Full Text
UDC 519.4 + 513.88
MATHEMATICS
M. G. RABINOVICH
ON A CLASS OF STRUCTURES WITH OPERATORS AND A NEW CHARACTERISTIC OF THE POSITIVE PART OF A \(K\)-SPACE
(Presented by Academician L. V. Kantorovich, 25 VII 1966)
In the work of A. G. Pinsker \((^1)\) a characteristic of the cone of positive elements of a \(K\)-space is given, and it is proved that, in essence, this object is determined only by the properties of the additive operation and of the order relation \((^2)\). In the present note it is shown that in the characteristic of the positive part of a \(K\)-space one may dispense altogether with the definition of the additive operation, relying on the properties of the order and of the operation of multiplication by a number. At the same time, some properties of one class of structures with operators are investigated. Where no special definitions are given, all terminology is borrowed by us from the book \((^3)\).
Definition 1. A set \(X\) is called an \((l,\lambda)\)-system if the following conditions are satisfied:
L1. \(X\) is a conditionally complete lattice with least element \(0\).
L2. On \(X\) there is defined an external law of composition \((\lambda,x)\to \lambda x\), \(R_+^1\times X\to X\), where \(R_+^1\) is the set of nonnegative real numbers, with
\[
\lambda(\mu x)=(\lambda\mu)x,\quad 1\cdot x=x.
\]
L3. From \(x\ge y,\ \lambda\ge \mu\) it follows that \(\lambda x\ge \mu y\), and, if \(nx\le y\) for every \(n\), then \(x=0\).
Example 1. The cone of positive elements of a \(K\)-space.
Example 2. The collection, ordered by inclusion, of all convex linearly bounded \((^4)\) subsets of a vector space that contain the zero vector.
Example 3. The collection, ordered by inclusion, of all closed linearly bounded star-shaped subsets of a topological vector space that contain the zero vector. A set \(A\) is called star-shaped if from \(a\in A\) it follows that \(\lambda a\in A\) for all \(0\le \lambda\le 1\).
Example 4. The set of real functions on \([0,1]\) generated by the two functions \(f_1(x)=1\) and \(f_2(x)=x\) by means of the operations of multiplication by a number, \(\sup\), and \(\inf\).
Lemma 1. a) If \(x=\sup_\alpha x_\alpha\), then \(\lambda x=\sup_\alpha \lambda x_\alpha\); b) if \(y=\inf_\beta y_\beta\), then \(\mu y=\inf_\beta \mu y_\beta\); c) if \(x\,d\,y\), i.e. \(x\wedge y=0\), then \(\lambda x\,d\,\mu y\) for arbitrary \(\lambda,\mu\in R_+^1\).
Lemma 2. a) If \(x\ne 0\), then \(\lambda x>\mu x\) for arbitrary \(\lambda>\mu\); b) if \(x>y\), then \(\lambda x>\lambda y\) for arbitrary \(\lambda>0\).
Definition 2. A subset \(X_1\) of an \((l,\lambda)\)-system \(X\) is called a component in \(X\) if \(X_1\) is a proper and normal sublattice in \(X\), and the operation of multiplication by a number does not lead outside \(X_1\).
In order that the concept of a component in an \((l,\lambda)\)-system be meaningful, we introduce the following axiom:
D. If \(x\le \sup_\alpha y_\alpha\) and \(x\,d\,y_{\alpha_0}\), then
\[
x\le \sup_{\alpha\ne \alpha_0} y_\alpha .
\]
Let us note that this condition is weaker than the infinite distributive law
\[
x\wedge \sup_\alpha y_\alpha=\sup_\alpha (x\wedge y_\alpha).
\]
Axiom D is satisfied in all the examples except Example 2.
Lemma 3. a) The disjoint complement of any set in \(X\) is a component in \(X\); b) in \(X\) there exists a complete set of pairwise disjoint components.
Definition 3. Let \(X_0\) be a component in \(X\). For any \(x \in X\) put
\[ \operatorname{pr}_{X_0} x=\sup \{y:\ y\in X_0,\ y\leq x\}. \]
Lemma 4. Let \(X_0\) be a component in \(X\). Then: a) \(\operatorname{pr}_{X_0}x\leq x\),
\(\operatorname{pr}_{X_0}\operatorname{pr}_{X_0}x=\operatorname{pr}_{X_0}x\), \(x\in X_0\) is equivalent to \(\operatorname{pr}_{X_0}x=x\); b) \(x\,d\,y\) implies \(\operatorname{pr}_{X_0}x\,d\,\operatorname{pr}_{X_0}y\); \(x\,d\,X_0\) is equivalent to \(\operatorname{pr}_{X_0}x=0\).
As examples show, the single condition D does not ensure “good” properties of the projection operator in an \((l,\lambda)\)-system. Therefore we introduce two more axioms.
A1. If \(z \vee y > y\), then there is a component \(X_\alpha \subset X\) such that \(\operatorname{pr}_\alpha y \leq z\).
A2. If \(x>y\), then there is \(z\): \(z\vee y>y\) and \(x>\lambda z\), where \(\lambda>1\).
Theorem 1. Let the \((l,\lambda)\)-system \(X\) satisfy D, A1, A2, and let the system of components \(\{X_\alpha\}\) form a decomposition of \(X\) \((^3)\). Then for every \(x\in X\) the equality \(x=\sup_\alpha\{\operatorname{pr}_\alpha x\}\) holds. If, however, \(x=\sup_\alpha x_\alpha\), where \(x_\alpha\in X_\alpha\), then \(x_\alpha=\operatorname{pr}_\alpha x\).
Definition 4. An element \(a\in X\) is called a unit if \(X_a=X\), where \(X_a\) is the component generated by \(a\) \((^3)\). The projection of a unit onto a component is called a unit element.
Lemma 5. In \(X\) there exists a complete set of pairwise disjoint components with units.
Let now \(X\) contain a unit.
Lemma 6. For every \(x\in X\) there exist such a unit element \(e\) and a number \(\alpha\geq 0\) that \(x>\alpha e\).
Let \(x\) be an arbitrary element of \(X\). Denote
\[ K_n(x)= \left\{ \bigvee_{i=1}^{n}\lambda_i e_i:\ e_i\,d\,e_j,\ \bigvee_{i=1}^{n}\lambda_i e_i\leq x \right\}. \]
Here \(\{e_i\}_{1\leq i\leq n}\) are all possible decompositions of the unit into \(n\) unit elements, and for a given \(e_i\)
\[ \lambda_i=\sup\{\lambda:\ \lambda e_i\leq x\}. \]
We note that it is possible that \(\lambda_i=0\). Denote
\[ K(x)=\bigcup_{n=1}^{\infty}K_n(x). \]
Theorem 2. For every \(x\in X\)
\[ x=\sup K(x). \]
We now define an addition operation in \(X\). Let \(x,y\in X\). Put
\[ K_n(x,y)= \left\{ \bigvee_{i=1}^{n}(\lambda_i+\mu_i)e_i:\ \bigvee_{i=1}^{n}\lambda_i e_i\in K_n(x),\ \bigvee_{i=1}^{n}\mu_i e_i\in K_n(y) \right\}; \]
\[ K(x,y)=\bigcup_{n=1}^{\infty}K_n(x,y). \]
Then, by definition,
\[ x+y=\sup K(x,y). \]
Lemma 7. The addition operation in \(X\) satisfies the following conditions:
\[ x+y=y+x,\qquad (x+y)+z=x+(y+z), \]
\[ (\lambda+\mu)x=\lambda x+\mu x,\qquad \lambda(x+y)=\lambda x+\lambda y,\qquad 0+x=x. \]
Lemma 8. a) \(x>y\) is equivalent to \(x=y+z,\ z\ne 0\); b) \(x+y\leq y+u\) implies \(x\leq u\).
Theorem 3. Let \(X\) be an \((l,\lambda)\)-system satisfying D, A1, A2. Then \(X\) is, up to isomorphism, the cone of positive elements of a \(K\)-space.
The proof of this theorem essentially relies on the theorem of the work \((^1)\) on a characterization of the positive part of a \(K\)-space.
Remark. The conditions D and A1, A2 are independent of one another, as follows from the examples given below.
Example 5. The inclusion-ordered collection of convex closed bounded subsets of the space \(R^n\) that contain zero is an \((l,\lambda)\)-system with conditions A1, A2, but without condition D.
Example 6. Let \(X\) be the join (3) of two components \(X_1\) and \(X_2\), where \(X_1\) is the function space of Example 4, and \(X_2\) is the set of nonnegative measurable functions on \([1,2]\). Then \(X\) is an \((l,\lambda)\)-system with D, but without A1 and A2.
Let us note that a trivial example of this kind is the \((l,\lambda)\)-system of Example 4 itself.
Example 7. The ordered collection of all bounded star-shaped subsets of \(R^n\) that contain zero is an \((l,\lambda)\)-system with D and A1, but without A2. Here the exact upper bound of a bounded collection of star-shaped subsets is taken to be their union (which is a bounded star-shaped subset), and the exact lower bound is their intersection.
Since, obviously, conditions D, A1, A2 are necessary in a \(K\)-space, Theorem 3 gives a characterization of the cone of positive elements of a \(K\)-space in terms of the order relation and multiplication by a number.
Leningrad Engineering-Economic Institute
named after Palmiro Togliatti
Received
12 VII 1966
CITED LITERATURE
- A. G. Pinsker, Uch. zap. Leningradsk. gos. ped. inst. im. A. I. Gertsena, 86, 285 (1949).
- A. G. Pinsker, ibid., 86, 235 (1949).
- L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semi-Ordered Spaces, Moscow—Leningrad, 1950.
- V. Klee, Proc. Am. Math. Soc., 7, 4 (1956).