the bicyclic semigroup ${\mathscr{C}}(p,q)$ are isolated points in the space $(\mathscr{C}^{0},\tau_{\operatorname{\textsf{Ac}}})$ and left and right translations are finite-to-one functions in ${\mathscr{C}}(p,q)$ (see [15, Lemma I.1]).

Lemma 3.5**.**

Let $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ be a locally compact semitopological semigroup with a compact group of units. Then the quotient semigroup $\textbf{B}(G,\theta)^{0}/\eta^{\natural}$ with the quotient topology is topologically isomorphic to the semigroup $\mathscr{C}^{0}$ with either the topology $\tau_{\operatorname{\textsf{Ac}}}$ or the discrete topology.

Proof.

By Lemma 3.1 every non-zero $\mathscr{H}$-class of the semigroup $\textbf{B}(G,\theta)^{0}$ is an open-and-closed subset of $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ and hence the quotient space $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)/\eta^{\natural}$ is Hausdorff. Lemma 3.3 implies that $\eta$ is an open map. By Theorem 3.3.15 from [17], $\textbf{B}(G,\theta)^{0}/\eta^{\natural}$ with the quotient topology is a locally compact space. Then Theorem 1 of [22] implies the statement of our lemma. ∎

Lemma 3.5 implies the following proposition.

Proposition 3.6**.**

*Let $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ be a locally compact semitopological semigroup with a compact group of units and an isolated zero 0. Then the topological space $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ is homeomorphic to topological sum of its $\mathscr{H}$-classes with induced from $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ topologies, i.e.,

---

[TABLE]

Lemma 3.7**.**

Let $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ be a locally compact semitopological semigroup with compact maximal subgroups and a non-isolated zero 0. Then for every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ there are only finitely many non-zero $\mathscr{H}$-classes $G_{i,j}$, $i,j\in\mathbb{N}_{0}$, non-intersecting with $U_{\textsf{0}}$.

Proof.

Suppose to the contrary that there exists an open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ such that $G_{i,j}\cap U_{\textsf{0}}=\varnothing$ for infinitely many non-zero $\mathscr{H}$-classes $G_{i,j}$, $i,j\in\mathbb{N}_{0}$. Then by Lemma 3.3 the quotient natural homomorphism $\eta\colon\textbf{B}(G,\theta)^{0}\to\mathscr{C}^{0}$ is an open map, and hence the quotient semigroup $\textbf{B}(G,\theta)^{0}/\eta^{\natural}$ with the quotient topology is neither compact nor discrete, which contradicts Lemma 3.5. ∎

For each subset $A$ of $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ and each $i,j\in\mathbb{N}_{0}$ put $\left[A\right]_{i,j}=A\cap G_{i,j}$.

Lemma 3.8**.**

Let $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ be a locally compact semitopological semigroup with a compact group of units and non-isolated zero 0. Then for every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ and every non-negative integer $i_{0}$ the sets

[TABLE]

are finite.

Proof.

Suppose to the contrary that there exist a open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ and a non-negative integer $i_{0}$ such that the set $\left\{j\in\mathbb{N}_{0}\colon G_{i_{0},j}\nsubseteq U_{\textsf{0}}\right\}$ is infinite. Since the topological space $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ is locally compact, we can take a regular open neighbourhood $U_{\textsf{0}}$ of the zero with compact closure.

We consider the following two cases:

• $(1)$

there exists a non-negative integer $j_{0}$ such that $\left[U_{\textsf{0}}\right]_{i_{0},j}\neq G_{i_{0},j}$ for all $j\geqslant j_{0}$;

• $(2)$

for every positive integer $k$ there exists a positive integer $n>k$ such that $\left[U_{\textsf{0}}\right]_{i_{0},n}=G_{i_{0},n}$.

Suppose case $(1)$ holds. Since every maximal subgroup of $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ is compact, by Proposition 2.4$(ii)$ without loss of generality we may assume that $j_{0}=0$. By Lemma 3.1 every $\mathscr{H}$-class $G_{i,j}$ is an open subset of $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ and hence the set

[TABLE]

is compact. By Lemma 3.1 the family $\mathscr{U}_{i_{0}}=\left\{\left\{U_{\textsf{0}}\right\},\left\{G_{i_{0},j}\colon j\in\mathbb{N}_{0}\right\}\right\}$ is an open cover of the compactum $\mathfrak{H}_{i_{0}}^{0}(U_{\textsf{0}})$ and hence there exists a positive integer $j_{1}$ such that $\left[U_{\textsf{0}}\right]_{i_{0},n}=\left[\operatorname{cl}_{\textbf{B}(G,\theta)^{0}}(U_{\textsf{0}})\right]_{i_{0},n}$ for all $n\geqslant j_{1}$. Since $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ is a semitopological semigroup, a set $V_{\textsf{0}}=\rho_{(1,1_{G},0)}^{-1}(U_{\textsf{0}})$ is an open neighbourhood of zero. By Lemma 3.1 the family $\mathscr{V}_{i_{0}}=\left\{\left\{V_{\textsf{0}}\right\},\left\{G_{i_{0},j}\colon j\in\mathbb{N}_{0}\right\}\right\}$ is an open cover of the compactum $\mathfrak{H}_{i_{0}}^{0}(U_{\textsf{0}})$ and hence there exists a positive integer $j_{2}\geqslant j_{1}$ such that $\left[V_{\textsf{0}}\right]_{i_{0},n}=\left[U_{\textsf{0}}\right]_{i_{0},n}=\left[\operatorname{cl}_{\textbf{B}(G,\theta)^{0}}(U_{\textsf{0}})\right]_{i_{0},n}$, for all integers $n\geqslant j_{2}$. Indeed, since $(i_{0},g,j)\cdot(1,1_{G},0)=(i_{0},g,j-1)$ for all positive integers $j$ and any $g\in G$, for all integers $n\geqslant j_{2}$, we have that

[TABLE]

We put

[TABLE]

By Lemma 3.1, $\widetilde{U}_{\textsf{0}}$ is an open neighbourhood of zero in $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ such that $\left[\widetilde{U}_{\textsf{0}}\right]_{i_{0},n}=\left[U_{\textsf{0}}\right]_{i_{0},n}=\left[\operatorname{cl}_{\textbf{B}(G,\theta)^{0}}(U_{\textsf{0}})\right]_{i_{0},n}$, for all integers $n\geqslant j_{2}$. Since $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ is locally compact, without loss of generality we may assume that the neighbourhood $U_{\textsf{0}}$ is a regular open set. This implies that $\widetilde{U}_{\textsf{0}}$ is a regular open set too. So there exist distinct $g,h\in G$ such that $(i_{0},g,n)\notin\left[U_{\textsf{0}}\right]_{i_{0},n}$ and $(i_{0},h,n)\in\left[U_{\textsf{0}}\right]_{i_{0},n}$, for all integers $n\geqslant j_{2}$. But $(i_{0},gh^{-1},i_{0})\cdot(i_{0},h,n)=(i_{0},g,n)$ for all integers $n\geqslant j_{2}$. Let W_{\textsf{0}}=\lambda_{(i_{0},gh^{-1},i_{0})}^{-1}\big{(}\widetilde{U}_{\textsf{0}}\big{)}. Hence we have that

[TABLE]

for all integers $n\geqslant j_{2}$. Then by Lemma 3.1 the family $\mathscr{W}_{i_{0}}=\left\{\left\{W_{\textsf{0}}\right\},\left\{G_{i_{0},j}\colon j\in\mathbb{N}_{0}\right\}\right\}$ is an open cover of $\mathfrak{H}_{i_{0}}^{0}(U_{\textsf{0}})$ which has not a finite subcover. This contradicts the compactness of $\mathfrak{H}_{i_{0}}^{0}(U_{\textsf{0}})$, and hence the set $\left\{j\in\mathbb{N}_{0}\colon G_{i_{0},j}\nsubseteq U_{\textsf{0}}\right\}$ is finite.

Suppose case $(2)$ holds. Then there are infinitely many non-negative integers $j$ such that $\left[U_{\textsf{0}}\right]_{i_{0},j}=G_{i_{0},j}$ but $\left[U_{\textsf{0}}\right]_{i_{0},j-1}\neq G_{i_{0},j-1}$. Since every maximal subgroup of $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ is compact, Lemma 3.1 implies that every $\mathscr{H}$-class $G_{i,j}$ is an open subset of $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ and hence the set

[TABLE]

is compact. Let $V_{\textsf{0}}=\rho_{(1,1_{G},0)}^{-1}(U_{\textsf{0}})$. By Lemma 3.1 the family $\mathscr{V}_{i_{0}}=\left\{\left\{V_{\textsf{0}}\right\},\left\{G_{i_{0},j}\colon j\in\mathbb{N}_{0}\right\}\right\}$ is an open cover of the compactum $\mathfrak{H}_{i_{0}}^{0}(U_{\textsf{0}})$. Then the continuity of the right shift $\rho_{(1,1_{G},0)}$ and the equality $(i_{0},g,j)\cdot(1,1_{G},0)=(i_{0},g,j-1)$ imply that $\left[V_{\textsf{0}}\right]_{i_{0},j}\neq G_{i_{0},j}$ for infinitely many non-negative integers $j$. Also, the equality $(i_{0},g,j)\cdot(1,1_{G},0)=(i_{0},g,j-1)$ and assumption of case $(2)$ imply that $\left[U_{\textsf{0}}\right]_{i_{0},j}\setminus\left[V_{\textsf{0}}\right]_{i_{0},j}\neq\varnothing$ for infinitely many non-negative integers $j$. Hence, the open cover $\mathscr{V}_{i_{0}}$ of $\mathfrak{H}_{i_{0}}^{0}(U_{\textsf{0}})$ does not contain a finite subcover, which contradicts the compactness of $\mathfrak{H}_{i_{0}}^{0}(U_{\textsf{0}})$, and hence the set $\left\{j\in\mathbb{N}_{0}\colon G_{i_{0},j}\nsubseteq U_{\textsf{0}}\right\}$ is finite.

The proof of the statement that the set $\left\{j\in\mathbb{N}_{0}\colon G_{j,i_{0}}\nsubseteq U_{\textsf{0}}\right\}$ is finite, is similar. ∎

Lemma 3.9**.**

Let $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ be a locally compact semitopological semigroup with a compact group of units and non-isolated zero 0. Then for every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ the set $A_{U_{\textsf{0}}}=\left\{(i,j)\in\mathbb{N}_{0}\times\mathbb{N}_{0}\colon G_{i,j}\nsubseteq U_{\textsf{0}}\right\}$ is finite.

Proof.

Suppose to the contrary that there exists an open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ such that the set $A_{U_{\textsf{0}}}$ is infinite. Since the topological space $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ is locally compact without loss of generality we may assume that the closure $\operatorname{cl}_{\textbf{B}(G,\theta)^{0}}(U_{\textsf{0}})$ of $U_{\textsf{0}}$ is compact and the neighbourhood $U_{\textsf{0}}$ is regular open. By Lemma 3.8, for every positive integer $k$ there exists $(i,j)\in A_{U(0)}$ such that $i>k$ and $j>k$.

By induction we define an infinite sequence $\left\{\left(i_{n},j_{n}\right)\right\}_{n\in\mathbb{N}}$ of elements of the set $A_{U_{\textsf{0}}}$ in the following way. By the assumption, there exists the smallest non-negative integer $i_{1}$ such that $G_{i_{1},j}\nsubseteq U_{\textsf{0}}$, $j\in\mathbb{N}_{0}$. By Lemma 3.8 there exits $j_{1}=\max\left\{j\in\mathbb{N}_{0}\colon G_{i_{1},j}\nsubseteq U_{\textsf{0}}\right\}$.

At $k+1$-th step of induction we define pair $\left(i_{k+1},j_{k+1}\right)\in A_{U_{\textsf{0}}}$ as follows. Let $i_{k+1}$ be the smallest non-negative integer $>i_{k}$ such that $G_{i_{k+1},j}\nsubseteq U_{\textsf{0}}$, $j\in\mathbb{N}_{0}$. By Lemma 3.8 there exits $j_{k+1}=\max\left\{j\in\mathbb{N}_{0}\colon G_{i_{k+1},j}\nsubseteq U_{\textsf{0}}\right\}$. Our assumption and Lemma 3.8 imply that so ordered pair $(i_{k+1},j_{k+1})$ there exists in $A_{U_{\textsf{0}}}$.

Now, by the separate continuity of the semigroup operation in $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(G,\theta)^{0},\tau_{\textbf{B}}^{0}\right)$ such that $V_{\textsf{0}}\cdot(1,1_{G},0)\subseteq U_{\textsf{0}}$. Then the construction of the sequence $\left\{\left(i_{n},j_{n}\right)\right\}_{n\in\mathbb{N}}$ implies that

[TABLE]

for each $\left(i_{n},j_{n}\right)$. By Lemma 3.1 the family $\mathscr{V}=\left\{\left\{V_{\textsf{0}}\right\},\left\{G_{i,j}\colon i,j\in\mathbb{N}_{0}\right\}\right\}$ is an open cover of the compact set $\operatorname{cl}_{\textbf{B}(G,\theta)^{0}}(U_{\textsf{0}})$. Then the continuity of the right shift $\rho_{(1,1_{G},0)}$ implies that $\left[V_{\textsf{0}}\right]_{i_{n},j_{n}+1}\neq G_{i_{n},j_{n}+1}$ for infinitely many pairs $\left(i_{n},j_{n}+1\right)$. This implies that $\left[U_{\textsf{0}}\right]_{i_{n},j_{n}+1}\setminus\left[V_{\textsf{0}}\right]_{i_{n},j_{n}+1}\neq\varnothing$ for infinitely many pairs $\left(i_{n},j_{n}+1\right)$. Then above arguments imply that the cover $\mathscr{V}$ has not a finite subcover, which contradicts the compactness of $\operatorname{cl}_{\textbf{B}(G,\theta)^{0}}(U_{\textsf{0}})$. ∎

Example 3.10**.**

Let $(G,\tau_{G})$ be a semitopological group, $\theta\colon G\to G$ be a continuous homomorphism and $\mathscr{B}_{G}(g)$ be a base of the topology $\tau_{G}$ at a point $g\in G$.

On the semigroup $\textbf{B}(G,\theta)^{0}$ we define a topology $\tau^{\oplus}_{\textbf{B}}$ in the following way:

• $(i)$

at any non-zero element $(i,g,j)\in G_{i,j}$ of the semigroup $\textbf{B}(G,\theta)^{0}$ the family

[TABLE]

is a base of the topology $\tau^{\oplus}_{\textbf{B}}$ at the point $(i,g,j)\in\textbf{B}(G,\theta)^{0}$;

• $(ii)$

zero $\textsf{0}\in\textbf{B}(G,\theta)^{0}$ is an isolated point in $\left(\textbf{B}(G,\theta)^{0},\tau^{\oplus}_{\textbf{B}}\right)$.

Simple verifications show that the semigroup operation in $\left(\textbf{B}(G,\theta)^{0},\tau^{\oplus}_{\textbf{B}}\right)$ is separately continuous.

In Example 3.11, we extend the construction proposed in Example 3.4 onto compact semitopological [math]-bisimple inverse $\omega$-semigroup with a compact maximal subgroup.

Example 3.11**.**

Let $(G,\tau_{G})$ be a compact Hausdorff topological group, $\theta\colon G\to G$ be a continuous homomorphism and $\mathscr{B}_{G}(g)$ be a base of the topology $\tau_{G}$ at a point $g\in G$.

On the semigroup $\textbf{B}(G,\theta)^{0}$ we define a topology $\tau^{\operatorname{\textsf{Ac}}}_{\textbf{B}}$ by pointing it base $\mathscr{B}^{\operatorname{\textsf{Ac}}}_{\textbf{B}}(x)$ at each point $x$ of $\textbf{B}(G,\theta)^{0}$, namely

• $(i)$

at any non-zero element $(i,g,j)\in G_{i,j}$ of the semigroup $\textbf{B}(G,\theta)^{0}$

[TABLE]

• $(ii)$

$\mathscr{B}(\textsf{0})=\left\{U_{(i_{1},j_{1}),\ldots,(i_{k},j_{k})}\colon(i_{1},j_{1}),\ldots,(i_{k},j_{k})\in\mathbb{N}_{0}\times\mathbb{N}_{0}\right\}$, where

[TABLE]

I.e., $\tau^{\operatorname{\textsf{Ac}}}_{\textbf{B}}$ is the topology of the Alexandroff one-point compactification of the locally compact space $\bigoplus\left\{G_{i,j}\colon i,j\in\mathbb{N}_{0}\right\}$ (where for any $i,j\in\mathbb{N}_{0}$ the space $G_{i,j}$ is homeomorphic to the compact group $(G,\tau_{G})$ by the map $(i,g,j)\mapsto g$), with the remainder $\{\textsf{0}\}$. Simple routine verifications show that the semigroup operation in $\left(\textbf{B}(G,\theta)^{0},\tau^{\operatorname{\textsf{Ac}}}_{\textbf{B}}\right)$ is separately continuous.

Proposition 3.6 and Lemma 3.8 imply the following dichotomy for locally compact semitopological [math]-bisimple inverse $\omega$-semigroups with a compact maximal subgroup:

Theorem 3.12**.**

Let $S$ be a locally compact semitopological [math]-bisimple inverse $\omega$-semigroup with a compact maximal subgroups $G$ distinct from zero. Then $S$ is topologically isomorphic either to $\left(\textbf{B}(G,\theta)^{0},\tau^{\oplus}_{\textbf{B}}\right)$ or to $\left(\textbf{B}(G,\theta)^{0},\tau^{\operatorname{\textsf{Ac}}}_{\textbf{B}}\right)$, for some continuous homomorphism $\theta\colon G\to G$.

Since the bicyclic monoid $\mathscr{C}(p,q)$ does not embed into any compact topological semigroup [2], Theorem 3.12 implies the following corollary.

Corollary 3.13**.**

If $S$ be a locally compact topological [math]-bisimple inverse $\omega$-semigroup with a compact maximal subgroup $G$ distinct from zero, then $S$ is topologically isomorphic to $\left(\textbf{B}(G,\theta)^{0},\tau^{\oplus}_{\textbf{B}}\right)$, for some continuous homomorphism $\theta\colon G\to G$.

Later we shall need the following trivial lemma, which follows from the separate continuity of the semigroup operation in semitopological semigroups.

Lemma 3.14**.**

Let $S$ be a semitopological semigroup and $I$ be a compact ideal in $S$. Then the Rees-quotient semigroup $S/I$ with the quotient topology is a semitopological semigroup.

Theorem 3.15**.**

Let $(\textbf{B}(G,\theta)^{I},\tau)$ be a locally compact semitopological bisimple inverse $\omega$-semigroup with a compact group of units $G$ and an adjoined compact ideal $I$, i.e., $\textbf{B}(G,\theta)^{I}=\textbf{B}(G,\theta)\sqcup I$. Then either $(\textbf{B}(G,\theta)^{I},\tau)$ is a compact semitopological semigroup or the ideal $I$ is open.

Proof.

Suppose that the ideal $I$ is not open. By Lemma 3.1 the Rees-quotient semigroup $\textbf{B}(G,\theta)^{I}/I$ with the quotient topology $\tau_{\operatorname{\textsf{q}}}$ is a semitopological semigroup. Let $\pi\colon\textbf{B}(G,\theta)^{I}\to\textbf{B}(G,\theta)^{I}/I$ be the natural homomorphism which is a quotient map. It is obvious that the Rees-quotient semigroup $\textbf{B}(G,\theta)^{I}/I$ is isomorphic to the Reilly semigroup $\textbf{B}(G,\theta)^{0}$ and the image $\pi(I)$ is zero 0 of the semigroup $\textbf{B}(G,\theta)^{0}$.

Now we shall show that the natural homomorphism $\pi\colon\textbf{B}(G,\theta)^{I}\to\textbf{B}(G,\theta)^{I}/I$ is a hereditarily quotient map.

We shall show that for every open neighbourhood $U(I)$ of the ideal $I$ in the space $(\textbf{B}(G,\theta)^{I},\tau)$ the image $\pi(U(I))$ is an open neighbourhood of the zero 0 in the space $\left(\textbf{B}(G,\theta)^{I}/I,\tau_{\operatorname{\textsf{q}}}\right)$. Indeed, $\textbf{B}(G,\theta)^{I}\setminus U(I)$ is a closed subset of $(\textbf{B}(G,\theta)^{I},\tau)$. Also, since the restriction $\pi|_{\textbf{B}(G,\theta)}\colon\textbf{B}(G,\theta)\to\pi(\textbf{B}(G,\theta))$ of the natural homomorphism $\pi\colon\textbf{B}(G,\theta)^{I}\to\textbf{B}(G,\theta)^{I}/I$ is one-to-one, $\pi(\textbf{B}(G,\theta)^{I}\setminus U(I))$ is a closed subset of $\left(\textbf{B}(G,\theta)^{I}/I,\tau_{\operatorname{\textsf{q}}}\right)$. So $\pi(U(I))$ is an open neighbourhood of the zero 0 of the semigroup $\left(\textbf{B}(G,\theta)^{I}/I,\tau_{\operatorname{\textsf{q}}}\right)$, and hence the natural homomorphism $\pi\colon\textbf{B}(G,\theta)^{I}\to\textbf{B}(G,\theta)^{I}/I$ is a hereditarily quotient map.

Since $I$ is a compact ideal of the semitopological semigroup $\left(\textbf{B}(G,\theta)^{I},\tau\right)$, $\pi^{-1}(y)$ is a compact subset of $\left(\textbf{B}(G,\theta)^{I},\tau\right)$ for every $y\in\textbf{B}(G,\theta)^{I}/I$. By Din’ N’e T’ong’s Theorem (see [14] or [17, 3.7.E]), $\left(\textbf{B}(G,\theta)^{I}/I,\tau_{\operatorname{\textsf{q}}}\right)$ is a Hausdorff locally compact space. Since $I$ is not open by Theorem 3.12 the semitopological semigroup $\left(\textbf{B}(G,\theta)^{I}/I,\tau_{\operatorname{\textsf{q}}}\right)$ is topologically isomorphic to $\left(\textbf{B}(G,\theta)^{0},\tau^{\operatorname{\textsf{Ac}}}_{\textbf{B}}\right)$ and hence it is compact.

At last we shall prove that the space $\left(\textbf{B}(G,\theta)^{I},\tau\right)$ is compact. Let $\mathscr{U}=\left\{U_{\alpha}\colon\alpha\in\mathscr{I}\right\}$ be an arbitrary open cover of $\left(\textbf{B}(G,\theta)^{I},\tau\right)$. Since $I$ is compact, it can be covered by a some finite subfamily $\mathscr{U}^{\prime}=\left\{U_{\alpha_{1}},\ldots,U_{\alpha_{n}}\right\}$ of $\mathscr{U}$. Put $U=U_{\alpha_{1}}\cup\cdots\cup U_{\alpha_{n}}$. Then $\textbf{B}(G,\theta)^{I}\setminus U$ is a closed subset of $\left(\textbf{B}(G,\theta)^{I},\tau\right)$. Also, since the restriction $\pi|_{\textbf{B}(G,\theta)}\colon\textbf{B}(G,\theta)\to\pi(\textbf{B}(G,\theta))$ of the natural homomorphism $\pi\colon\textbf{B}(G,\theta)^{I}\to\textbf{B}(G,\theta)^{I}/I$ is one-to-one, $\pi(\textbf{B}(G,\theta)^{I}\setminus U)$ is a closed subset of $(\textbf{B}(G,\theta)^{I}/I,\tau_{\operatorname{\textsf{q}}})$, and hence the image $\pi(\textbf{B}(G,\theta)^{I}\setminus U)$ is compact, because the semigroup $\left(\textbf{B}(G,\theta)^{I}/I,\tau_{\operatorname{\textsf{q}}}\right)$ is compact. Thus, the set $\textbf{B}(G,\theta)^{I}\setminus U$ is compact, so there exists a finite subfamily $\mathscr{U}^{\prime\prime}$ of $\mathscr{U}$, covering $\textbf{B}(G,\theta)^{I}\setminus U$. Then $\mathscr{U}^{\prime}\cup\mathscr{U}^{\prime\prime}$ is a finite cover of $\left(\textbf{B}(G,\theta)^{I},\tau\right)$. Hence the space $\left(\textbf{B}(G,\theta)^{I},\tau\right)$ is compact as well. ∎

Since every Bruck–Reilly extension contains the bicyclic monoid ${\mathscr{C}}(p,q)$ and compact topological semigroup does not contain the semigroup ${\mathscr{C}}(p,q)$, Theorem 3.15 implies the following corollary.

Corollary 3.16**.**

Let $(\textbf{B}(G,\theta)^{I},\tau)$ be a locally compact topological bisimple inverse $\omega$-semigroup with a compact group of units $G$ and an adjoined compact ideal $I$, i.e., $\textbf{B}(G,\theta)^{I}=\textbf{B}(G,\theta)\sqcup I$. Then the ideal $I$ is open in $(\textbf{B}(G,\theta)^{I},\tau)$.

4. On locally compact semitopological [math]-bisimple inverse $\omega$-semigroups which contain the additive group of integers as a maximal subgroup

The structure of a locally compact topological group with adjoined non-isolated zero was described by Hofmann in [32]:

Theorem 4.1** ([32, Theorem I]).**

Let $S$ be a locally compact group with non-isolated zero and $G$ its maximal subgroup $S\setminus\{0\}$. Then

• $(i)$

$G$* contains a unique characteristic maximal compact subgroup $C$.*

• $(ii)$

If [math] and $1$ lie in some connected subspace, then $S$ contains in its center a locally compact group with zero $M_{0}=M\cup\{0\}$ which is isomorphic to the multiplicative semigroup of all non-negative real numbers and $S$ is isomorphic to the quotient semigroup of the direct product $M_{0}\times C$ modulo the congruence relation identifying all points of $\{0\}\times C$.

If [math] and $1$ do not lie in any connected subspace of $S$, then $S$ contains a locally compact group with zero $M_{0}=M\setminus\{0\}$ which is isomorphic to the set of real numbers $\{0\}\cup\left\{2^{n}\colon n{=}0,\pm 1,\pm 2,\ldots\right\}$ under multiplication and S is isomorphic to the union of [math] and the semidirect product $M\rtimes_{\alpha}C$, where $\alpha$ is the inner automorphism $c\longmapsto g^{-1}cg$ with the generator $g$ of $M$ whose powers converge to [math].

• $(iii)$

If $C$ is any compact group and a any automorphism of it, then there exists a locally compact group with zero whose maximal compact group is isomorphic to $C$ and whose maximal group can be made isomorphic to $M\times C$ where $M$ are positive reals under multiplication or to $M\rtimes_{\alpha}C$ where in this case $M$ is infinite cyclic.

When $G^{0}$ is a semitopological semigroup which is a locally compact group with adjoined non-isolated zero then its structure is more complicated than Hofmann’s results. Thus in this section we consider a simple case of the additive group of integers $\mathbb{Z}_{+}$.

We shall denote by $\mathbb{Z}_{+}^{0}$ the additive group of integers $\mathbb{Z}_{+}$ with adjoined zero [math] which is a semitopological semigroup and by $0_{\mathbb{Z}_{+}}$ the identity of $\mathbb{Z}_{+}$.

Remark 4.2**.**

If the space $\mathbb{Z}_{+}^{0}$ is locally compact then by Theorem 3.3.8 from [17] the subspace $\mathbb{Z}_{+}$ is locally compact. Then by the Baire Category Theorem (see [17, Theorem 3.9.3]) the space $\mathbb{Z}_{+}$ has an isolated point, and since translations in $\mathbb{Z}_{+}$ are homeomorphisms, the subspace $\mathbb{Z}_{+}$ is discrete.

We need two technical lemmas.

Lemma 4.3**.**

Let $\mathbb{N}_{+}^{0}$ be a locally compact semitopological additive semigroup of positive integers with adjoined zero [math]. Then $\mathbb{N}$ is a discrete subspace of $\mathbb{N}_{+}^{0}$ and the space $\mathbb{N}_{+}^{0}$ is either compact or discrete.

Proof.

Fix an arbitrary $n\in\mathbb{N}$. Since $\mathbb{N}_{+}^{0}$ is locally compact Theorem 3.3.8 of [17] implies $\mathbb{N}$ is locally compact too. Now Hausdorffness of the space $\mathbb{N}$ implies that the set $[n+1)=\mathbb{N}\setminus\{1,\ldots,n\}$ is open in $\mathbb{N}_{+}^{0}$, and hence by Theorem 3.3.8 from [17] it is locally compact. By the Baire Category Theorem the space $[n+1)$ contains an isolated point $m$. It is obvious that $m$ is isolated in $\mathbb{N}$. Since $n<m$ the separate continuity of the semigroup operation in $\mathbb{N}_{+}^{0}$ implies that $n$ is an isolated point of $\mathbb{N}$, and hence by Hausdorffness of $\mathbb{N}_{+}^{0}$ the point $n$ is isolated in $\mathbb{N}_{+}^{0}$.

Suppose that the space $\mathbb{N}_{+}^{0}$ is not compact and its point [math] is not isolated. Fix an arbitrary open neighbourhood $U(0)$ of [math] in $\mathbb{N}_{+}^{0}$ with the compact closure $\operatorname{cl}_{\mathbb{N}_{+}^{0}}(U(0))$. Since $\mathbb{N}$ is a discrete subspace of $\mathbb{N}_{+}^{0}$, $\operatorname{cl}_{\mathbb{N}_{+}^{0}}(U(0))=U(0)$. Since the space $\mathbb{N}_{+}^{0}$ is not compact and its subspace $\mathbb{N}$ is discrete, $U(0)=\mathbb{N}_{+}^{0}\setminus\{n_{i}\colon i\in\mathbb{N}\}$, where $\left\{n_{i}\colon i\in\mathbb{N}\right\}$ is an infinite increasing sequence of positive integers. The separate continuity of the semigroup operation of $\mathbb{N}_{+}^{0}$ implies that there exists an open neighbourhood $V(0)\subseteq U(0)$ of the zero [math] in $\mathbb{N}_{+}^{0}$ such that $1+V(0)\subseteq U(0)$.

Now, by the existence of the infinite sequence $\left\{n_{i}\colon i\in\mathbb{N}\right\}$ and since the set $U(0)\setminus V(0)$ is infinite, we have that the family $\mathscr{V}=\left\{V(0),\left\{\{i\}\colon i\in\mathbb{N}\right\}\right\}$ is an open cover of $U(0)$ which does not contain a finite subcover. This contradicts the compactness of $U(0)$. The obtained contradiction implies the second statement of the lemma. ∎

Lemma 4.4**.**

Let $S$ be a semitopological semigroup and $M$ be a dense subsemigroup of $S$. If $1_{M}$ and $0_{M}$ are identity and zero in $M$, respectively, then $1_{M}$ and $0_{M}$ so are in $S$.

Proof.

Let $\operatorname{id}$ be the identity map on $S$ and $\bar{O}$ be a constant map $S\to\{0_{M}\}$. Clearly, these maps are continuous. Since the shifts (on $S$) by elements $1_{M}$ and $0_{M}$ are continuous too, and $1_{M}x=x1_{M}=x=\operatorname{id}(x)$ and $0_{M}x=x0_{M}=0_{M}=\bar{0}(x)$ for each element $x$ of a dense subset $M$ of a Hausdorff space $S$, by [17, Theorem 1.5.4] these equalities hold for each $x\in S$. ∎

Since the subsemigroup $\mathbb{N}_{-}^{0}=\left\{0\right\}\cup\left\{-n\colon n\in\mathbb{N}\right\}$ of $\mathbb{Z}_{+}^{0}$ is isomorphic to $\mathbb{N}_{+}^{0}$, Lemmas 4.3 and 4.4 and Remark 4.2 imply the following proposition.

Proposition 4.5**.**

Let $\mathbb{Z}_{+}^{0}$ be a locally compact semitopological semigroup. Then all non-zero points of $\mathbb{Z}_{+}^{0}$ are isolated and exactly one of the following conditions holds:

• $(i)$

zero [math] is isolated in $\mathbb{Z}_{+}^{0}$;

• $(ii)$

the family $\mathscr{B}_{\mathbf{cf}}=\left\{U_{F}=\mathbb{Z}_{+}^{0}\setminus F\colon F\hbox{~{}is a finite subset of~{}}\mathbb{Z}\right\}$ is a base at zero;

• $(iii)$

the family $\mathscr{B}^{+}=\left\{U_{F}^{+}=\mathbb{N}_{+}^{0}\setminus F\colon F\hbox{~{}is a finite subset of~{}}\mathbb{N}\right\}$ is a base at zero;

• $(iv)$

the family $\mathscr{B}^{-}=\left\{U_{F}^{-}=\mathbb{Z}_{+}^{0}\setminus\left(\mathbb{N}\cup F\right)\colon F\hbox{~{}is a finite subset of~{}}\mathbb{Z}\right\}$ is a base at zero.

Next we define four topologies $\tau_{\mathfrak{d}}$, $\tau_{\mathbf{cf}}$, $\tau_{+}$ and $\tau_{-}$ on the Reilly semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)$, where $\theta\colon\mathbb{Z}_{+}\to\mathbb{Z}_{+}\colon z\mapsto 0_{\mathbb{Z}_{+}}$ is the annihilating homomorphism.

Example 4.6**.**

Let $\tau_{\mathfrak{d}}$ be the discrete topology on $\textbf{B}(\mathbb{Z}_{+},\theta)$. It is obvious that $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{\mathfrak{d}}\right)$ is a topological inverse semigroup.

Example 4.7**.**

We define a topology $\tau_{\mathbf{cf}}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ in the following way. Let $(i,g,j)$ be an isolated point of $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{\mathbf{cf}}\right)$ in the following cases:

• $(1)$

$g\neq 0_{\mathbb{Z}_{+}}$ and $i,j\in\mathbb{N}_{0}$;

• $(2)$

$i=0$ or $j=0$.

The family

[TABLE]

defines the base of the topology $\tau_{\mathbf{cf}}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ at the point $(i,0_{\mathbb{Z}_{+}},j)$, for all $i,j\in\mathbb{N}$. Simple verifications show that $\tau_{\mathbf{cf}}$ is a Hausdorff locally compact topology on $\textbf{B}(\mathbb{Z}_{+},\theta)$.

Next we shall prove that $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{\mathbf{cf}}\right)$ is a semitopological inverse semigroup with continuous inversion. The definition of the topology $\tau_{\mathbf{cf}}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ implies that it is sufficient to show that the semigroup operation on $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{\mathbf{cf}}\right)$ is separately continuous in the following two cases:

[TABLE]

where $i,j\in\mathbb{N}_{0}$, $m,n\in\mathbb{N}$, $F$ is an arbitrary finite subset of $\mathbb{Z}$ and $g$ is an arbitrary element of the group $\mathbb{Z}_{+}$, because in cases $(1)$ and $(2)$ the point $(i,g,j)$ is isolated in $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{\mathbf{cf}}\right)$.

Simple calculations show that

[TABLE]

[TABLE]

and

[TABLE]

where $g+F=F+g=\left\{f+g\colon f\in F\right\}$ and $-F=\left\{-f\colon f\in F\right\}$.

Example 4.8**.**

We define a topology $\tau_{+}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ in the following way. Let $(i,g,j)$ be an isolated point of $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{+}\right)$ in the following cases:

• $(1)$

$g\neq 0_{\mathbb{Z}_{+}}$ and $i,j\in\mathbb{N}_{0}$;

• $(2)$

$i=0$ or $j=0$.

The family

[TABLE]

defines the base of the topology $\tau_{+}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ at the point $(i,0_{\mathbb{Z}_{+}},j)$, for all $i,j\in\mathbb{N}$. Simple verifications show that $\tau_{+}$ is a Hausdorff locally compact topology on $\textbf{B}(\mathbb{Z}_{+},\theta)$.

Next we shall prove that $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{+}\right)$ is a topological semigroup. Now, the definition of the topology $\tau_{+}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ implies that it is sufficient to show that the semigroup operation on $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{+}\right)$ is continuous in the following three cases:

[TABLE]

where $i,j,k.l\in\mathbb{N}_{0}$, $m,n\in\mathbb{N}$, $F$ and $F^{\prime}$ are arbitrary finite subsets of $\mathbb{Z}$ and $g$ is an arbitrary element of the group $\mathbb{Z}_{+}$, because in cases $(1)$ and $(2)$ the point $(i,g,j)$ is isolated in $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{+}\right)$.

Without loss of generality we may assume that in formula (2) the set $F$ is an initial interval of the set of positive integers, because for every finite subset $F$ of $\mathbb{N}$ there exists an initial interval $\{1,\ldots,m\}$ of $\mathbb{N}$ such that $F\subseteq\{1,\ldots,m\}$.

Simple calculations show that

[TABLE]

[TABLE]

and

[TABLE]

where ${F-g}=\left\{f-g\colon f\in F\right\}\cap\mathbb{N}$ and ${F+F}=\left\{f+g\colon f,g\in F\right\}$.

Example 4.9**.**

We define a topology $\tau_{-}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ in the following way. Let $(i,g,j)$ be an isolated point of $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{-}\right)$ in the following cases:

• $(1)$

$g\neq 0_{\mathbb{Z}_{+}}$ and $i,j\in\mathbb{N}_{0}$;

• $(2)$

$i=0$ or $j=0$.

The family

[TABLE]

where $-F=\{-m\colon m\in F\}$, defines the base of the topology $\tau_{-}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ at the point $(i,0_{\mathbb{Z}_{+}},j)$, for all $i,j\in\mathbb{N}$. Simple verifications show that $\tau_{-}$ is a Hausdorff locally compact topology on $\textbf{B}(\mathbb{Z}_{+},\theta)$.

We observe that the map $\Phi\colon\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{+}\right)\to\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{-}\right)$ defined by the formula $\Phi(i,g,j)=(i,-g,j)$ is an isomorphism of the semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)$ which is a homeomorphism of topological spaces $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{+}\right)$ and $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{-}\right)$, and hence $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{-}\right)$ is a topological semigroup which is topologically isomorphic to the topological semigroup $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{+}\right)$.

The following theorem describes all shift-continuous locally compact topologies on the Reilly semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)$ in the case when $\theta\colon\mathbb{Z}_{+}\to\mathbb{Z}_{+}$, $z\mapsto 0_{\mathbb{Z}_{+}}$ is the annihilating homomorphism.

Theorem 4.10**.**

Let $\theta\colon\mathbb{Z}_{+}\to\mathbb{Z}_{+}$, $z\mapsto 0_{\mathbb{Z}_{+}}$ be the annihilating homomorphism and $\tau$ be a shift-continuous locally compact topology on $\textbf{B}(\mathbb{Z}_{+},\theta)$. Then only one of the following conditions holds:

• $(i)$

$\tau$* is discrete;*

• $(ii)$

$\tau=\tau_{\mathbf{cf}}$;

• $(iii)$

$\tau=\tau_{+}$;

• $(iv)$

$\tau=\tau_{-}$.

Proof.

Since

[TABLE]

for arbitrary $i,j\in\mathbb{N}_{0}$ and the right shift $\rho_{(1,0_{\mathbb{Z}_{+}},1)}\colon\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)\to\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$, $x\mapsto x\cdot(1,0_{\mathbb{Z}_{+}},1)$ is continuous, the subspace $\left(\mathbb{Z}_{+}\right)_{0,0}^{0}=\rho_{(1,0_{\mathbb{Z}_{+}},1)}^{-1}\left((1,0_{\mathbb{Z}_{+}},1)\right)$ of $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$ is closed, and hence the local compactness of $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$ and Theorem 3.3.8 from [17] imply that $\left(\mathbb{Z}_{+}\right)_{0,0}^{0}$ is locally compact. It is obvious that $\left(\mathbb{Z}_{+}\right)_{0,0}^{0}$ with the induced semigroup operation from $\textbf{B}(\mathbb{Z}_{+},\theta)$ is isomorphic to $\mathbb{Z}_{+}^{0}$, and hence the statement of Proposition 4.5 for the locally compact subsemigroup $\left(\mathbb{Z}_{+}\right)_{0,0}^{0}$ holds. Proposition 2.4$(iv)$ completes the proof of our theorem. ∎

The following corollary follows from Theorem 4.10 and it describes all semigroup Hausdorff locally compact topologies on the Reilly semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)$ in the case when $\theta$ is the annihilating homomorphism.

Corollary 4.11**.**

For a locally compact topological semigroup $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$ with the annihilating homomorphism $\theta$ exactly one of the following conditions holds:

• $(i)$

$\tau$* is discrete;*

• $(ii)$

$\tau=\tau_{+}$;

• $(iii)$

$\tau=\tau_{-}$.

Proposition 4.12**.**

If $\theta$ is the annihilating homomorphism and $\tau$ is a shift-continuous locally compact topology on $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ such that the induced topology of $\tau$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ is discrete, then $\tau$ is discrete.

Proof.

Suppose to the contrary that the zero 0 is not an isolated point of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$.

Since all non-zero elements of the semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ are isolated points in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$, the binary relation $\eta^{\natural}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ defined by formula (1) is a closed congruence on $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. Next we shall show that if the natural homomorphism $\eta\colon\textbf{BR}(\mathbb{Z}_{+},\theta)^{0}\to\mathscr{C}^{0}$ is a quotient map then $\eta$ is open. Since all non-zero elements in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ are isolated points it suffices to prove that the image $\eta(U_{\textbf{0}})$ is open for every open neighbourhood $U_{\textbf{0}}$ of zero. Put $U_{\textbf{0}}^{*}=\eta^{-1}\left(\eta(U_{\textbf{0}})\right)$. Clearly, $U^{*}_{\textbf{0}}=\eta^{-1}\left(\eta(U^{*}_{\textbf{0}})\right)$. Since $\eta\colon\textbf{BR}(\mathbb{Z}_{+},\theta)^{0}\to\mathscr{C}^{0}$ is a natural homomorphism,

[TABLE]

The last equality and Lemma 3.1 implies that $U^{*}_{0}$ is an open subset of the space $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. Since $\eta$ is a quotient map and $U^{*}_{0}=\eta^{-1}\left(\eta(U^{*}_{0})\right)$, $\eta(U_{0})$ is an open subset of the space $\mathscr{C}^{0}$. This implies that the quotient semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}/\eta^{\natural}$ with the quotient topology is a Hausdorff semitopological semigroup.

By Theorem 3.3.15 from [17], $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}/\eta^{\natural}$ with the quotient topology is a locally compact space with non-isolated zero. Then by Theorem 1 from [22] the quotient semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}/\eta^{\natural}$ with the quotient topology is topologically isomorphic to the compact semitopological semigroup $(\mathscr{C}^{0},\tau_{\operatorname{\textsf{Ac}}})$ (see Example 3.4). Now, since the natural homomorphism $\eta\colon\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)\to(\mathscr{C}^{0},\tau_{\operatorname{\textsf{Ac}}})$ is an open map, for every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$, there exist finitely many subsets $(\mathbb{Z}_{+})_{i,j}$ in $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ such that $(\mathbb{Z}_{+})_{i,j}\cap U_{\textsf{0}}=\varnothing$.

Next we shall show that for any open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exist finitely many subsets $(\mathbb{Z}_{+})_{0,j}$ in $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$, $j\in\mathbb{N}_{0}$, such that $(\mathbb{Z}_{+})_{0,j}\nsubseteq U_{\textsf{0}}$. Suppose to the contrary that there exist an open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and infinitely many subsets $(\mathbb{Z}_{+})_{0,j}$ in $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$, $j\in\mathbb{N}_{0}$, such that $(\mathbb{Z}_{+})_{0,j}\nsubseteq U_{\textsf{0}}$. Since $\textbf{B}(\mathbb{Z}_{+},\theta)$ is a discrete subspace of the locally compact space $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ without loss of generality we may assume that the neighbourhood $U_{\textsf{0}}$ is compact. Then the separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ such that $(0,1,0)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. Also, the assumption that there exist infinitely many subsets $(\mathbb{Z}_{+})_{0,j}$ in $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$, $j\in\mathbb{N}_{0}$, such that $(\mathbb{Z}_{+})_{0,j}\nsubseteq U_{\textsf{0}}$ implies that the following open cover $\mathscr{V}=\left\{\left\{V_{\textsf{0}}\right\},\left\{(\mathbb{Z}_{+})_{i,j}\colon i,j\in\mathbb{N}_{0}\right\}\right\}$ of the set $U_{\textsf{0}}$ does not contain a finite subcover, because our assumption implies that there does not exist finitely many subsets $(\mathbb{Z}_{+})_{0,j}$ in $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$, $j\in\mathbb{N}_{0}$, which cover the set $\left\{(\mathbb{Z}_{+})_{0,j}\colon j\in\mathbb{N}_{0}\right\}\cap\left((0,1,0)\cdot V_{\textsf{0}}\right)\setminus V_{\textsf{0}}$. This contradicts the compactness of $U_{\textsf{0}}$.

Fix an arbitrary compact open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and a set $(\mathbb{Z}_{+})_{0,j_{0}}\subseteq U_{\textsf{0}}$. Then the following formula

[TABLE]

$i,j\in\mathbb{N}_{0}$, implies that the set $(\mathbb{Z}_{+})_{0,j_{0}}^{0}$ is open-and-closed in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ because $\textbf{B}(\mathbb{Z}_{+},\theta)$ is a discrete subspace of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and $\tau$ is shift-continuous. Then the neighbourhood $U_{\textsf{0}}$ contains the open-and-closed discrete subspace $(\mathbb{Z}_{+})_{0,j_{0}}$ which contradicts the compactness of $U_{\textsf{0}}$. ∎

For all non-negative integers $k$ and $l$ we put $(\widetilde{\mathbb{Z}}_{+})_{k,l}^{0}=(\mathbb{Z}_{+})_{k,l}^{0}\setminus\left\{(k,0_{\mathbb{Z}_{+}},l)\right\}$, $\widetilde{\mathfrak{Z}}=\left\{(\widetilde{\mathbb{Z}}_{+})_{i,j}^{0}\colon i,j\in\mathbb{N}_{0}\right\}$ and $\widetilde{\mathfrak{O}}=\left\{\left\{(i,0_{\mathbb{Z}_{+}},j)\right\}\colon i,j\in\mathbb{N}_{0}\hbox{~{}and~{}}ij=0\right\}$. It is obvious that $\textbf{B}(\mathbb{Z}_{+},\theta)=\bigcup\widetilde{\mathfrak{Z}}\cup\bigcup\widetilde{\mathfrak{O}}$.

Proposition 4.13**.**

Let $\theta$ be the annihilating homomorphism and $\tau$ be a locally compact topology on $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ such that $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is a semitopological semigroup with non-isolated zero 0 and the induced topology of $\tau$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ coincides with the topology $\tau_{\mathbf{cf}}$. Then the following assertions hold:

• $(i)$

for any non-negative integers $i$ and $j$ the set $(\widetilde{\mathbb{Z}}_{+})_{i,j}^{0}$ is open and compact in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and hence it is closed;

• $(ii)$

every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$;

• $(iii)$

for every open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure and any open neighbourhood $V_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ each of sets $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})\setminus V_{\textsf{0}}$ and $U_{\textsf{0}}\setminus V_{\textsf{0}}$ intersects finitely many elements of the family $\widetilde{\mathfrak{Z}}$;

• $(iv)$

for any non-negative integer $i_{0}$ every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$ of the form $(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}$;

• $(v)$

for any non-negative integer $i_{0}$ every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$ of the form $(\widetilde{\mathbb{Z}}_{+})_{j,i_{0}}^{0}$;

• $(vi)$

for any non-negative integer $i_{0}$ and every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ the family $\mathfrak{H}_{i_{0}}=\left\{(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}\in\widetilde{\mathfrak{Z}}\colon(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}\cap U_{\emph{{0}}}=\varnothing\right\}$ is finite;

• $(vii)$

for any non-negative integer $i_{0}$ and every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ the family $\mathfrak{V}_{i_{0}}=\left\{(\widetilde{\mathbb{Z}}_{+})_{j,i_{0}}^{0}\in\widetilde{\mathfrak{Z}}\colon(\widetilde{\mathbb{Z}}_{+})_{j,i_{0}}^{0}\cap U_{\emph{{0}}}=\varnothing\right\}$ is finite;

• $(viii)$

for any non-negative integer $i_{0}$ and every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ the set $\left\{(i_{0},0_{\mathbb{Z}_{+}},j)\colon j\in\mathbb{N}_{0}\right\}\setminus U_{\emph{{0}}}$ is finite;

• $(ix)$

for any non-negative integer $j_{0}$ and every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ the set $\left\{(i,0_{\mathbb{Z}_{+}},j_{0})\colon i\in\mathbb{N}_{0}\right\}\setminus U_{\emph{{0}}}$ is finite;

• $(x)$

for any non-negative integer $i_{0}$ and every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exist finitely many subsets of the form $(\mathbb{Z}_{+})_{i_{0},j}^{0}$ such that $(\mathbb{Z}_{+})_{i_{0},j}^{0}\nsubseteq U_{\emph{{0}}}$;

• $(xi)$

for any non-negative integer $i_{0}$ and every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exist finitely many subsets of the form $(\mathbb{Z}_{+})_{j,i_{0}}^{0}$ such that $(\mathbb{Z}_{+})_{i_{0},j}^{0}\nsubseteq U_{\emph{{0}}}$;

• $(xii)$

for every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exist finitely many subsets $(\mathbb{Z}_{+})_{i_{1},j_{1}}^{0},\ldots,(\mathbb{Z}_{+})_{i_{k},j_{k}}^{0}$ such that $(\mathbb{Z}_{+})_{i_{1},j_{1}}^{0}\cup\ldots\cup(\mathbb{Z}_{+})_{i_{k},j_{k}}^{0}\cup U_{\textsf{0}}=\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$.

Proof.

$(i)$ Since $\textbf{B}(\mathbb{Z}_{+},\theta)$ is an open subset of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ the set $(\widetilde{\mathbb{Z}}_{+})_{i,j}^{0}$ is open for any non-negative integers $i$ and $j$. Moreover, the definition of the topology $\tau_{\mathbf{cf}}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ implies that $(\widetilde{\mathbb{Z}}_{+})_{i,j}^{0}$ is compact, and since $\tau$ is Hausdorff, the set $(\widetilde{\mathbb{Z}}_{+})_{i,j}^{0}$ is closed.

$(ii)$ Suppose to the contrary that there exists an open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ which intersects finitely many elements of the family $\widetilde{\mathfrak{Z}}$. The definition of the topology $\tau_{\mathbf{cf}}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ implies that the set $(\widetilde{\mathbb{Z}}_{+})_{i,j}^{0}$ is compact for all non-negative integers $i$ and $j$, and hence our assumption implies that there exists finitely many $(\widetilde{\mathbb{Z}}_{+})_{i_{1},j_{1}}^{0},\ldots,(\widetilde{\mathbb{Z}}_{+})_{i_{k},j_{k}}^{0}\in\widetilde{\mathfrak{Z}}$ such that $U_{\textsf{0}}\setminus\{\textsf{0}\}\subseteq\widetilde{\mathbb{Z}}_{+})_{i_{1},j_{1}}^{0}\cup\ldots\cup(\widetilde{\mathbb{Z}}_{+})_{i_{k},j_{k}}^{0}$. This implies that zero 0 is an isolated point of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$, a contradiction.

$(iii)$ Fix an arbitrary open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure and an open neighbourhood $V_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. Then the family

[TABLE]

is an open cover of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. Since $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ is compact, $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})\setminus V_{\textsf{0}}$ intersects finitely many elements of the family $\widetilde{\mathfrak{Z}}$, and hence $U_{\textsf{0}}\setminus V_{\textsf{0}}$ intersects finitely many elements of the family $\widetilde{\mathfrak{Z}}$ as well.

$(iv)$ We claim that for every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists a non-negative integer $i_{0}$ such that $U_{\textsf{0}}$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$ of the form $(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}$. Indeed, suppose to the contrary and pick an open neighbourhood $U_{\textsf{0}}$ of zero which hasn’t this property. Without loos of generality we may assume that the closure $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ is a compact set. Then the separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $V_{\textsf{0}}\cdot(1,0_{\mathbb{Z}_{+}},0)\subseteq U_{\textsf{0}}$. Since

[TABLE]

our assumption implies that the set $U_{\textsf{0}}\setminus V_{\textsf{0}}$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$, which contradicts assertion $(iii)$.

Next we suppose that that for an arbitrary open neighbourhood $W_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists a non-negative integer $i_{0}$ such that $W_{\textsf{0}}$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$ of the form $(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}$. We shall prove that for an arbitrary non-negative integer $k$ the neighbourhood $W_{\textsf{0}}$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$ of the form $(\widetilde{\mathbb{Z}}_{+})_{k,j}^{0}$. The separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq W_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(k,0_{\mathbb{Z}_{+}},i_{0})\cdot V_{\textsf{0}}\subseteq W_{\textsf{0}}$, and hence our assertion holds.

The proof of statement $(v)$ is similar to $(iv)$.

$(vi)$ Suppose to the contrary that there exist a non-negative integer $i_{0}$ and an open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that the family $\mathfrak{H}_{i_{0}}$ is infinite. Then the separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $V_{\textsf{0}}\cdot(1,0_{\mathbb{Z}_{+}},0)\subseteq U_{\textsf{0}}$. By $(iv)$ the neighbourhood $U_{\textsf{0}}$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$ of the form $(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}$. Then our assumption and the following formula

[TABLE]

imply that the set $U_{\textsf{0}}\setminus V_{\textsf{0}}$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$, which contradicts assertion $(iii)$.

The proof of statement $(vii)$ is similar to $(vi)$.

$(viii)$ Fix an arbitrary non-negative integer $i_{0}$. Suppose to the contrary that there exists an open neighbourhood $U_{\textsf{0}}$ of zero with compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that the set $\left\{(i_{0},0_{\mathbb{Z}_{+}},j)\colon j\in\mathbb{N}_{0}\right\}\setminus U_{\textsf{0}}$ is infinite. Since for any non-negative integer $j$ the subspace $(\mathbb{Z}_{+})_{0,j}$ is open and discrete in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$, our above assumption implies that the set $\left\{(i_{0},0_{\mathbb{Z}_{+}},j)\colon j\in\mathbb{N}_{0}\right\}\setminus\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ is infinite too. By $(i)$ for every non-negative integer $j$ the set $(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}$ is compact and hence the discreetness of $(\mathbb{Z})_{i_{0},j}$ implies that $(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}\cap U_{\textsf{0}}$ is compact, as well. Then the separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(i_{0},1_{\mathbb{Z}_{+}},i_{0})\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$, where $1_{\mathbb{Z}_{+}}$ is a generator of the additive group of integers $\mathbb{Z}_{+}$. By $(vi)$ there exist finitely many $(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}\in\widetilde{\mathfrak{Z}}$ such that $(\widetilde{\mathbb{Z}}_{+})_{i_{0},j}^{0}\cap U_{\emph{{0}}}=\varnothing$. Then our assumption and the equality

[TABLE]

imply that the set $U_{\textsf{0}}\setminus V_{\textsf{0}}$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$, which contradicts assertion $(iii)$.

The proof of statement $(ix)$ is similar to $(viii)$.

$(x)$ Suppose to the contrary that there exist a non-negative integer $i_{0}$, an open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and infinitely many subsets of the form $(\mathbb{Z}_{+})_{i_{0},j}^{0}$ such that $(\mathbb{Z}_{+})_{i_{0},j}^{0}\nsubseteq U_{\textsf{0}}$. Then the separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(i_{0},1_{\mathbb{Z}_{+}},i_{0})\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$, where $1_{\mathbb{Z}_{+}}$ is a generator of the additive group of integers $\mathbb{Z}_{+}$. By statement $(viii)$ there exist finitely many subsets of the form $(\mathbb{Z}_{+})_{i_{0},j}^{0}$ such that $(\mathbb{Z}_{+})_{i_{0},j}^{0}\cap U_{\textsf{0}}=\varnothing$. Then our assumption and the equality

[TABLE]

imply that the set $U_{\textsf{0}}\setminus V_{\textsf{0}}$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$, which contradicts statement $(iii)$.

The proof of statement $(xi)$ is similar to $(x)$.

$(xii)$ Suppose to the contrary that there exists an open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that

[TABLE]

for any finite family $\left\{(\mathbb{Z}_{+})_{i_{1},j_{1}}^{0},\ldots,(\mathbb{Z}_{+})_{i_{k},j_{k}}^{0}\right\}$. This assumption, statements $(x)$ and $(xi)$ imply that there exists an infinite sequence $\left\{(i_{n},g_{n},j_{n})\colon n\in\mathbb{N}\right\}$ in $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}\setminus U_{\textsf{0}}$ such that $g_{n}\in\mathbb{Z}_{+}$ for $n\in\mathbb{N}$, $\left\{i_{n}\colon n\in\mathbb{N}\right\}$ and $\left\{j_{n}\colon n\in\mathbb{N}\right\}$ are increasing sequences of the positive integers.

Also, statements $(x)$ and $(xi)$ imply that without loss of generality we may assume that for elements of the sequence $\left\{(i_{n},g_{n},j_{n})\colon n\in\mathbb{N}\right\}$ the following condition holds:

[TABLE]

By the separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that

[TABLE]

Then statements $(x)$ and $(xi)$, our assumption and the equalities

[TABLE]

imply that the set $U_{\textsf{0}}\setminus V_{\textsf{0}}$ intersects infinitely many elements of the family $\widetilde{\mathfrak{Z}}$, which contradicts $(iii)$. ∎

Theorem 4.14**.**

Let $\theta$ be the annihilating homomorphism and $\tau$ be a shift-continuous locally compact topology on $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ such that the induced topology of $\tau$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ coincides with the topology $\tau_{\mathbf{cf}}$. Then either $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is compact or zero is an isolated point in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. Moreover, $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is compact, then the family

[TABLE]

where for any positive integer $p$ the set $M_{p}$ is one of the following sets: $\left\{\left(0,0_{\mathbb{Z}_{+}},j\right)\right\}$, $\left\{\left(i,0_{\mathbb{Z}_{+}},0\right)\right\}$ or $(\widetilde{\mathbb{Z}}_{+})_{m,n}^{0}$, $i,j\in\mathbb{N}_{0}$, $m,n\in\mathbb{N}$, defines a base of the topology $\tau$ at zero 0 of $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$.

Proof.

It is obvious that $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau_{\mathbf{cf}}\right)$ with the adjoined isolated zero is a locally compact semitopological semigroup.

Later we assume that zero 0 is non-isolated point of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$.

By Proposition 4.13$(i)$ for any non-negative integers $i$ and $j$ the set $(\widetilde{\mathbb{Z}}_{+})_{i,j}^{0}$ is open and compact in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$, and hence by Proposition 2.4$(i)$ and the definition of the topology $\tau_{\mathbf{cf}}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ we have that $(0,0_{\mathbb{Z}_{+}},i)$ and $(j,0_{\mathbb{Z}_{+}},0)$ are isolated point in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ for all non-negative integers $i$ and $j$. For any non-negative integers $i$ and $j$ put

[TABLE]

Then for any non-negative integers $i$ and $j$ we have that $(\mathbb{Z}_{+})_{i,j}^{0}\subseteq A_{i,j}$ and $A_{i,j}$ is a compact open subset of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. By Proposition 4.13$(xii)$ for an arbitrary open neighbourhood $U_{\textbf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exist finitely many subsets $(\mathbb{Z}_{+})_{i_{1},j_{1}}^{0},\ldots,(\mathbb{Z}_{+})_{i_{k},j_{k}}^{0}$ such that $(\mathbb{Z}_{+})_{i_{1},j_{1}}^{0}\cup\ldots\cup(\mathbb{Z}_{+})_{i_{k},j_{k}}^{0}\cup U_{\textsf{0}}=\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$. Then $O(U_{\textbf{0}})=\textbf{B}(\mathbb{Z}_{+},\theta)^{0}\setminus\left(A_{i_{1},j_{1}}\cup\cdots\cup A_{i_{k},j_{k}}\right)$ is an open-and-closed neighbourhood of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $O(U_{\textbf{0}})\subseteq U_{\textbf{0}}$. This implies that if the family $\left\{U_{\textbf{0}}^{\alpha}\right\}_{\alpha\in\Omega}$ is a base of the topology $\tau$ at zero of $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$, then so constructed family $\left\{O(U_{\textbf{0}}^{\alpha})\right\}_{\alpha\in\Omega}$ is a base of the topology $\tau$ at zero of $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ too. This completes the last statement of the theorem. ∎

For all non-negative integers $k$ and $l$ we put $\left(-\mathbb{N}\right)_{k,l}=\left\{(k,-n,l)\colon n\in\mathbb{N}\right\}$, $\mathfrak{N}^{+}=\left\{\mathbb{N}_{i,j}^{0}\colon i,j\in\mathbb{N}_{0}\right\}$ and $\mathfrak{N}^{-}=\left\{\left(-\mathbb{N}\right)_{i,j}\colon i,j\in\mathbb{N}_{0}\right\}$.

Proposition 4.15**.**

If $\theta$ is the annihilating homomorphism, $\tau$ is a locally compact shift-continuous topology on $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ such that zero 0 is a non-isolated point in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and the induced topology of $\tau$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ coincides with $\tau_{+}$, then the following assertions hold:

• $(i)$

for any non-negative integers $i$ and $j$ the set $\mathbb{N}^{0}_{i,j}$ is open and compact in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and hence it is closed;

• $(ii)$

every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ intersects infinitely many elements of the family $\mathfrak{N}^{+}$;

• $(iii)$

for any non-negative integer $i_{0}$ every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ intersects almost all elements of the family $\mathfrak{N}^{+}$ of the form $\mathbb{N}_{i_{0},j}^{0}$;

• $(iv)$

every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ contains almost all elements of the form $(0,0_{\mathbb{Z}_{+}},j)$, $j\in\mathbb{N}_{0}$;

• $(v)$

every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ contains almost all subsets of the form $(\mathbb{Z}_{+})_{0,j}^{0}$, $j\in\mathbb{N}_{0}$.

Proof.

$(i)$ Since $\textbf{B}(\mathbb{Z}_{+},\theta)$ is an open subset of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$, by the definition of $\tau_{+}$ the set $\mathbb{N}_{i,j}^{0}$ is open for any non-negative integers $i$ and $j$. Moreover, the definition of the topology $\tau_{+}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ implies that $\mathbb{N}_{i,j}^{0}$ is compact, and since $\tau$ is Hausdorff, $\mathbb{N}_{i,j}^{0}$ is closed.

$(ii)$ Suppose to the contrary that there exists an open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ which intersects finitely many elements of the family $\mathfrak{N}^{+}$. By the definition of the topology $\tau_{+}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ the set $\mathbb{N}_{i,j}^{0}$ is compact for all non-negative integers $i$ and $j$, and hence without loss of generality we may assume that $U_{\textsf{0}}\cap\bigcup\mathfrak{N}^{+}=\varnothing$. By the equality

[TABLE]

the separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and assertion $(i)$ the set

[TABLE]

is open-and-closed in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. This and the definition of the topology $\tau_{+}$ imply that the set $(\widetilde{\mathbb{Z}}_{+})_{k,l}^{0}$ is opens-and-closed and the points $(0,0_{\mathbb{Z}_{+}},k)$ and $(k,0_{\mathbb{Z}_{+}},0)$ are isolated in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ for all $k,l\in\mathbb{N}_{0}$. By the definition of the topology $\tau_{+}$ and statement $(i)$ we have that $\left(-\mathbb{N}\right)_{k,l}=(\widetilde{\mathbb{Z}}_{+})_{k,l}^{0}\setminus\mathbb{N}_{k,l}^{0}$ is an open-and-closed discrete subspace of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ for all $k,l\in\mathbb{N}_{0}$. This implies that the set $U_{\textsf{0}}\cap\left(-\mathbb{N}\right)_{k,l}$ is finite for all $k,l\in\mathbb{N}_{0}$. By the separate continuity of the semigroup operation of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $V_{\textsf{0}}\cdot(1,0_{\mathbb{Z}_{+}},0)\subseteq U_{\textsf{0}}$. Then by the equality

[TABLE]

we have that $U_{\textsf{0}}\setminus V_{\textsf{0}}$ is an infinite subset of the compactum $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U(0))$, and the definition of the topology $\tau_{+}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ implies that the set $U_{\textsf{0}}\setminus V_{\textsf{0}}$ does not have an accumulation point in $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$, which contradicts the compactness of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$.

$(iii)$ We claim that for every open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists a non-negative integer $i_{0}$ such that $U_{\textsf{0}}$ intersects infinitely many elements of the form $\mathbb{N}_{i_{0},j}^{0}$. Indeed, assume the contrary and pick $U_{\textsf{0}}$ which has not this property. Without loss of generality we may assume that the neighbourhood $U_{\textsf{0}}$ has the compact closure $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. By item $(ii)$ there exists an increasing sequence $\left\{i_{n}\right\}_{n\in\mathbb{N}}$ of positive integers such that $\mathbb{N}_{i_{n},j_{n}}^{0}\cap U_{\textsf{0}}\neq\varnothing$ for some sequence of non-negative integers $\left\{j_{n}\right\}_{n\in\mathbb{N}}$. Then for every element $i_{p}$ of the sequence $\left\{i_{n}\right\}_{n\in\mathbb{N}}$ there exits a maximum non-negative integer $j_{i_{p}}$ such that $\mathbb{N}_{i_{p},j_{i_{p}}}^{0}\cap U_{\textsf{0}}\neq\varnothing$ and $\mathbb{N}_{i_{p},j_{i_{p}}+k}^{0}\cap U_{\textsf{0}}=\varnothing$ for any positive integer $k$. By the separate continuity of the semigroup operation of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $V_{\textsf{0}}\cdot(0,0_{\mathbb{Z}_{+}},1)\subseteq U_{\textsf{0}}$. The above arguments imply that there exists a sequence of distinct points $\left\{\left(i_{n},z_{n},j_{i_{n}}\right)\right\}_{n\in\mathbb{N}}\subseteq U_{\textsf{0}}\setminus V_{\textsf{0}}\subset\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. Then the definition of the topology $\tau_{+}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ implies that this sequence has not an accumulation point in the set $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ which contradicts the compactness of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$.

By the above for every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists a non-negative integer $i_{0}$ such that $U_{\textsf{0}}\cap\mathbb{N}_{i_{0},j}^{0}\neq\varnothing$ for infinitely many elements of the family $\mathfrak{N}^{+}$ of the form $\mathbb{N}_{i_{0},j}^{0}$. Since the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is separately continuous, for any non-negative integer $i$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(i,0_{\mathbb{Z}_{+}},i_{0})\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. Thus $U_{\textsf{0}}\cap\mathbb{N}_{i,j}^{0}\neq\varnothing$ for infinitely many elements of the family $\mathfrak{N}^{+}$ of the form $\mathbb{N}_{i,j}^{0}$. Hence we get that $U_{\textsf{0}}\cap\mathbb{N}_{0,j}^{0}\neq\varnothing$ for infinitely many elements of the family $\mathfrak{N}^{+}$ of the form $\mathbb{N}_{0,j}^{0}$.

Suppose that there exists an infinite increasing sequence $\left\{j_{n}\right\}_{n\in\mathbb{N}}$ such that $U_{\textsf{0}}\cap\mathbb{N}_{0,j_{n}}^{0}=\varnothing$ for any elements $j_{n}$ of $\left\{j_{n}\right\}_{n\in\mathbb{N}}$. Without loss of generality we may assume that the sequence $\left\{j_{n}\right\}_{n\in\mathbb{N}}$ is maximal, i.e., $U_{\textsf{0}}\cap\mathbb{N}_{0,j}^{0}\neq\varnothing$ for any non-negative integer $j\notin\left\{j_{n}\right\}_{n\in\mathbb{N}}$. Then there exists a subsequence $\left\{j_{n_{k}}\right\}_{k\in\mathbb{N}}$ of $\left\{j_{n}\right\}_{n\in\mathbb{N}}$ such that $U_{\textsf{0}}\cap\mathbb{N}_{0,j_{n_{k}}}^{0}=\varnothing$ and $U_{\textsf{0}}\cap\mathbb{N}_{0,j_{n_{k}}+1}^{0}\neq\varnothing$. The separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(1,0_{\mathbb{Z}_{+}},0)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. It is obvious that our above arguments imply that there exists a sequence of distinct points $\left\{\left(0,z_{n_{k}},j_{n_{k}}\right)\right\}_{n\in\mathbb{N}}\subseteq U_{\textsf{0}}\setminus V_{\textsf{0}}$ which is a subset of the compactum $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. Then the definition of the topology $\tau_{+}$ on $\textbf{B}(\mathbb{Z}_{+},\theta)$ implies that this sequence has not an accumulation point in $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ which contradicts the compactness of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. Hence we have that every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ intersects almost all elements of the form $\mathbb{N}_{0,j}^{0}$. Again, since the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is separately continuous, for any non-negative integer $i$ there exists an open neighbourhood $W_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(i,0_{\mathbb{Z}_{+}},i_{0})\cdot W_{\textsf{0}}\subseteq U_{\textsf{0}}$. The last inclusion implies assertion $(iii)$.

$(iv)$ Fix an arbitrary open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. Without loss of generality we may assume that the neighbourhood $U_{\textsf{0}}$ has the compact closure $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. By statement $(iii)$ the neighbourhood $U_{\textsf{0}}$ intersects almost all elements of the family $\mathfrak{N}^{+}$ of the form $\mathbb{N}_{0,j}^{0}$ and by the separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(1,0_{\mathbb{Z}_{+}},1)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. Since $(1,0_{\mathbb{Z}_{+}},1)\cdot(0,z,k)=(0,0_{\mathbb{Z}_{+}},k)$ the inclusion $(1,0_{\mathbb{Z}_{+}},1)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$ implies our assertion.

$(v)$ Fix an arbitrary open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. The separate continuity of the semigroup operation of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that for every element $k$ of the additive group of integers $\mathbb{Z}_{+}$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(0,k,0)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. By statement $(iv)$ the neighbourhood $U_{\textsf{0}}$ contains almost all elements of the form $(0,0_{\mathbb{Z}_{+}},j)$ and hence the above arguments imply statement $(v)$. ∎

Theorem 4.16**.**

Let $\theta$ be the annihilating homomorphism and $\tau$ be a shift-continuous locally compact topology on $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ such that the induced topology of $\tau$ onto $\textbf{B}(\mathbb{Z}_{+},\theta)$ coincides with the topology $\tau_{+}$. Then zero 0 of $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ is an isolated point in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$.

Proof.

First we observe that Example 4.8 and Proposition 4.15 imply that for all non-negative integers $i$ and $j$ and any $n\in\mathbb{Z}_{+}$ the point $(i,n,j)$ is isolated in $(\mathbb{Z}_{+})_{i,j}^{0}$. Then the proof of assertion $(ii)$ of Proposition 4.15 implies that the set $(\mathbb{Z}_{+})_{0,j}^{0}$ is open-and-closed in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ for any non-negative integer $j$.

Fix an arbitrary open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. By Proposition 4.15$(v)$ there exists a positive integer $j_{0}$ such that $(\mathbb{Z}_{+})_{0,j_{0}}^{0}\subseteq U_{\textsf{0}}$ and hence $(\mathbb{Z}_{+})_{0,j}^{0}$ is an open-and-closed subset of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U(0))$. But $(\mathbb{Z}_{+})_{0,j}^{0}$ is not compact, which contradicts the compactness of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U(0))$. ∎

The proof of the following theorem is similar to Theorem 4.16.

Theorem 4.17**.**

Let $\theta$ be the annihilating homomorphism and $\tau$ be a shift-continuous locally compact topology on $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ such that the induced topology of $\tau$ onto $\textbf{B}(\mathbb{Z}_{+},\theta)$ coincides with the topology $\tau_{-}$. Then zero 0 of $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ is an isolated point in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$.

Later we need the following two folklore lemmas.

Lemma 4.18**.**

Let $\theta\colon\mathbb{Z}_{+}\to\mathbb{Z}_{+}$ be a homomorphism. Then the image $\theta(\mathbb{Z}_{+})$ is isomorphic to the additive group of integers $\mathbb{Z}_{+}$ if and only if $\theta$ is non-annihilating.

Proof.

The implication $(\Rightarrow)$ is trivial.

$(\Leftarrow)$ Suppose that a homomorphism $\theta$ is non-annihilating. Since the group $\mathbb{Z}_{+}$ is generated by the element $1$ we have that $\theta(1)=n\neq 0_{\mathbb{Z}_{+}}\in\theta(\mathbb{Z}_{+})\subseteq\mathbb{Z}_{+}$ for some integer $n$. This implies that the image $\theta(\mathbb{Z}_{+})$ generated by the element $n$ as a subgroup of $\mathbb{Z}_{+}$, and since $\mathbb{Z}_{+}$ is isomorphic to the free group over a singleton set, $\theta(\mathbb{Z}_{+})$ is isomorphic to the subgroup $n\mathbb{Z}_{+}=\{nk\colon k\in\mathbb{Z}_{+}\}$ of $\mathbb{Z}_{+}$. It is obvious that $n\mathbb{Z}_{+}$ is isomorphic to $\mathbb{Z}_{+}$. ∎

Lemma 4.19**.**

If $\theta\colon\mathbb{Z}_{+}\to\mathbb{Z}_{+}$ is an arbitrary non-annihilating homomorphism then for arbitrary positive integer $i$ and an arbitrary $a\in\mathbb{Z}_{+}$ the equation $\theta^{i}(x)=a$ has at most one solution.

Proof.

Suppose that $\theta(1)=n$ for some $n\in\mathbb{Z}_{+}\setminus\{0_{\mathbb{Z}_{+}}\}$. Since the homomorphism $\theta\colon\mathbb{Z}_{+}\to\mathbb{Z}_{+}$ is non-annihilating, it is obvious that the equation $\theta^{i}(x)=a$ has a unique solution if and only if $a=n^{i}b$ for some integer $b$, and in the other case the equation $(x)\theta^{i}=a$ hasn’t a solution. ∎

Proposition 4.20**.**

Let $\theta\colon\mathbb{Z}_{+}\to\mathbb{Z}_{+}$ be an arbitrary non-annihilating homomorphism. Then both equations $\alpha\cdot\chi=\beta$ and $\chi\cdot\gamma=\delta$ have finitely many solutions in $\textbf{B}(\mathbb{Z}_{+},\theta)$.

Proof.

We consider only the case of $\alpha\cdot\chi=\beta$. The proof in the other case is similar.

Put

[TABLE]

Then the semigroup operation of $\textbf{B}(\mathbb{Z}_{+},\theta)$ implies that

[TABLE]

Then

• $(1)$

in the case when $m_{1}<n$ we have that $n=n_{2}-n_{1}+m_{1}$, $m=m_{2}$ and $z=z_{2}-\theta^{n-m_{1}}(z_{1})$;

• $(2)$

in the case when $m_{1}=n$ we have that $m=m_{2}$ and $z=z_{2}-z_{1}$;

• $(3)$

in the case when $m_{1}>n$ we have that $n_{1}=n_{2}$, $n-m=m_{2}-m_{1}$ and $\theta^{m_{1}-n}(z)=z_{2}-z_{1}$.

Now, the above three cases and Lemma 4.19 imply the statement of the proposition. ∎

We recall that a topological space $X$ is said to be Baire, if for each sequence $A_{1},A_{2},\ldots,A_{i},\ldots$ of open dense subsets of $X$ the intersection $\bigcap_{i=1}^{\infty}A_{i}$ is dense in $X$. [30]. It is well known that every Čech-complete (and hence every locally compact) space is Baire (see [17, Section 3.9]).

The following theorem describes Baire $T_{1}$-semitopological semigroups $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$ with a non-annihilating homomorphism $\theta$.

Theorem 4.21**.**

If $\theta$ is an arbitrary non-annihilating homomorphism then every shift-continuous Baire $T_{1}$-topology on $\textbf{B}(\mathbb{Z}_{+},\theta)$ is discrete.

Proof.

Since the space $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$ is Baire and countable, $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$ contains an isolated point $(i_{0},z_{0},j_{0})$, where $i_{0}$ and $j_{0}$ are non-negative integers and $z_{0}\in\mathbb{Z}_{+}$. Fix an arbitrary element $(i_{1},z_{1},j_{1})\in\textbf{B}(\mathbb{Z}_{+},\theta)$. Then

[TABLE]

Proposition 4.20 implies that the equation

[TABLE]

has a finite non-empty set of solutions. Since $\tau$ is a $T_{1}$-topology on $\textbf{B}(\mathbb{Z}_{+},\theta)$ the separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$ implies that the map $f\colon\textbf{B}(\mathbb{Z}_{+},\theta)\to\textbf{B}(\mathbb{Z}_{+},\theta)$, $f(x)=(i_{0},z_{0}-z_{1},i_{1})\cdot x\cdot(j_{1},0_{\mathbb{Z}_{+}},j_{0})$ is continuous and hence $(i_{1},z_{1},j_{1})$ is an isolated point of $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$. Therefore, $\left(\textbf{B}(\mathbb{Z}_{+},\theta),\tau\right)$ is the discrete space. ∎

Theorem 4.21 implies the following corollary.

Corollary 4.22**.**

If $\theta$ is an arbitrary non-annihilating homomorphism then every shift-continuous locally compact $T_{1}$-topology on $\textbf{B}(\mathbb{Z}_{+},\theta)$ is discrete.

Now we obtain the description of Baire $T_{1}$-semitopological semigroups $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ with a non-annihilating homomorphism $\theta$.

Theorem 4.23**.**

Let $\theta$ be an arbitrary non-annihilating homomorphism and $\tau$ be a shift-continuous Baire $T_{1}$-topology on $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$. Then every non-zero element of the semigroup $\textbf{B}(\mathbb{Z}_{+}^{0},\theta)^{0}$ is an isolated point in $\left(\textbf{B}(\mathbb{Z}_{+}^{0},\theta)^{0},\tau\right)$.

Proof.

Since $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is a Baire $T_{1}$-space, $\textbf{B}(\mathbb{Z}_{+},\theta)$ is its open subspace, and hence by Proposition 1.14 from [30], $\textbf{B}(\mathbb{Z}_{+},\theta)$ is Baire, too. Next we apply Theorem 4.21. ∎

Theorem 4.23 implies the following corollary.

Corollary 4.24**.**

Let $\theta$ be an arbitrary non-annihilating homomorphism and $\tau$ be a shift-continuous locally compact $T_{1}$-topology on $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$. Then every non-zero element of the semigroup $\textbf{B}(\mathbb{Z}_{+}^{0},\theta)^{0}$ is an isolated point in $\left(\textbf{B}(\mathbb{Z}_{+}^{0},\theta)^{0},\tau\right)$.

Proposition 4.25**.**

Let $\theta$ be an arbitrary non-annihilating homomorphism and $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ be a locally compact semitopological semigroup with non-isolated zero 0. Then the following assertions hold:

• $(i)$

every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ intersects infinitely many sets of the form $\mathbb{Z}_{i,j}$, $i,j\in\mathbb{N}_{0}$;

• $(ii)$

for any non-negative integer $i_{0}$ every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ intersects almost all sets of the form $\mathbb{Z}_{i_{0},j}$, $j\in\mathbb{N}_{0}$;

• $(iii)$

every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ contains almost all elements of the form $(0,0_{\mathbb{Z}_{+}},j)$, $j\in\mathbb{N}_{0}$;

• $(iv)$

every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ contains almost all subsets of the form $(\mathbb{Z}_{+})_{0,j}$, $j\in\mathbb{N}_{0}$;

• $(v)$

for any non-negative integer $i_{0}$ every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ contains almost all subsets of the form $(\mathbb{Z}_{+})_{i_{0},j}$, $j\in\mathbb{N}_{0}$;

• $(vi)$

for any non-negative integer $j_{0}$ every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ contains almost all subsets of the form $(\mathbb{Z}_{+})_{i,j_{0}}$, $i\in\mathbb{N}_{0}$;

• $(vii)$

every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ contains almost all subsets of the form $(\mathbb{Z}_{+})_{i,j}$, $i,j\in\mathbb{N}_{0}$;

• $(viii)$

for every open neighbourhood $U_{\emph{{0}}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ the set $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}\setminus U_{\textsf{0}}$ is finite.

Proof.

$(i)$ Suppose to the contrary that there exists an open neighbourhood $U_{\textsf{0}}$ of zero the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ which intersects finitely many sets of the form $\mathbb{Z}_{i,j}$. By the separate continuity of the semigroup operation of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $V_{\textsf{0}}\cdot(1,0_{\mathbb{Z}_{+}},0)\subseteq U_{\textsf{0}}$. Then the equality

[TABLE]

implies that $U_{\textsf{0}}\setminus V_{\textsf{0}}$ is an infinite subset of the compactum $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ and by Theorem 4.21, $\textbf{B}(\mathbb{Z}_{+},\theta)$ is a discrete subspace of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. This implies that the set $U_{\textsf{0}}\setminus V_{\textsf{0}}$ does not have an accumulation point in $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$, which contradicts the compactness of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$.

$(ii)$ We claim that for every open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists a non-negative integer $i_{0}$ such that $U_{\textsf{0}}$ intersects infinitely many sets of the form $\mathbb{Z}_{i_{0},j}$. If we assume to the contrary then by $(i)$ there exists an increasing sequence $\left\{i_{n}\right\}_{n\in\mathbb{N}}$ of positive integers such that $\mathbb{Z}_{i_{n},j_{n}}\cap U_{\textsf{0}}\neq\varnothing$ for some sequence of non-negative integers $\left\{j_{n}\right\}_{n\in\mathbb{N}}$. Then for every element $i_{p}$ of the sequence $\left\{i_{n}\right\}_{n\in\mathbb{N}}$ there exits a maximum non-negative integer $j_{i_{p}}$ such that $\mathbb{Z}_{i_{p},j_{i_{p}}}\cap U_{\textsf{0}}\neq\varnothing$ and $\mathbb{Z}_{i_{p},j_{i_{p}}+k}\cap U_{\textsf{0}}=\varnothing$ for any positive integer $k$. By the separate continuity of the semigroup operation of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $V_{\textsf{0}}\cdot(0,0_{\mathbb{Z}_{+}},1)\subseteq U_{\textsf{0}}$. Then the equality

[TABLE]

implies that there exists a sequence of distinct points $\left\{\left(i_{n},z_{n},j_{i_{n}}\right)\right\}_{n\in\mathbb{N}}\subseteq U_{\textsf{0}}\setminus V_{\textsf{0}}$ which is a subset of the compactum $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. By Corollary 4.22, $\textbf{B}(\mathbb{Z}_{+},\theta)$ is a discrete subspace of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$, and hence this sequence has not an accumulation point in $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ which contradicts the compactness of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$.

By the previous part of the proof for every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ there exists a non-negative integer $i_{0}$ such that $U_{\textsf{0}}\cap\mathbb{Z}_{i_{0},j}\neq\varnothing$ for infinitely many sets of the form $\mathbb{Z}_{i_{0},j}$, $j\in\mathbb{N}\cup\{0\}$. Since the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is separately continuous, for any non-negative integer $i$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(i_{1},0_{\mathbb{Z}_{+}},i_{0})\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. The last inclusion implies that for every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and any non-negative integer $i_{1}$ we have that $U_{\textsf{0}}\cap\mathbb{Z}_{i,j}\neq\varnothing$ for infinitely many sets of the form $\mathbb{Z}_{i_{1},j}$, $j\in\mathbb{N}_{0}$. Hence we get that $U_{\textsf{0}}\cap\mathbb{Z}_{0,j}\neq\varnothing$ for infinitely many sets of the form $\mathbb{Z}_{0,j}$, $j\in\mathbb{N}_{0}$.

Suppose that there exists an infinite increasing sequence $\left\{j_{n}\right\}_{n\in\mathbb{N}}$ of non-negative integers such that $U_{\textsf{0}}\cap\mathbb{Z}_{0,j_{n}}=\varnothing$ for any element $j_{n}$ of $\left\{j_{n}\right\}_{n\in\mathbb{N}}$. Without loss of generality we may assume that the sequence $\left\{j_{n}\right\}_{n\in\mathbb{N}}$ is maximal, i.e., $U_{\textsf{0}}\cap\mathbb{Z}_{0,j}\neq\varnothing$ for any non-negative integer $j\notin\left\{j_{n}\right\}_{n\in\mathbb{N}}$. Then there exists a subsequence $\left\{j_{n_{k}}\right\}_{k\in\mathbb{N}}$ in $\left\{j_{n}\right\}_{n\in\mathbb{N}}$ such that $U_{\textsf{0}}\cap\mathbb{Z}_{0,j_{n_{k}}}=\varnothing$ and $U_{\textsf{0}}\cap\mathbb{Z}_{0,j_{n_{k}}+1}\neq\varnothing$. The separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(1,0_{\mathbb{Z}_{+}},0)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. Then the equality

[TABLE]

implies that there exists a sequence of distinct points $\left\{\left(0,z_{n_{k}},j_{n_{k}}\right)\right\}_{n\in\mathbb{N}}\subseteq U_{\textsf{0}}\setminus V_{\textsf{0}}$ which is a subset of the compactum $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. By Corollary 4.22, $\textbf{B}(\mathbb{Z}_{+},\theta)$ is a discrete subspace of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and hence this sequence has not an accumulation point in $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ which contradicts the compactness of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. This implies that every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ intersects almost all sets of the form $\mathbb{Z}_{i_{0},j}$, $j\in\mathbb{N}_{0}$. Again, since the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is separately continuous, for any non-negative integer $i$ there exists an open neighbourhood $W_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(i,0_{\mathbb{Z}_{+}},i_{0})\cdot W_{\textsf{0}}\subseteq U_{\textsf{0}}$. The last inclusion implies assertion $(ii)$.

$(iii)$ Fix an arbitrary open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. Then by $(ii)$ the set $U_{\textsf{0}}$ intersects almost all sets of the form $\mathbb{Z}_{i_{0},j}$, $j\in\mathbb{N}_{0}$. By the separate continuity of the semigroup operation of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(1,0_{\mathbb{Z}_{+}},1)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. Since $(1,0_{\mathbb{Z}_{+}},1)\cdot(0,z,k)=(0,0_{\mathbb{Z}_{+}},k)$ the inclusion $(1,0_{\mathbb{Z}_{+}},1)\cdot V(\textsf{0})\subseteq U_{\textsf{0}}$ implies our assertion.

$(iv)$ Fix an arbitrary open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. The separate continuity of the semigroup operation of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that for every element $k$ of the additive group of integers $\mathbb{Z}_{+}$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(0,k,0)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. By statement $(iii)$ the neighbourhood $U_{\textsf{0}}$ contains almost all elements of the form $(0,0_{\mathbb{Z}_{+}},j)$ and hence the inclusion $(0,k,0)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$ implies the statement.

$(v)$ Fix an arbitrary open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$. By $(iv)$ the neighbourhood $U_{\textsf{0}}$ contains almost all subsets of the form $(\mathbb{Z}_{+})_{0,j}$, $j\in\mathbb{N}_{0}$. Since the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is separately continuous, for any non-negative integer $i$ there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(i_{0},0_{\mathbb{Z}_{+}},0)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$, which implies the assertion.

The proof of assertion $(vi)$ is similar to $(v)$.

$(vii)$ If we assume to the contrary then by items $(v)$ and $(vi)$ there exist an open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and an increasing sequence $\left\{(i_{n},i_{n})\right\}_{n\in\mathbb{N}}$ of ordered pairs of positive integers such that $\mathbb{Z}_{i_{n},j_{n}}\cap U_{\textsf{0}}=\varnothing$ and $\mathbb{Z}_{i_{n},j_{n}+k}\cap U_{\textsf{0}}=\varnothing$ for any positive integer $k$. The separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(1,0_{\mathbb{Z}_{+}},0)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. Then the equality

[TABLE]

implies that there exists a sequence of distinct points $\left\{\left(0,z_{n_{k}},j_{n_{k}}\right)\right\}_{n\in\mathbb{N}}\subseteq U_{\textsf{0}}\setminus V_{\textsf{0}}$ which is a subset of the compactum $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. By Corollary 4.22, $\textbf{B}(\mathbb{Z}_{+},\theta)$ is a discrete subspace of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and hence this sequence has not an accumulation point in $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ which contradicts the compactness of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$.

$(viii)$ Suppose to the contrary that there exists an open neighbourhood $U_{\textsf{0}}$ of zero with the compact closure in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that the set $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}\setminus U_{\textsf{0}}$ is infinite. By assertion $(vii)$ there exist non-negative integers $i_{0}$ and $j_{0}$ such that the set $\mathbb{Z}_{i_{0},j_{0}}\setminus U_{\textsf{0}}$ is infinite and the set $\mathbb{Z}_{i_{0},j_{0}+1}\setminus U_{\textsf{0}}$ is finite. The separate continuity of the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ implies that there exists an open neighbourhood $V_{\textsf{0}}\subseteq U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ such that $(1,0_{\mathbb{Z}_{+}},0)\cdot V_{\textsf{0}}\subseteq U_{\textsf{0}}$. Then the equality

[TABLE]

implies that there exists a sequence of distinct points $\left\{\left(i_{0},z_{n},j_{0}+1\right)\right\}_{n\in\mathbb{N}}\subseteq\mathbb{Z}_{i_{0},j_{0}+1}\setminus V_{\textsf{0}}\subseteq U_{\textsf{0}}\setminus V_{\textsf{0}}$ which is a subset of the compactum $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. By Corollary 4.22, $\textbf{B}(\mathbb{Z}_{+},\theta)$ is a discrete subspace of $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ and hence this sequence has not an accumulation point in $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$ which contradicts the compactness of $\operatorname{cl}_{\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)}(U_{\textsf{0}})$. ∎

The following example shows that the Reilly semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ with a non-annihilating homomorphism $\theta$ admits the structure of a Hausdorff compact semitopological inverse semigroup with continuous inversion.

Example 4.26**.**

Let $\theta\colon\mathbb{Z}_{+}\to\mathbb{Z}_{+}$ be an arbitrary non-annihilating homomorphism. We define a topology $\tau_{\textsf{AC}}$ on the semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ in the following way:

• $(i)$

all non-zero elements of the semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ are isolated points in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau_{\textsf{AC}}\right)$;

• $(ii)$

the family $\mathscr{B}_{\textsf{AC}}(\textsf{0})=\left\{U\colon\textsf{0}\in U\hbox{~{}and~{}}\textbf{B}(\mathbb{Z}_{+},\theta)^{0}\setminus U\hbox{~{}is finite}\right\}$ is a base of the topology $\tau_{\textsf{AC}}$ at zero $\textsf{0}\in\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$.

It is obvious that the space $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau_{\textsf{AC}}\right)$ is the one-point Alexandroff compactification of the discrete space $\textbf{B}(\mathbb{Z}_{+},\theta)$ with the remainder $\{\textsf{0}\}$. Proposition 4.20 implies that both equations $\alpha\cdot\chi=\beta$ and $\chi\cdot\gamma=\delta$ have finitely many solutions in $\textbf{B}(\mathbb{Z}_{+},\theta)$. Then for every open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau_{\textsf{AC}}\right)$ and any non-zero element $(i,z,j)\in\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ the sets

[TABLE]

are finite. Hence for any open neighbourhood $U_{\textsf{0}}$ of zero in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau_{\textsf{AC}}\right)$ and any non-zero element $(i,z,j)\in\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ there exists a neighbourhood $V_{\textsf{0}}$ of zero such that

[TABLE]

which implies that the semigroup operation in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau_{\textsf{AC}}\right)$ is separately continuous. It is easy to see that the space $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau_{\textsf{AC}}\right)$ is Hausdorff and inversion in $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau_{\textsf{AC}}\right)$ is continuous.

Corollary 4.24, Proposition 4.25 and Example 4.26 imply the following dichotomy for a locally compact $T_{1}$-semitopological semigroup $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ with a non-annihilating homomorphism $\theta$.

Theorem 4.27**.**

Let $\theta$ be an arbitrary non-annihilating homomorphism and $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ be a locally compact $T_{1}$-semitopological semigroup. Then either $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ is topologically isomorphic to $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau_{\textsf{AC}}\right)$ or $\tau$ is discrete.

Theorem 4.27 implies the following corollary.

Corollary 4.28**.**

Every locally compact $T_{1}$-topological semigroup $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau\right)$ with a non-annihilating homomorphism $\theta$ is the discrete space.

5. On the closure of the discrete semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)$ with a non-annihilating homomorphism $\theta$

The following proposition extends Theorem I.3 from [15].

Proposition 5.1**.**

Let $\theta$ be an arbitrary non-annihilating homomorphism and $\textbf{B}(\mathbb{Z}_{+},\theta)$ is a discrete dense subsemigroup of a semitopological monoid $S$ such that $I=S\setminus\textbf{B}(\mathbb{Z}_{+},\theta)\neq\varnothing$. Then $I$ is a two-sided ideal of $S$.

Proof.

Fix an arbitrary element $y\in I$. If $xy=z\notin I$ for some $x\in\textbf{B}(\mathbb{Z}_{+},\theta)$, then there exists an open neighbourhood $U(y)$ of the point $y$ in the space $S$ such that $\{x\}\cdot U(y)=\{z\}\subset\textbf{B}(\mathbb{Z}_{+},\theta)$. The neighbourhood $U(y)$ contains infinitely many elements of the semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)$. This contradicts Proposition 4.20. The obtained contradiction implies that $xy\in I$ for all $x\in\textbf{B}(\mathbb{Z}_{+},\theta)$ and $y\in I$. The proof of the statement that $yx\in I$ for all $x\in\textbf{B}(\mathbb{Z}_{+},\theta)$ and $y\in I$ is similar.

Suppose to the contrary that $xy=w\notin I$ for some $x,y\in I$. Then $w\in\textbf{B}(\mathbb{Z}_{+},\theta)$ and the separate continuity of the semigroup operation in $S$ implies that there exist open neighbourhoods $U(x)$ and $U(y)$ of the points $x$ and $y$ in $S$, respectively, such that $\{x\}\cdot U(y)=\{w\}$ and $U(x)\cdot\{y\}=\{w\}$. Since both neighbourhoods $U(x)$ and $U(y)$ contain infinitely many elements of the semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)$, both equalities $\{x\}\cdot U(y)=\{w\}$ and $U(x)\cdot\{y\}=\{w\}$ contradict mentioned above Proposition 4.20. The obtained contradiction implies that $xy\in I$. ∎

Later we need the following trivial lemma, which follows from separate continuity of the semigroup operation in semitopological semigroups.

Lemma 5.2**.**

Let $S$ be a semitopological semigroup and $I$ be a compact ideal in $S$. Then the Rees-quotient semigroup $S/I$ with the quotient topology is a Hausdorff semitopological semigroup.

Theorem 5.3**.**

Let $\theta$ be an arbitrary non-annihilating homomorphism and $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$ be a locally compact semitopological semigroup, where $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}=\textbf{B}(\mathbb{Z}_{+},\theta)\sqcup I$ and $I$ is a compact ideal of $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}$. Then either $(\textbf{B}(\mathbb{Z}_{+})_{I},\tau)$ is compact or $I$ is an open subset of $(\textbf{B}(\mathbb{Z}_{+})_{I},\tau)$.

Proof.

Since $I$ is a compact ideal of $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}$, Corollary 3.3.10 of [17] implies that $\textbf{B}(\mathbb{Z}_{+},\theta)$ is a locally compact subspace of $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$, and hence by Corollary 4.22, $\textbf{B}(\mathbb{Z}_{+},\theta)$ is the discrete space.

Suppose that $I$ is not open. By Lemma 5.2 the Rees-quotient semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I$ with the quotient topology $\tau_{\operatorname{\textsf{q}}}$ is a semitopological semigroup. Let $\pi\colon\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\to\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I$ be the natural homomorphism which is a quotient map. It is obvious that the Rees-quotient semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I$ is isomorphic to the semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$ and the image $\pi(I)$ corresponds zero 0 of $\textbf{B}(\mathbb{Z}_{+},\theta)^{0}$. Now we shall show that the natural homomorphism $\pi\colon\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\to\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I$ is a hereditarily quotient map. Since $\pi(\textbf{B}(\mathbb{Z}_{+},\theta))$ is a discrete subspace of $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I,\tau_{\operatorname{\textsf{q}}})$, it is sufficient to show that for every open neighbourhood $U(I)$ of the ideal $I$ in the space $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$ we have that the image $\pi(U(I))$ is an open neighbourhood of the zero 0 in the space $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I,\tau_{\operatorname{\textsf{q}}})$. Indeed, $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\setminus U(I)$ is an open-and-closed subset of $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$, because by Corollary 4.22 the elements of the semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)$ are isolated point of $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$. Also, since the restriction $\pi|_{\textbf{B}(\mathbb{Z}_{+},\theta)}\colon\textbf{B}(\mathbb{Z}_{+},\theta)\to\pi(\textbf{B}(\mathbb{Z}_{+},\theta))$ of the natural homomorphism $\pi\colon\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\to\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I$ is one-to-one, $\pi(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\setminus U(I))$ is an open-and-closed subset of $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I,\tau_{\operatorname{\textsf{q}}})$. So $\pi(U(I))$ is an open neighbourhood of the zero 0 of the semigroup $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I,\tau_{\operatorname{\textsf{q}}})$, and hence the natural homomorphism $\pi\colon\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\to\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I$ is a hereditarily quotient map. Since $I$ is a compact ideal of the semitopological semigroup $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$, $\pi^{-1}(y)$ is a compact subset of $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$ for every $y\in\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I$. By Din’ N’e T’ong’s Theorem (see [14] or [17, 3.7.E]), $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I,\tau_{\operatorname{\textsf{q}}})$ is a Hausdorff locally compact space. If $I$ is not open then by Theorem 4.27 the semitopological semigroup $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I,\tau_{\operatorname{\textsf{q}}})$ is topologically isomorphic to $(\textbf{B}(\mathbb{Z}_{+},\theta)^{0},\tau_{\operatorname{\textsf{Ac}}})$ and hence it is compact. Next we shall prove that the space $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$ is compact. Let $\mathscr{U}=\left\{U_{\alpha}\colon\alpha\in\mathscr{I}\right\}$ be an arbitrary open cover of $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$. Since $I$ is compact, there exist $U_{\alpha_{1}},\ldots,U_{\alpha_{n}}\in\mathscr{U}$ such that $I\subseteq U_{\alpha_{1}}\cup\cdots\cup U_{\alpha_{n}}$. Put $U=U_{\alpha_{1}}\cup\cdots\cup U_{\alpha_{n}}$. Then $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\setminus U$ is an open-and-closed subset of $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$. Also, since the restriction $\pi|_{\textbf{B}(\mathbb{Z}_{+},\theta)}\colon\textbf{B}(\mathbb{Z}_{+},\theta)\to\pi(\textbf{B}(\mathbb{Z}_{+},\theta))$ of the natural homomorphism $\pi\colon\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\to\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I$ is one-to-one, $\pi(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\setminus U(I))$ is an open-and-closed subset of $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I,\tau_{\operatorname{\textsf{q}}})$, and hence the image $\pi(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\setminus U(I))$ is finite, because the semigroup $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I}/I,\tau_{\operatorname{\textsf{q}}})$ is compact. Thus, the set $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}\setminus U$ is finite and hence the space $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$ is compact as well. ∎

Theorem 5.3 implies the following corollary.

Corollary 5.4**.**

Let $\theta$ be an arbitrary non-annihilating homomorphism and $(\textbf{B}(\mathbb{Z}_{+},\theta)_{I},\tau)$ be a locally compact topological semigroup, where $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}=\textbf{B}(\mathbb{Z}_{+},\theta)\sqcup I$ and $I$ is a compact ideal of $\textbf{B}(\mathbb{Z}_{+},\theta)_{I}$. Then $I$ is an open subset of $(\textbf{B}(\mathbb{Z}_{+})_{I},\tau)$.

Acknowledgements

The author acknowledges Alex Ravsky and the referee for their important comments and suggestions.