L. B. Shneperman

Semigroups of Continuous Transformations and Homeomorphisms of a Simple Arc

(Presented by Academician A. I. Mal’tsev on 23 V 1962)

1°. Semigroups of continuous transformations of topological spaces were studied in \((^{1-3})\). In \((^{1,2})\), in particular, an abstract characterization was obtained of the topological semigroup of all continuous transformations of a compactum, and in \((^3)\) also of the algebraic semigroup for a broad class of compacta; for this purpose the corresponding semigroups are embedded in certain supersemigroups. In the present work, for the case of a simple arc, the abstract characterization can be obtained intrinsically, without embeddings.

The semigroup of homeomorphisms of a simple arc was studied in detail in \((^4)\). The fact that it does not contain the semigroup \(\mathfrak H\) \((^6)\) is, probably, the principal reason making it difficult to obtain its abstract characterization. Here some additional information is given, and a construction of an adjoined ideal, which make it possible to give an abstract characterization of the semigroup of homeomorphisms of a simple arc.

2°. The necessary information from the general theory of semigroups is contained in \((^5)\); notation for semigroups with topology, the definitions of an idempotent element, of an inner shift, and some others—in \((^3)\). We recall only that semigroups \(\mathfrak A\) with operation \(XY=X\) \((X,Y\in\mathfrak A)\), and only they, belong to the class of semigroups \(\Sigma\).

3°. Let \(\mathfrak L\) be the minimal two-sided ideal of the semigroup \(\mathfrak A\). Define a family \(\Gamma\) of subsets of the set of elements \(\mathfrak L\): \(\mathfrak R\in\Gamma\) if and only if there exist elements \(L\in\mathfrak L\) and \(A\in\mathfrak A\) such that \(A\mathfrak R=L\) and \(A(\mathfrak L\setminus\mathfrak R)\ni L\).

By \(\tau\) we shall denote the topology on \(\mathfrak L\) for which the family \(\Gamma\) serves as a pseudobasis of closed sets.

4°. On \(\mathfrak L\) \((3^\circ)\) another inner topology will also be considered (i.e., a topology constructed only from the operation on the semigroup), for which the family of subsets
\[ M=\{A\mathfrak L\}_{A\in\mathfrak A} \]
is taken as a pseudobasis of closed sets. We shall denote this topology by \(\sigma\).

5°. Let now \(t\) be some topology on the minimal two-sided ideal \(\mathfrak L\) of the semigroup \(\mathfrak A\), and let \(\mathfrak L_t\) be a compact locally connected topological space. Extend the topology \(t\) from \(\mathfrak L_t\) to \(\mathfrak A\).

Denote by \(\Phi\) the set of all regions of the topological space \(\mathfrak L_t\). It is easy to show that there exists a finite system of regions covering \(\mathfrak L_t\). Let \(\{U_i\}\) be such a finite system. To an open neighborhood \(V_{\{U_i\}}(A)\) of an element \(A\in\mathfrak A\) we assign precisely those elements \(X\in\mathfrak A\) for which, for every \(L\in\mathfrak L_t\), the point \(XL\) belongs to at least one of the neighborhoods \(U_i(AL)\) of the point \(AL\). We take the set
\[ \{V_{\{U_i\}}(A)\}_{A\in\mathfrak A,\ \{U_i\}\in\Phi} \]
as a pseudobasis of open sets of a topology on \(\mathfrak A\). This topology will henceforth be denoted by \(\widetilde t\).

6°. Let \(\Omega\) be a simple arc; \(\mathfrak C\) the semigroup of all continuous transformations of \(\Omega\); \(\mathfrak H\) the subsemigroup of the semigroup \(\mathfrak C\) consisting of those and only those transformations which map \(\Omega\) into one point;

\(\mathfrak{H}\) is the semigroup of all homeomorphisms of \(\Omega\) into itself; \(\mathfrak{B}\) is the group of all homeomorphisms of \(\Omega\) onto itself.

The semigroup \(\mathfrak{H}\) belongs to the class of semigroups \(\Sigma\) (2°) and is a minimal two-sided ideal of the semigroup \(\mathfrak{C}\).

Let \(r\) be some fixed metric on \(\Omega\), compatible with the topology. Starting from this metric, define a metric \(r_0\) on the semigroup \(\mathfrak{C}\): for any \(S_1, S_2 \in \mathfrak{C}\) put
\[ r_0(S_1,S_2)=\max_{\xi\in\Omega} r(S_1\xi,S_2\xi). \]
The metric \(r_0\) induces on \(\mathfrak{C}\) a topology \(t_0\), which does not depend on the choice of the metric \(r\) on \(\Omega\). The topological spaces \(\mathfrak{H}_{t_0}\) and \(\Omega\) are homeomorphic.

7°. Let \(\Omega_t\) be a minimal two-sided ideal of the topological semigroup \(\mathfrak{A}_t\), \(\Omega \in \Sigma\) (2°); \(\Omega_t\) is a simple arc; the semigroup \(\mathfrak{A}\) contains no elements acting identically on the left on \(\Omega\), and on \(\mathfrak{A}\) the topologies \(t\) and \(\widetilde t\) coincide. If \(r\) is some metric on \(\Omega_t\), compatible with the topology, then, putting
\[ r(A_1,A_2)=\max_{L\in\Omega} r(A_1L,A_2L), \]
we extend it to \(\mathfrak{A}\). In this case the metric \(r\) on \(\mathfrak{A}_t\) will be compatible with the topology. If \(r'\) is another metric on \(\mathfrak{A}\), constructed in the same way, then the metric spaces \(\mathfrak{A}_r\) and \(\mathfrak{A}_{r'}\) will be simultaneously complete or incomplete. This makes it possible to speak of the completeness of the topological space \(\mathfrak{A}_t\), understanding by this the completeness of the metric space \(\mathfrak{A}_r\).

On the semigroup \(\mathfrak{C}_{t_0}\) this notion of completeness coincides with the usual one.

8°. Having fixed on \(\Omega_t\) (7°) one of the two possible natural orderings, the semigroup \(\mathfrak{A}\) can be ordered: for any \(A_1,A_2\in\mathfrak{A}\) put
\[ A_1\leqslant A_2 \leftrightarrow \forall_{L\in\Omega} A_1L\leqslant A_2L. \]
If, in this case, it turns out that \(\mathfrak{A}\) is a lattice (6), then under the other natural ordering of \(\Omega_t\) the semigroup \(\mathfrak{A}\) also forms a lattice.

On the semigroup \(\mathfrak{C}_{t_0}\) this ordering coincides with the usual one.

9°. The results of §§ 6°—8°, (3) and one proposition from (7) (p. 36) make it possible to give an abstract characterization of the topological semigroup \(\mathfrak{C}_{t_0}\).

Theorem. A topological semigroup \(\mathfrak{A}_t\) is topologically isomorphic to the topological semigroup \(\mathfrak{C}_{t_0}\) if and only if:

1) the semigroup \(\mathfrak{A}\) contains a minimal two-sided ideal \(\Omega\);
2) the semigroup \(\Omega\) belongs to the class of semigroups \(\Sigma\) (2°);
3) the semigroup \(\mathfrak{A}\) contains no elements acting identically on \(\Omega\) on the left;
4) for any four elements \(L_1,L_2,L',L''\in\Omega\), \(L_1\ne L_2\), there exists an element \(A\in\mathfrak{A}\) such that \(AL_1=L'\) and \(AL_2=L''\);
5) the topological space \(\Omega_t\) is a simple arc;
6) on the semigroup \(\mathfrak{A}\) the topology \(t\) coincides with the topology \(\widetilde t\) (5°);
7) the space \(\mathfrak{A}_t\) is complete (7°);
8) the semigroup \(\mathfrak{A}\) forms a lattice (8°).

10°. On the minimal two-sided ideal \(\mathfrak{H}\) of the semigroup \(\mathfrak{C}\) we construct the topology \(\tau\) (3°) and extend it to \(\mathfrak{C}\) (5°).

Theorem. On the semigroup \(\mathfrak{C}\) the topologies \(t_0\) (6°) and \(\widetilde{\tau}\) (5°) coincide.

11°. Theorem 10° makes it possible to assert that from the abstract characterization of the topological semigroup \(\mathfrak{C}_{t_0}\) one can always obtain an abstract characterization of the semigroup \(\mathfrak{C}\). For this it is enough to introduce on \(\mathfrak{C}\) the topology \(\widetilde{\tau}\), and then to require that the resulting topological semigroup satisfy the conditions of the corresponding theorem, for example Theorem 9°.

12°. Everywhere below \(\mathfrak{B}\) is a semigroup with left cancellation, containing an identity. It is easy to show that \(\mathfrak{B}\) is a semigroup with a separating group part, i.e.
\[ \mathfrak{B}=\mathfrak{G}\cup\mathfrak{R}, \]
where \(\mathfrak{G}\) is the universal maximal subgroup of the semigroup \(\mathfrak{B}\), and \(\mathfrak{R}\) is the maximal proper ideal of the semigroup \(\mathfrak{B}\), with
\[ \mathfrak{G}\cap\mathfrak{R}=\varnothing . \]

13°. Let \(\lambda\) be the relation of left divisibility on the semigroup \(\mathfrak B\): \((X,Y)\in\lambda \leftrightarrow \exists_{B\in\mathfrak B} X=YB\) for any \(X,Y\in\mathfrak B\).

Denote by \(\mathfrak P\) the set of all sequences \(\mathfrak X=\{X_1,X_2,\ldots\}\) from \(\mathfrak B\) satisfying the conditions: a) for all \(n\), \((X_{n+1},X_n)\in\lambda\); b) there is no element \(Z\in\mathfrak B\) such that, for all \(n\), \((Z,X_n)\in\lambda\).

It is not hard to verify that
\[ \{X_1,X_2,\ldots\}\in\mathfrak P \leftrightarrow \{BX_1,BX_2,\ldots\}\in\mathfrak P \]
for any \(B\in\mathfrak B\).

14°. On the set \(\mathfrak D=\mathfrak B\cap\mathfrak P\) introduce a multiplication operation: a) elements of \(\mathfrak B\) are multiplied as in the semigroup \(\mathfrak B\); b) if \(B\in\mathfrak B\) and \(\mathfrak X\in\mathfrak P\), then \(B\mathfrak X=\{BX_1,BX_2,\ldots\}\in\mathfrak P\) (13°); c) if \(Q\in\mathfrak D\) and \(\mathfrak X\in\mathfrak P\), then \(\mathfrak X Q=\mathfrak X\).

It is easy to verify that \(\mathfrak D\) is a semigroup with respect to the multiplication operation introduced. Moreover, the subsemigroup \(\mathfrak P\in\Sigma\) (2°) and is a minimal two-sided ideal of the semigroup \(\mathfrak D\).

15°. On the semigroup \(\mathfrak D\) introduce a binary relation \(\mu\): for any \(\mathfrak X,\mathfrak Y\in\mathfrak P\),
\[ (\mathfrak X,\mathfrak Y)\in\mu \leftrightarrow \forall_n\ (X_n,Y_n)\in\lambda . \]
Let \(\bar\mu\) be the equivalence closure of the relation \(\mu\) on \(\mathfrak D\) (8°).

Lemma. The equivalence relation \(\bar\mu\) is two-sided stable on the semigroup \(\mathfrak D\).

16°. Let \(\varphi\) be the natural homomorphism of the semigroup \(\mathfrak D\) onto the quotient semigroup \(\mathfrak D/_{\bar\mu}\). It is easy to show that, for all \(B_1,B_2\in\mathfrak B\),
\[ \varphi(B_1)=\varphi(B_2)\leftrightarrow B_1=B_2. \]
Therefore one may put \(\varphi(B)=B\). Denoting \(\varphi(\mathfrak P)=\mathfrak K\), we obtain \(\varphi(\mathfrak D)=\mathfrak B\cup\mathfrak K\).

It follows from 14° that the subsemigroup \(\mathfrak K\in\Sigma\) (2°) and is a minimal two-sided ideal of the semigroup \(\mathfrak B\). We shall call \(\mathfrak K\) the adjoined ideal of the semigroup \(\mathfrak B\).

17°. On the minimal two-sided ideal \(\mathfrak K\) of the semigroup \(\mathfrak B\cup\mathfrak K\) construct the topology \(\sigma\) (4°).

Lemma. If the topological space \(\mathfrak K_\sigma\) is compact, then every inner left shift of the semigroup \(\mathfrak B\cup\mathfrak K\) generated by an element \(B\in\mathfrak B\) is a homeomorphism of the space \(\mathfrak K_\sigma\) onto itself.

18°. Let \(\mathfrak K_\sigma\) be a compact locally connected topological space. Then the topology may be extended from \(\mathfrak K_\sigma\) to \(\mathfrak B\) (5°). In what follows, when considering a semigroup with left cancellation that contains an identity and its subsemigroups, we shall always assume that the topology \(\tilde\sigma\) on them is constructed precisely in this way.

19°. We pass to the study of the semigroup \(\mathfrak D\) (6°). Since \(\mathfrak D\) is a semigroup with left cancellation and contains the identity transformation, all the results of §§ 12°—18° can be carried over to it.

Moreover, note that
\[ (X,Y)\in\lambda \leftrightarrow X\Omega\subset Y\Omega \]
for any \(X,Y\in\mathfrak D\). Hence it is easy to obtain that a sequence \(\mathfrak X=\{X_1,X_2,\ldots\}\subset\mathfrak D\) belongs to the set of sequences \(\mathfrak P\) (13°) if and only if
\[ X_1\Omega \supset X_2\Omega \supset \cdots \]
and \(\bigcap X_n\Omega\) is a point. And from this assertion it follows:
\[ (\mathfrak X,\mathfrak Y)\in\mu \leftrightarrow \bigcap X_n\Omega=\bigcap Y_n\Omega \]
for any \(\mathfrak X,\mathfrak Y\in\mathfrak P\subset\mathfrak D\).

Thus, to every equivalence class of \(\bar\mu\) (15°) on \(\mathfrak P\) there corresponds a point \(\xi\in\Omega\), and conversely.

20°. Theorem. Let \(\mathfrak K\) be the adjoined ideal of the semigroup \(\mathfrak D\) (16°). The semigroups \(\mathfrak D\cup\mathfrak H\) (6°) and \(\mathfrak D\cup\mathfrak K\) are isomorphic.

21°. Theorem. On the semigroup \(\mathfrak D\), the topologies \(\tilde\sigma\) (18°) and \(t_0\) (6°) coincide.

22°. We shall call a topological group \(\mathfrak G_t\) complete in itself if, for every sequence \(G_1,G_2,\ldots\in\mathfrak G_t\), from the convergence of the sequences \(\{G_nG_m^{-1}\}\) and \(\{G_n^{-1}G_m\}\) to the identity of the group it follows that the sequence \(G_1,G_2,\ldots\) converges to \(G_0\in\mathfrak G_t\).

23°. Let \(\mathfrak K\) be the adjoined ideal of the semigroup \(\mathfrak B\) (16°), and let the topological space \(\mathfrak K_\sigma\) (17°) be a simple arc. Denote by \(K_0,K_1\)

its endpoints, and by \(\mathfrak G^{+}\) the subgroup of the group \(\mathfrak G\) (12°) consisting of elements \(X\) such that \(XK_0=K_0\). We also note that on the simple arc \(\mathfrak K_{\sigma}\) we shall regard as chosen and fixed one of the two possible natural orderings.

In the following proposition an abstract characterization of the semigroup \(\mathfrak D\) is given.

Theorem. The semigroup \(\mathfrak B\) is isomorphic to the semigroup \(\mathfrak D\) if and only if:

1) \(\mathfrak B\) is a semigroup with left cancellation, containing an identity;

2) the semigroup \(\mathfrak B\) contains no elements acting identically from the left on its adjoined ideal \(\mathfrak K\) (16°);

3) the topological space \(\mathfrak K_{\sigma}\) (17°) is a simple arc;

4) \(\mathfrak G\setminus \mathfrak G^{+}\ne\varnothing\);

5) for any \(2n\) elements \(K_0<K'_1<\cdots<K'_n<K_I\) and \(K_0<K''_1<\cdots<K''_n<K_I\) from \(\mathfrak K\) there exists an element \(G\in\mathfrak G^{+}\) such that, for all \(i\),
\[ GK'_i=K''_i; \]

6) the topological group \(\mathfrak G_{\widetilde{\sigma}}^{+}\) (18°) is complete in itself (22°).

24°. From Theorem 23° one easily obtains an abstract characterization of the semigroup \(\mathfrak D_{t_0}\).

Theorem. The topological semigroup \(\mathfrak B_t\) is topologically isomorphic to the topological semigroup \(\mathfrak D_{t_0}\) if and only if conditions 1)–6) of Theorem 23° are satisfied and

7) on the semigroup \(\mathfrak B\) the topology \(t\) coincides with the topology \(\widetilde{\sigma}\) (18°).

I express my gratitude to E. S. Lyapin for his constant attention and useful advice.

Leningrad State
Pedagogical Institute
named after A. I. Herzen

Received
19 V 1962

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