Boolean-Valued Introduction to the Theory of Vector Lattices

The book treats Boolean valued analysis. This term signifies the technique of studying properties of an arbitrary mathematical object by means of comparison between its representations in two different set-theoretic models whose construction utilizes principally distinct Boolean algebras. As these models, we usually take the classical Cantorian paradise in the shape of the von Neumann universe and a specially-trimmed Boolean valued universe in which the conventional set-theoretic concepts and propositions acquire bizarre interpretations. Exposition focuses on the fundamental properties of order bounded operators in vector lattices. This volume is intended for the classical analyst seeking new powerful tools and for the model theorist in search of challenging applications of nonstandard models of set theory.

Vector Lattices and Integral Operators, 1996

Illinois journal of mathematics

BOOLEAN ALGEBRAS IN ANALYSIS foundation and always stand on the grounds of naive set theory, using the axiom of choice and its equivalents with no circumlocution. The author tried to avoid any significant intersections with other books on the theory of Boolean algebras. Among the latter we must mention the celebrated monographs by R. Sikorski and P. Halmos as well as the huge recent treatise Handbook of Boolean Algebras edited by D. Monk and R. Bonnet. The existence of these books allows the author to concentrate on the metric aspects of the theory, especially in the concluding chapters. Unfortunately, the so-called "Boolean valued analysis," a rapidly developed area of nonstandard analysis residing at the frontiers with algebra and model theory, falls completely beyond the limits of this book. However, it is beyond a doubt that more monographs and textbooks on Boolean valued analysis will appear soon, since this area of analysis is undergoing intensive study. The author intended to make the book comprehensible to readers with diverse mathematical interests, that is why he abstained from attempts to make the exposition concise and to include the maximal number of results in a minimal volume. The main facts are usually furnished with complete proofs; however, we expect an active attitude from the reader. As was mentioned, the first two chapters of the book are comprehensible to a novice; the remaining part of the book presumes two years of the academic training that includes acquaintance with the basics of measure theory and general topology. Formulating a theorem, we do not mention its author often; "mathematical folklore" occupies ample room in this book. I used the content of my previous book published in 1969 under the title Boolean Algebras mainly in compiling the first part of the present edition. D. A. Vladimirov Chapter 0 PRELIMINARIES ON BOOLEAN ALGEBRAS Each Boolean algebra is a partially ordered set of a special form. Therefore, we start with some general facts and concepts relating to order. 2 Indeed, assume for instance that x, y ∈ E s ∩ E si. Then x ≤ y since y ∈ E s and x ∈ E si. Analogously, we see that the reverse inequality y ≤ x holds. Hence, x = y. 3 We thus use the definite article the speaking of the least upper bound and greatest lower bound or the supremum and infimum of a set. (S. S. Kutateladze) 4 The terms "join" and "meet" are in common parlance for finite families. (S. S. Kutateladze) Preliminaries on Boolean Algebras 9 require that the entire Q and the empty set ∅ belong 8 to Σ, then Q and ∅ obviously play the roles of zero and unity in Σ. The zero and unity of a poset X are usually denoted by 0 and 1 and sometimes by 0 X and 1 X. However, even if we simultaneously consider several posets, the same symbols 0 and 1 are applied to each of these posets. 9 Moreover, throughout the book, we conventionally denote by E + the set of all positive elements of E. In a poset with zero and unity, it is natural to assume that the supremum of the empty set equals 0 and the infimum of this set equals 1. (Of course, for a nonempty set E, we always have the inequality sup E ≥ inf E.) 1.9 Examples We turn again to Example A (see 0.1.5) and suppose that the basic set Q is an interval [a, b] (with a < b) and the system Σ is constituted by all intervals lying in Q (open, half-open, and closed). The empty set also belongs to the system (as any interval of the form (p, p)). Denote the resulting partially ordered set by I. Since the intersection of each family of intervals is again an interval, each (not necessarily finite) subset E of I possesses the infimum that coincides with the intersection of E. A union of intervals need not be an interval; however, this does not mean the absence of suprema: for each system E ⊂ I , there is a least interval including all e ∈ E; this interval is the supremum of E. Therefore, the requirements of the definition of a lattice are redundantly satisfied in our case and I is a lattice. Obviously, I possesses zero and unity. However, this lattice is clearly not distributive. Another example: the inclusion-ordered totality of all open sets in an arbitrary topological space R is a lattice. The supremum is now interpreted as the union of sets, while the infimum of a finite family is its intersection. As above, there are zero and unity, namely, the empty set ∅ and the entire R. This lattice is obviously distributive. The closed sets, ordered by inclusion, constitute a lattice dually isomorphic to the former. 1.10 Disjoint elements. Complements Let X be a partially ordered set with zero 0. Elements x, y ∈ X are called disjoint 10 if x ∧ y = 0. For disjointness, it is necessary and

FLAP, 2020

This is an overview of the basic techniques and applications of Boolean valued analysis. Exposition focuses on the Boolean valued transfer principle for vector lattices and positive operators, Banach spaces and injective Banach lattices, AW ∗-modules and AW ∗-algebras, etc.

Journal of Soviet Mathematics, 1991

The survey is devoted to the presentation of the state of the art of a series of directions of the theory of order-bounded operators in vector lattices and in spaces of measurable functions. The theory of disjoint operators, the generalized Hewitt-Yosida theorem, the connection with p-absolutely summing operators are considered in detail.

Communications in Mathematical Physics, 1990

This is the first of a planned series of investigations on the theory of ordered spaces based upon four axioms. Two of these, the order (1.1.1) and the local structure (II.5.1) axioms provide the structure of the theory, and the other two [the identification (1.1.11) and cone (1.2.7) axioms] eliminate pathologies or excessive generality. In the present paper the axioms are supplemented by the nontriviality conditions (1.1.9) and a regularity property (II.4.2). The starting point is a nonempty set M and a family of distinguished subsets, called light rays, which are totally ordered. The order axiom provides the properties of this order. Positive and negative cones at a point are defined in terms of increasing and decreasing subsets and are used to extend the total order on the light rays to a partial order over all of M. The first significant result is the polygon lemma (1.2.3) which provides an essential constructive tool. A non-topological definition is found for the interiors of the cones; it leads to a "more homogeneous" partial order relation on M. In Sect. II, subsets called D-sets (Def. Π.2.2), possessing certain desirable properties, are studied. The key concept of perpendicularity of light rays is isolated (Def. II.3.1) and used to derive the basic "separation properties," provided that the interiors of cones are nonempty. It is shown that, in a D-set, "good" properties of one cone can be transported along light rays, so that the structure of a D-set is homogeneous. In particular, if one cone has nonempty interior, so have all others. However, the existence of even one cone with nonempty interior does not follow from the axioms, but has to be imposed as an additional regularity condition. The local structure axiom now states that every point lies in a regular D-set. It is proved that the family of regular D-sets is closed under finite intersections. The order topology is defined as the topology which has this family as a base. This topology is Hausdorlf, and coincides with the usual topology for Minkowski spaces. 0. Introduction Ordered spaces play a central role in theoretical physics. The ordering of time, for instance, induces an ordering of the space of events by the future light cone.

Contemporary Mathematics, 2017

We study the model theoretic strength of various lattices that occur naturally in topology, like closed (semi-linear or semi-algebraic or convex) sets. The method is based on weak monadic second order logic and sharpens previous results by Grzegorczyk. We also answers a question of Grzegorczyk on the 'algebra of convex sets'.