Abstract
Full Text
UDC 519.46
MATHEMATICS
L. B. SHNEPERMAN
REPRESENTATIONS OF TOPOLOGICAL SEMIGROUPS BY CONTINUOUS TRANSFORMATIONS
(Presented by Academician A. I. Mal’cev on July 3, 1965)
1. A representation of a topological semigroup \(A_t\) (2) in a topological semigroup \(A_{t'}'\) will mean a continuous homomorphic mapping of \(A_t\) into \(A_{t'}'\). We shall say that a topological semigroup \(A_t\) is representable in a class of topological semigroups \(\Sigma\) if there exists a representation of \(A_t\) in some semigroup from the class \(\Sigma\). If there exists a topological isomorphism (2) of \(A_t\) onto some topological semigroup from the class \(\Sigma\), then we shall say that \(A_t\) is \(i\)-representable in the class \(\Sigma\).
In this paper it is established that every topological semigroup is \(i\)-representable in the class of topological semigroups of continuous transformations of topological spaces (4). Sufficient conditions are found for an \(i\)-representation of topological semigroups in the class of semigroups of continuous transformations of bicompacts (7), and it is shown that complete regularity is a necessary condition for such a representation (5). Further, it is established that every bicompact topological semigroup is representable in the class of semigroups of continuous transformations of compact spaces (8), and at the same time in the class of semigroups of continuous transformations of compact subsets of the Hilbert cube (10). The sufficiency of the system of such representations (9) is shown, which makes it possible to solve Ulam’s problem (5) on the construction of a universal bicompact topological semigroup (11).
2. A semigroup \(A\), on the set of elements of which a topology \(t\) is defined, will be denoted by \(A_t\). In the case when the semigroup operation is continuous with respect to this topology, the semigroup \(A_t\) is called topological. The semigroup \(A_t\) is called bicompact, completely regular, etc., when its topological space is bicompact, completely regular, etc. The semigroups \(A_t\) and \(A_{t'}'\) are called topologically isomorphic if there exists a mapping \(\varphi\) which is an isomorphism of the algebraic semigroups \(A\) and \(A'\) and a homeomorphism of their topological spaces.
Let \(A_t\) be a semigroup with topology \(t\). An isomorphism \(\varphi\) of the (algebraic) semigroup \(A\) onto the semigroup \(A'\) induces on the set of elements of the semigroup \(A'\) a topology, which will be denoted by \(\varphi(t)\).
3. Let \(X\) be a topological space with topology \(t\), and let \(S\) be some semigroup of continuous transformations of \(X\). The topology \(t\) induces on the semigroup \(S\) the bicompact-open topology (2). Namely, let \(B_1,\ldots,B_n; U_1,\ldots,U_n\), where \(n\) is a finite number, be subsets of \(X\); all \(B_i\) bicompact, and the \(U_i\) open. By \((B_1,\ldots,B_n; U_1,\ldots,U_n)\) we shall denote the set of all transformations \(s \in S\) for which \(\forall_{1 \le i \le n} sB_i \subset U_i\). The system of all sets \((B_1,\ldots,B_n; U_1,\ldots,U_n)\) may be taken as a base of open sets of the topology on \(S\). This topology will everywhere below be denoted by \(t\). In the case when \(X\) is a compact metric space, this topology on \(S\) coincides with the natural one generated by the metric on \(X\).
If the semigroup \(S^{\hat t}\) is topological, then we shall call it a topological semigroup of continuous transformations of the space \(X\).
Note that the semigroup \(S^{\hat t}\) is not always topological. In this connection, the following theorem is interesting and important.
Theorem 1. If \(X\) is bicompact, then the semigroup \(S^{\hat t}\) is topological.
This theorem makes it possible to speak of the semigroup of continuous transformations of a bicompactum without indicating each time that it is topological.
4. Let \(A_t\) be a topological semigroup. Without restricting the generality of the arguments, henceforth we shall everywhere assume that \(A_t\) contains an identity. Otherwise the identity \(e\) could be adjoined to the semigroup \(A\) externally \((^3)\), and to the topological space \(A_t\) as an isolated point.
To each element \(a \in A\) we assign the continuous transformation \(f_a\) of the space of the topological semigroup \(A_t\):
\[ \forall_{x \in A_t} f_a x = ax . \]
It is clear that, with respect to the operation of superposition of transformations, the set \(F(A_t)=\bigcup_{a\in A} f_a\) is a semigroup of continuous transformations of the space of the topological semigroup \(A_t\).
It is known that the mapping \(\varphi(a)=f_a\) of the semigroup \(A\) onto the semigroup \(F(A_t)\) is an isomorphism \((^3)\). This isomorphism induces on the semigroup \(F(A_t)\) the topology \(\varphi(t)\) (§ 2). On the other hand, on the semigroup \(F(A_t)\) one can construct the bicompact-open topology \(\hat t\) (§ 3).
Lemma 1. On the semigroup \(F(A_t)\) the topologies \(\varphi(\hat t)\) and \(\hat t\) coincide.
From Lemma 1 we immediately obtain
Corollary. The semigroup \(F(A_t)\) is topological.
From the same lemma it follows that
Theorem 2. Every topological semigroup is \(i\)-representable in the class of topological semigroups of continuous transformations of topological spaces.
Corollary. A bicompact topological semigroup is \(i\)-representable in the class of semigroups of continuous transformations of bicompacta.
5. The question naturally arises of the \(i\)-representation of topological semigroups in the class of semigroups of continuous transformations of bicompacta. First of all, let us note that not every topological semigroup is \(i\)-representable in this class, as is shown by the following
Theorem 3. A semigroup of continuous transformations of a bicompactum is completely regular.
6. We shall now find some sufficient conditions for the possibility of an \(i\)-representation of topological semigroups in the class of semigroups of continuous transformations of bicompacta. We shall need a number of auxiliary assertions.
Lemma 2. Let \(X^*\) be the maximal bicompact extension of the topological space \(X\). If \(B\) is a closed set and \(U\) an open set in \(X^*\), and \(B\cap X\) and \(X\setminus (U\cap X)\) are functionally separable, then \(B\subset U\).
From the results \((^{1,4})\) one can obtain
Lemma 3. Let \(X\) be a completely regular topological space, \(X^*\) its maximal bicompact extension, and \(f\) a continuous mapping of \(X\) into \(X^*\). Then \(f\) can be extended to a continuous mapping \(f^*\) of the bicompactum \(X^*\) into itself.
7. Let \(U\) be open, \(Y\) an arbitrary subset of the space of the topological semigroup \(A_t\), and \(a\in A\). We shall call the topological semigroup \(A_t\) strongly continuous on the left if whenever \(aY\) and \(A_t\setminus U\) are functionally separable, there exists a neighborhood \(V\)
point \(a\), such that for any \(a' \in V\) the closed sets \(\overline{a'Y}\) and \(A_t \setminus U\) are functionally separated.
Lemma 4. If \(A_t\) is a completely regular strongly left-continuous topological semigroup, \(U\) is open, \(Y\) is an arbitrary subset of its topological space, and \(f_a \in F(A_t)\) (Sec. 4) is such that \(\overline{f_aY}\) and \(A_t \setminus U\) are functionally separated, then there exists an open neighborhood \(V\) of the element \(f_a\) in \(\hat F_t(A_t)\) such that, for any \(f \in V\), the closed sets \(\overline{fY}\) and \(A_t \setminus U\) are functionally separated.
With the aid of Lemmas 2, 3, and 4 one proves
Theorem 4. A completely regular strongly left-continuous topological semigroup is \(i\)-representable in the class of semigroups of continuous transformations of bicompacta.
Sec. 8. We now turn to the question of representations of bicompact topological semigroups.
In what follows we shall need the following
Lemma 5. If a semigroup \(A'_{t'}\), with separable topology \(t'\), is the image of a bicompact topological semigroup \(A_t\) under a continuous homomorphic mapping, then \(A'_{t'}\) is a bicompact topological semigroup.
Let \(A_t\) be a bicompact topological semigroup, and let \(\varphi\) be some continuous mapping of the space \(A_t\) into the interval \([0,1]\). To each element \(a\) of the semigroup \(A\) we assign the continuous mapping \(\varphi_a\) of the space \(A_t\) into the interval \([0,1]\):
\[ \forall_{x \in A}\varphi_a(x)=\varphi(xa). \]
The set \(J_\varphi=\{\varphi_a\}a\in A\) is a metric space with respect to the metric \(r\):
\[ r(\varphi_a,\varphi_b)=\max_{x\in A}|\varphi_a(x)-\varphi_b(x)|, \]
and the metric space \(J_\varphi\) is compact.
On the compactum \(J_\varphi\) we define multiplication:
\[ \forall_{\varphi_a,\varphi_b\in J_\varphi}\varphi_a\cdot\varphi_b=\varphi_{ab}. \]
Then the mapping \(f(a)=\varphi_a\) of the topological semigroup \(A_t\) onto \(J_\varphi\) will be a continuous homomorphism and, by Lemma 5, the semigroup \(J_\varphi\) is topological.
From all that has been said it follows that
Theorem 5. For every bicompact topological semigroup there exists a representation in a compact metric semigroup.
Sec. 9. We shall say that a topological semigroup \(A_t\) admits a sufficient system of representations in the class of topological semigroups \(\Sigma\), if for any two elements \(x,y\in A_t\) there exists a representation \(f\) of the topological semigroup \(A_t\) in a topological semigroup from the class \(\Sigma\), for which \(f(x)\ne f(y)\).
Theorem 6. A bicompact topological semigroup \(A_t\) admits a sufficient system of representations in the class of metrizable compact semigroups.
Sec. 10. From the results of Sec. 4 it follows directly that a compact metric semigroup is \(i\)-representable in the class of semigroups of continuous transformations of compacta. In turn, every compactum is homeomorphic to some subset of the Hilbert cube. Together with Theorem 6 this gives the following theorem:
Theorem 7. A bicompact topological semigroup \(A_t\) admits a sufficient system of representations in the class of semigroups of continuous transformations of compact subsets of the Hilbert cube.
Sec. 11. S. Ulam \((^5)\) proposed the following problem: does there exist a universal bicompact topological semigroup, i.e. a topological-
topological semigroup \(A_t\) such that every bicompact topological semigroup is continuously isomorphic to a subsemigroup of it?
From item 10 it follows immediately:
Theorem 8. Every bicompact topological semigroup \(A_t\) is topologically isomorphic to some subsemigroup of the direct product of all semigroups of continuous transformations of compact subsets of the Hilbert cube.
Barnaul State
Pedagogical Institute
Received
28 VI 1965
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