Mathematics

A. I. VEKSLER

ON THE ARCHIMEDEAN PRINCIPLE IN PARTIALLY ORDERED FACTOR-LINEALS

(Presented by Academician P. S. Aleksandrov on 10 IV 1958)

This note investigates the question of when the factor-lineal (X/N) of an Archimedean (K)-lineal (X) by its normal sublineal (N) turns out to be an Archimedean (K)-lineal*.

It is known that if (X) is an arbitrary (K)-lineal and (N) is its normal sublineal, then, identifying all elements of (X) that belong to one adjacency class modulo (N), we again obtain a (K)-lineal (X/N) (if in (X/N) the algebraic operations are defined in the natural way, and the partial order as follows: let (\tilde{x}\in X/N); we regard (\tilde{x}>0) if in (\tilde{x}) there is at least one (x'\in X) such that (x'>0) and (x'\in N)). This result is contained, for example, in ((^3)) (in the exercises to Ch. II, § 1). Here the mapping (X\to X/N), assigning to each (x\in X) that class (\tilde{x}\in X/N) in which (x) is contained, is a homomorphism with respect to the algebraic and lattice operations.

Let (X) be an Archimedean (K)-lineal; then (N) is also an Archimedean (K)-lineal; however (X/N) may fail to be Archimedean. Consider, for example, the factor (X/N) in the case where (X) is the extended (K)-space and (N=B(X)) is the subspace of its bounded elements. For any real (\lambda,\mu) we have (\lambda^2+\mu^2\geq \lambda\mu). Replacing (\lambda) in the inequality by (x), and (\mu) by (n1), and using the property of preservation of relations (((^1),) p. 126), we obtain (x^2+n^2 1\geq x\cdot n1=nx). Therefore, passing in this inequality to the images in (X/N), we shall have for any (x)
[
(\tilde{x}^{\,2})\geq \widetilde{nx}\quad\text{for } n=1,2,\ldots\quad (\text{since } n^2 1\in N),
]
i.e. in (X/N) every element turns out to be non-Archimedean.

Theorem 1 gives a necessary and sufficient condition for the factor-lineal (X/N) to be Archimedean. In Theorems 2–6 this general condition is simplified for various particular types of (K)-lineals.

Theorem 1. Let (X) be an Archimedean (K)-lineal, and (N) its normal sublineal. Then, in order that the factor-lineal (X/N) be Archimedean, it is necessary and sufficient that (N) satisfy the following condition:

A(_1). Let (x_n\in N), (x_n\geq 0) ((n=1,2,\ldots)), and let the sequence ({x_n}) be bounded in (X). Let, further, the numbers (\lambda_n\geq 0) and (\lambda_n\to 0). Then, if (x\in X) and (0\leq x'\leq y) for every (y) that is an upper bound of the set ({\lambda_n x_n}), then (x\in N).

A set (X') of a (K)-lineal (X) is called a component in (X) if there exists a set (E\subset X) such that (X') is the totality of all elements (x\in X) disjoint from (E). The component generated by some set (H\subset X) is the smallest component in (X) containing (H). In an Archimedean (K)-lineal the definition of component given here is equivalent to the definition of component in a (K)-space introduced in ((^1)) (p. 62).

Let (X') be a component in the (K)-lineal (X), (x\in X), (x\geq 0). Then (\sup x') over all (x'\in X'), (x'\leq x) (if such a supremum exists), is called the projection of (x) in (X').

* For the definition of a (K)-lineal and the formulation of the Archimedean principle, see, for example, ((^1)).

For an arbitrary (x \in X), its projection in (X') is the difference of the projections in (X') of its positive and negative parts.

Theorem 2. Let (X) be an Archimedean (K)-lineal in which there exists the projection of each element (y \in X) onto any component generated by an arbitrary element (x \in X); let (N) be its normal sublineal. Then, in order that the factor-lineal (X/N) be Archimedean, it is necessary and sufficient that (N) satisfy the following condition:

(A_2). Let (x_n \in N), (x_n \geqslant 0) ((n=1,2,\ldots)), (x_n d x_p) for (n \ne p), and let there exist (\sup x_n \in X). Let, further, (\lambda_n \geqslant 0) and (\lambda_n \to 0). Then, if (x \in X) and (0 \leqslant x \leqslant y) for any (y) which is an upper bound of the set ({\lambda_n x_n}), then (x \in N).

Remark. Denote by (A') ((A'')) the weakened condition which is obtained from (A_1) if in the latter one considers only pairwise disjoint (x_n) ((x_n) for which (\sup x_n) exists). Then it can be shown that in Theorem 1 the condition (A_1) cannot be replaced either by (A') or by (A''), i.e. in this sense the condition (A_1) in Theorem 1 cannot be brought closer to (A_2). Namely, let (X) be the (K)-lineal of all functions on ([0,1]) representable in the form
[
\alpha\left(\frac{1}{t-\frac14}+\frac{1}{t-\frac34}\right)+\varphi(t),
]
where (\varphi \in C_{[0,1]}), (\alpha) is a real number; (N) is the totality of all functions from (X) which vanish on ([0,1/2]) and for (t=3/4). Then one can verify that the normal sublineal (N) in (X) satisfies the conditions (A_2), (A'), and (A''), but does not satisfy (A_1).

From Theorem 2 Theorem 3 immediately follows.

Theorem 3. Let (X) be a (K')-space, (N) its normal (K')-subspace. Then, in order that the factor (X/N) be Archimedean, it is necessary and sufficient that (N) satisfy the following condition:

(A_3). Let (x_n \in N), (x_n \geqslant 0) ((n=1,2,\ldots)), (x_n d x_p) for (n \ne p), and let (\sup x_n) exist in (X). Let, further, (\lambda_n \geqslant 0) and (\lambda_n \to 0). Then (\sup \lambda_n x_n \in N).

Theorem 3 is valid, in particular, also for (K)-spaces and gives a condition under which the factor (X/N) turns out to be Archimedean; at the same time the factor nevertheless may fail to be a (K')-space (and even a (K)-space), as is seen from the example given at the end of the note.

Let (X) be a structure and (H \subset X). We shall say that (H) is structurally (\sigma)-closed if (H) is closed with respect to the operations (\sup) and (\inf), i.e. if the following condition is fulfilled in (H): let (x_n \in H) and let (x=\sup x_n) or (x=\inf x_n) exist in (X); then (x \in H).

Theorem 4. Let (X) be an extended (K)-space, (N) its normal subspace. Then, in order that the factor (X/N) be Archimedean, it is necessary and sufficient that (N) be structurally (\sigma)-closed. In this case (X/N) turns out to be a (K')-space.

If (X) is an extended (K)-space of countable type, then, in order that (X/N) be Archimedean, it is necessary and sufficient that (N) be a component in (X).

It is clear that if (X) is a (K)-space and (N) is its component, then the factor (X/N) is isomorphic to the complementary component and, consequently, will also be a (K)-space.

Theorem 5. Let (X) be a (K)-space with convergence with regulator*; let (N) be its normal subspace. Then, in order that the factor (X/N) be Archimedean, it is necessary and sufficient that (N) be structurally (\sigma)-closed.

If (X) is, moreover, of countable type, then, in order that the factor (X/N) be Archimedean, it is necessary and sufficient that (N) be a component in (X).

Remark. If one postulates the continuum hypothesis, then a (K)-space with convergence with regulator turns out to be of countable type (([^1],) p. 176). Then the theorem can be formulated as follows: if (X)—

* That is, ((o))-convergence in (X) has the property formulated in (([^1]), Chap. V, Theorem 1.24).

(K)-space with convergence with a regulator, and (N) is its normal subspace, then in order that (X/N) be Archimedean it is necessary and sufficient that (N) be a component.

We shall call a sublinear (T) of a (KB)-lineal (X) ((b))-closed if from (x_n \in T) and (x_n \xrightarrow{(b)} x \in X) it follows that (x \in T).

Theorem 6. Let (X) be an Archimedean (K)-lineal of bounded elements with unit, and let (N) be its normal sublinear. Then, in order that the quotient (X/N) be Archimedean, it is necessary and sufficient that (N) be ((b))-closed. Moreover, (X/N) turns out to be an Archimedean (K)-lineal of bounded elements with unit.

Let the (K)-lineal (X) under consideration be realized in the form of the space (\mathfrak{C}(Q)) of certain bounded continuous functions on a bicompact (Q) (such a realization is possible; see, for example, ((^2)), Theorem 1*). In this case Theorem 6 may be given the following form:

In order that (X/N) be Archimedean, it is necessary and sufficient that (N) satisfy the following condition:

In (N) there is the set of all functions from (\mathfrak{C}(Q)) that vanish on some (obviously, closed) set (Q_0 \subseteq Q).

Remark. For the (K)-space of bounded elements, the set of ((b))-closed normal subspaces considered in Theorem 6 coincides with the set of ((b))-closed ideals ((^4)).

In conclusion let us consider one more example. Let (X=m), i.e. (X) is the (K)-space of bounded sequences; (N=c_0) is the subspace of all sequences tending to zero. Obviously, (c_0) is a normal ((b))-closed subspace in (m); hence, by Theorem 6, (m/c_0) is an Archimedean (K)-lineal. We shall show that the quotient (m/c_0) is not even a (K)-space.

Let (x_n) be the unit element in (m), (x_n \overline{\in} c_0) ((n=1,2,\ldots)), and (x_n d x_p) for (n \ne p). Suppose that in (m/c_0) there exists (\widetilde{x}=\sup \widetilde{x}n). It is clear that (0<\widetilde{x}\leq 1), where (1) is the image of the unit (1\in m) in (m/c_0). Take (x\in \widetilde{x}) such that (01/2) ((\xi^{(n)}_p) is the (p)-th coordinate of the element (x'_n)). Denote by (e_p) the (p)-th unit vector in (m) and put (z_n=\alpha_n e). On the other hand,}), and (x''_n=x'_n-z_n). It is clear that (z_n\in c_0), (z_n>0), (x''_n\in \widetilde{x}_n), and (x''_n d x''_p) ((n\ne p)). Let (x''=\sup x''_n), (x_0=\sup z_n); obviously, (x_0\overline{\in} c_0). Now, on the one hand, we have (x''\geq x_n), whence (\widetilde{x}''\geq \widetilde{x
(x''=\sup x''_n=\sup{x'_n-z_n}=\sup x'_n-\sup z_n) (for (-x'_n>z_n>0) and (x'_n d x'_p) when (n\ne p)), (x''=\sup x'_n-x_0\leq x-x_0), i.e. (\widetilde{x}''\leq \widetilde{x}-\widetilde{x}_0<\widetilde{x}). The contradiction obtained proves our assertion.

The author expresses deep gratitude to his scientific adviser, Prof. B. Z. Vulikh, for valuable advice.

Leningrad State Pedagogical Institute
named after A. I. Herzen

Received
10 IV 1958

REFERENCES

1. L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semiordered Spaces, 1950.
2. B. Z. Vulikh, Izv. AN SSSR, Ser. Matem., 17, 4, 365 (1953).
3. N. Bourbaki, Éléments de mathématique, 13, livre VI, L’intégration, 1952.
4. T. T. Domracheva, Generalized Semiordered Rings and Their Ideals, Dissertation, Leningrad Pedagogical Institute named after Herzen, 1955.

* In ((^2)) a realization of (X) is given in the form (\mathfrak{C}(Q)) on a “minimal” bicompact (Q); namely, all points of the bicompact (Q) are functionally separated by means of (\mathfrak{C}(Q)), i.e. for any (t_1,t_2\in Q) ((t_1\ne t_2)) there exists a continuous function (x(t)) from (\mathfrak{C}(Q)) for which (x(t_1)\ne x(t_2)).