Reports of the Academy of Sciences of the USSR

1962, Volume 144, No. 5

MATHEMATICS

R. A. Demarr

TRANSFORMATIONS IN SEMIGROUPS AND MULTIVALUED MAPPINGS

(Presented by Academician P. S. Aleksandrov on 26 I 1962)

Let \(\Omega\) be an arbitrary topological space and \(R\) the ring of all bounded continuous functions defined on \(\Omega\); then to each continuous mapping \(f:\Omega \to \Omega\) there corresponds a homomorphism \(T:R\to R\), defined as follows: \(T(x)(\sigma)=X(f(\sigma))\) for all \(X(\cdot)\in R\) and \(\sigma\in\Omega\). Conversely, under certain conditions, to each homomorphism \(T:R\to R\) there corresponds a continuous mapping \(f:\Omega\to\Omega\). Thus, under these conditions the homomorphisms \(T:R\to R\) represent continuous mappings \(f:\Omega\to\Omega\). The aim of the present paper is to show how multivalued mappings may be represented as “generalized homomorphisms” in semigroups of a special type.

1. In what follows we shall need an additive semigroup \(S\), in which the following conditions are satisfied:

1) \(x+y=y+x;\ x,y\in S;\)

2) \((x+y)+z=x+(y+z);\ x,y,z\in S;\)

3) there exists a unique element \(0\in S\) such that \(x+0=x\) for every \(x\in S\).

We also assume that \(S\) is partially ordered:

4) \(0\leqslant x\) for all \(x\in S;\)

5) from the inequality \(x\leqslant y\) follows the inequality \(x+z\leqslant y+z;\ x,y,z\in S;\)

6) there exists an element \(e\in S\) \((e\ne 0)\) such that for any \(x\in S\) there is an integer \(n\) such that
\[ x\leqslant ne=\underbrace{e+\cdots+e}_{n\ \text{times}}; \]

7) if \(H\) is a nonempty subset of \(S\), then \(\inf H\) exists.

Remark. From condition 7) it follows that if \(H\) is a nonempty subset of \(S\) bounded above, then \(\sup H\) exists.

Examples. Let \(\Omega\) be an arbitrary topological space. Define the semigroup \(S_1=S_1(\Omega)\) as the collection of all bounded lower semicontinuous functions taking only nonnegative integer values. Thus, if \(X(\cdot)\in S_1\) and \(\sigma\in\Omega\), then \(X(\sigma)\in\{0,1,2,\ldots\}\), and the set \(\{\tau:x(\tau)>\alpha\}\) is open (\(\alpha\) is an arbitrary real number).

If the order and addition of elements in \(S_1\) are defined in the usual way, and if we choose \(0(\cdot), e(\cdot)\in S_1\) so that \(0(\sigma)=0\) and \(e(\sigma)=1\) for every \(\sigma\in\Omega\), then conditions 1)—7) are satisfied in \(S_1\).

The semigroup \(S_2=S_2(\Omega)\) is defined as the collection of all bounded upper semicontinuous functions taking only nonnegative integer values. If addition of elements and the order in \(S_2\) are defined in the usual way, then we take \(0(\cdot), e(\cdot)\in S_2\) so that \(0(\sigma)=0\) and \(e(\sigma)=1\) for any \(\sigma\in\Omega\). In this case conditions 1)—7) are satisfied in \(S_2\).

Definition. A nonempty set \(I\subset S\) is called an ideal if: 1) \(x+y\in I\), when \(x,y\in I\); 2) \(y\in I\), when \(y\leqslant x\) and \(x\in I\). The ideal \(I\) is called proper if \(I\ne S\).

Lemma 1. If \(I\) is an ideal, then \(I=S\) if and only if \(e\in I\).

It follows from this lemma that maximal proper ideals always exist. Further, if \(\Omega\) is a \(T_1\)-space and \(S_1=S_1(\Omega)\), then the collection of all maximal proper ideals with the appropriate topology is the Urysohn compactification of the space \(\Omega\).

Definition. A transformation \(T\) in a semigroup \(S\) is any mapping \(T\) of the semigroup \(S\) into itself having the following properties: 1) \(T(x+y)\leqslant T(x)+T(y)\) for all \(x,y\in S\); 2) \(T(nx)=nT(x)\) for every \(x\in S\) and \(n=0,1,2,\ldots\); 3) \(T(x)\leqslant T(y)\), if \(x\leqslant y\).

We shall call a transformation \(T:S\to S\) a generalized homomorphism if \(T^{-1}(I)\) is a proper ideal in the case where \(I\) is a proper ideal. The term “homomorphism” is justified by the fact that, if \(T\) were a homomorphism of a ring \(R\) (with identity) into itself, then \(T^{-1}(I)\) would be a proper ideal whenever \(I\) is a proper ideal.

It is easy to prove the following lemma:

Lemma 2. If \(T\) is a transformation in the semigroup \(S\) and \(e\leqslant T(e)\), then \(T\) is a generalized homomorphism.

Definition. A transformation \(T\) in the semigroup \(S\) is called lower semicontinuous if, for every nonempty \(H\subset S\), when \(H\) is directed upward and bounded, we have \(T(\sup H)=\sup T(H)\). A transformation \(T\) is called upper semicontinuous if, for every nonempty \(H\subset S\), we have \(T(\inf H)=\inf T(H)\), when \(H\) is directed downward.

1. We shall call \(g(\cdot)\) a multivalued mapping if to each \(\tau\in\Omega\) there corresponds a nonempty closed set \(g(\tau)\subset\Omega\).

Definition. A multivalued mapping \(g(\cdot)\) is called lower semicontinuous if, for all \(\sigma_0\in\Omega\) and any open set \(U\) \((g(\sigma_0)\cap U\ne\varnothing)\), there exists an open set \(V\) such that \(\sigma_0\in V\) and \(g(\sigma)\cap U\ne\varnothing\) for all \(\sigma\in V\).

Definition. A multivalued mapping \(g(\cdot)\) is called upper semicontinuous\(^*\) if, for all \(\sigma_0\in\Omega\) and any open set \(U\) \((g(\sigma_0)\subset U)\), there exists an open set \(V\) such that \(\sigma_0\in V\) and \(g(\sigma)\subset U\) for all \(\sigma\in V\).

Remark. If \(g(\cdot)\) is a lower (or upper) semicontinuous multivalued mapping and the set \(g(\sigma)\) consists of one point for every \(\sigma\in\Omega\), then \(g(\cdot)\) is equivalent to a single-valued continuous mapping.

Theorem 1. Let \(\Omega\) be an arbitrary topological space, \(S_1=S_1(\Omega)\), and let \(g(\cdot)\) be a lower semicontinuous multivalued mapping. If, for all \(x(\cdot)\in S_1\), \(\sigma\in\Omega\), we put

\[ T(x)(\sigma)=\sup\{x(\tau):\tau\in g(\sigma)\}, \]

then \(T\) will be a lower semicontinuous transformation of \(S_1\) into itself. \(T\) is a generalized homomorphism, since \(T(e)=e\). Conversely, to every lower semicontinuous generalized homomorphism \(T\) there corresponds a lower semicontinuous multivalued mapping \(g(\cdot)\).

Theorem 2. Let \(\Omega\) be a bicompact Hausdorff space, \(S_2=S_2(\Omega)\), and let \(g(\cdot)\) be an upper semicontinuous multivalued mapping. If, for each \(x(\cdot)\in S_2\) and each \(\sigma\in\Omega\), we put

\[ T(x)(\sigma)=\sup\{x(\tau):\tau\in g(\sigma)\}, \]

then \(T\) will be an upper semicontinuous transformation of \(S_2\) into itself. Since \(T(e)=e\), \(T\) is a generalized homomorphism. Conversely, to every upper semicontinuous generalized homomorphism \(T\) there corresponds an upper semicontinuous multivalued mapping \(g(\cdot)\).

\[ \text{* V. Gurevich }(1)\text{ introduced these mappings under the name continuous.} \]

3. As an application, let us consider the problem of extending multivalued mappings. Let \(\Omega\) be a bicompact Hausdorff space and \(S_2 = S_2(\Omega)\). Let \(\Omega_0\) be an everywhere dense subset of \(\Omega\), and let \(g_0(\cdot)\) be a multivalued mapping whose domain of definition is \(\Omega_0\). It is assumed that \(g_0(\cdot)\) is upper semicontinuous in the sense that, for every \(\sigma_0 \in \Omega_0\) and every open set \(U\) \((g_0(\sigma_0) \subset U)\), there exists an open set \(V\) such that \(\sigma_0 \in V\) and \(g_0(\sigma) \subset U\) for every \(\sigma \in V \cap \Omega_0\). The question arises of the possibility of extending \(g_0(\cdot)\) to a multivalued mapping \(g(\cdot)\) defined on the whole space \(\Omega\) and upper semicontinuous. The possibility of such an extension is shown as follows: for every \(x(\cdot) \in S_2\) and every \(\sigma \in \Omega_0\), put

\[ T_0(x)(\sigma) = \sup \{x(\tau): \tau \in g_0(\sigma)\}. \]

Then put

\[ T(x) = \inf \{y: y(\cdot) \in S_2,\ T_0(x)(\sigma) \leq y(\sigma)\ \text{for all } \sigma \in \Omega_0\}. \]

It can be shown that \(T\) is an upper semicontinuous generalized homomorphism, to which, according to Theorem 2, there corresponds an upper semicontinuous multivalued mapping \(g(\cdot)\), and moreover \(g(\sigma) = g_0(\sigma)\) for all \(\sigma \in \Omega_0\).

Thus, we have obtained an extension \(g(\cdot)\) of the multivalued mapping \(g_0(\cdot)\). It can also be shown that \(g(\cdot)\) is the minimal extension in the sense that, if \(h(\cdot)\) is any extension of the mapping \(g_0(\cdot)\), then \(g(\sigma) \subset h(\sigma)\) for every \(\sigma \in \Omega\).

REFERENCES

1. W. Hurewicz, Proc. Acad. Amsterdam, 29, 1014 (1926).