On monoids of monotone injective partial selfmaps of integers with cofinite domains and images
semigroup of bijective partial transformations
To state the main results of this paper, first we need to recall some necessary definitions. Then we state the main results. For a topological space \(X\), a family \(\{A_s|s\in\mathcal{A}\}\) of subsets of \(X\) is called \textit{locally finite} if for every point \(x\in X\), there exists an open neighborhood \(U\) of \(x\) in \(X\) such that the set \(\{s\in \mathcal{A}|U\cap A_s\neq \emptyset\}\) is finite. A subset \(A\) of \(X\) is said to be \textit{codense} in \(X\) if \(X\setminus A\) is dense in \(X\). A space \(X\) is said to be a \textit{Baire space} if for each sequence \(A_1,\cdots,A_i,\cdots\) of nowhere dense subsets of \(X\), the union \(\bigcup_{i= 1}^\infty A_i\) is a codense subset of \(X\). A semigroup \(S\) is called \textit{bisimple} if \(S\) has only one \(\mathcal{D}\)-class, where \(\mathcal{D}=\mathcal{L}\circ \mathcal{R}\) (\(a\mathcal{R} b\) (\(a\mathcal{L} b\)) if and only if \(aS^1=bS^1\) (\(S^1a=S^1b\))). A \textit{semitopological} (resp. \textit{topological}) \textit{semigroup} is a Hausdorff topological space together with a separately (resp. jointly) continuous semigroup operation. An inverse topological semigroup with continuous inversion is called \textit{topological inverse semigroup}. A Hausdorff topology \(\tau\) on an (inverse) semigroup \(S\) such that \((S,\tau)\) is a topological inverse semigroup is called (\textit{inverse}) \textit{semigroup topology}. Let \(\mathbb{Z}\) be the set of integers with the usual order \(\leq\). A partial map \(\alpha:Z\to Z\) is said to be \textit{monotone} if \(n\leq m\) implies that \((n)\alpha\leq (m)\alpha\) for \(n,m\in\mathbb{Z}\). By \(\mathcal{F}^\nearrow_\infty(\mathbb{Z})\) we denote a subsemigroup of injective partial monotone selfmaps such that \(|\mathbb{Z}\setminus \mathrm{dom} \alpha|\) and \(|\mathbb{Z}\setminus \mathrm{ran} \alpha|\) are finite.NEWLINENEWLINEIn this paper, the authors study the semigroup \(\mathcal{F}^\nearrow_\infty(\mathbb{Z})\) and they show that \(\mathcal{F}^\nearrow_\infty(\mathbb{Z})\) is bisimple and all of its non-trivial semigroup homomorphisms are either isomorphisms or group homomorphisms. Also they prove that every Baire topology \(\tau\) on \(\mathcal{F}^\nearrow_\infty(\mathbb{Z})\) such that \((\mathcal{F}^\nearrow_\infty(\mathbb{Z}),\tau)\) is a Hausdorff semitopological semigroup is discrete and they construct a non-discrete Hausdorff inverse semigroup topology \(\tau_w\) on \(\mathcal{F}^\nearrow_\infty (\mathbb{Z})\). As the main results in this note, the authors prove that the discrete semigroup \(\mathcal{F}^\nearrow_\infty(\mathbb{Z}) \) cannot be embedded into some classes of compact-like topological semigroups and they prove the following:NEWLINENEWLINETheorem. Let \(S\) be a topological semigroup which contains \(\mathcal{F}^\nearrow_\infty (\mathbb{Z})\) as a dense discrete subsemigroup. If \(I=S\setminus \mathcal{F}^\nearrow_\infty (\mathbb{Z})\neq\emptyset\) then \(I\) is an ideal of \(S\).NEWLINENEWLINECorollary. If a topological semigroup \(S\) satisfies one of the following conditions: (i) \(S\) is compact; (ii) the square \(S\times S\) is countably compact; (iii) \(S\) is a countably compact topological inverse semigroup, or (iv) the square \(S\times S\) is a Tychonoff pseudocompact space, then \(S\) does not contain the semigroup \(\mathcal{F}^\nearrow_\infty (\mathbb{Z})\) (and hence the semigroup).
scientific article; zbMATH DE number 6092469