We present the notions of category theory which are of use in Boolean valued analysis and describe the Boolean toposes that form the backstages of Boolean valued universes.

The aim of the present article is to extend the Stone-Weierstrass theorem to functions ranging in a lattice normed space and admitting order rather than topological approximation. We proceed with the machinery of Boolean-valued transfer from lattice normed space to normed space.

We implement the Boolean valued analysis of Banach spaces. The realizations of Banach spaces in a Boolean valued universe are lattice normed spaces. We present the basic techniques of studying these objects as well as the Boolean valued approach to injective Banach lattices.

An erased Kantorovich space is the lattice ordered additive groupof a Kantorovich space. We study the special role of erased Kantorovich spacesin extending positive, dominated, and lattice homomorphismsand also the existence of unbounded polar-preserving group homomorphisms.Our method of study is Boolean-valued analysis.

— Based on long-term observations over the Ossetian regional GNSS network, which currently consists of 59 compaign sites and 7 stations, estimates of the strain rate and spatial distribution in the Ossetian part of the Greater Caucasus are obtained. The primary data analysis involves constraining the coordinates of the monitoring GNSS compaign sites by combining the regional network solution with the global solution calculated in global data centers. As many as 45 IGS stations included into the International Terrestrial Reference Frame (ITRF) are selected as fiducial stations. The strain-rate tensor is calculated using the Shen method. To interpret the geodynamic settings, information on regional seismicity acquired from the Earthquakes of Russia database of the Federal Research Center Geophysical Survey, Russian Academy of Sciences (RAS), is used. The results show that, in general, the Ossetian region is in the state of tectonic compression on top of which several characteristics of particular structures of the region are revealed. The Main Caucasian Ridge and the deflection of the southern slope are in the state of not only submeridional compression, but also sublatitudinal extension, which leads to an intense dilatant expansion of the eastern segment of this area. The strain pattern of the northern part of the Ossetian region differs from the southern one. The northern slope of the Main Caucasian Ridge zone and the foothill trough, including the Vladikavkaz Fault Zone, are in the state of compression with moderate intensity. At the same time, an analysis of the distribution of earthquake epicenters has shown that the Northern Branch is currently aseismic in the central and eastern parts of the Vladikavkaz fault. This geodynamic feature indicates the high seismic potential of the Vladikavkaz Fault Zone.

The article deals with a Boolean valued approach to some algebraic problems arising from functional analysis. The main results are as follows. (1) A universally complete vector lattice without locally one-dimensional bands contains an infinite direct sum of order dense sublattices each of which is a band preserving linear isomorphic (but not lattice isomorphic) copy of the whole lattice. (2) Every separated injective module over a semiprime rationally complete commutative ring admits a direct sum decomposition with homogeneous summands. (3) A semiprime rationally complete commutative ring properly embedded in a ring with projections K is a homogeneity ring of an additive mapping between appropriate K-modules.

The paper provides a brief overview of the origins, methods and results of Boolean valued analysis. Boolean valued representations of some mathematical structures and mappings are given in tabular form. A list of some problems arising outside the theory of Boolean valued models, but solved using the Boolean valued approach, is given. The relationship between the Kantorovich’s heuristic principle and the Boolean valued transfer principle is also discussed.

This is a continuation of the authors' previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an AL-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding AL-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices.

The paper is devoted to a description of compact disjointness preserving homogeneous polynomials on quasi-Banach lattices. De Pagter and Wickstead found independently that a compact linear operator between Banach lattices is a lattice homomorphism if and only if it is representable as the sum of a norm convergent series of one-dimensional lattice homomorphisms with pairwise disjoint ranges. It is shown that this result also holds for a wider class of operators and spaces, namely, for compact disjointness preserving homogeneous polynomials acting from one quasi-Banach lattice into another. As an auxiliary result it is proved that a lattice homomorphism between quasi-Banach lattices dominated by a positive compact operator is compact.

The paper deals with the study of Banach spaces whose duals are injective Banach lattices. Davies in 1967 proved that an ordered Banach space is an L1-predual space if and only if it is a simplex space. In 2007 Duan and Lin proved that a real Banach space is an L1-predual space if and only if its every four-point subset is centerable. We prove the counterparts of these remarkable results for injectives by the new machinery of Boolean valued transfer from L1-spaces to injective Banach lattices.

The paper aims to survey the recent progress towards functional representation of injective Banach lattices as well as to present some new results and open questions. A concrete functional description of injective Banach lattices in the spirit of Kakutani–Maharam representation for AL‐spaces is given.

Boolean valued analysis, the term coined by Takeuti, signifies a branch of functional analysis which uses a special technique of Boolean valued models of set theory. The fundamental result of Boolean valued analysis is Gordon’s Theorem stating that each internal field of reals of a Boolean valued model descends into a universally complete vector lattice. Thus, a remarkable opportunity opens up to expand and enrich the mathematical knowledge by translating information about the reals to the language of other branches of functional analysis. This is a brief overview of the mathematical events around the Gordon Theorem. The relationship between the Kantorovich’s heuristic principle and Boolean valued transfer principle is also discussed.

The paper contains two main results that are obtained by using Boolean valued analysis. The first asserts that a universally complete vector lattice without locally one-dimensional bands can be decomposed into a direct sum of two vector sublattices that are laterally complete and invariant under all band projections and there exists a band preserving linear isomorphism of each of these sublattices onto the original lattice. The second result establishes a counterpart of the Ando Theorem on the joint characterization of ALp and c0 (Γ) for the class of the so-called $$\mathbb{B}$$ -cyclic Banach lattices, using the Boolean valued transfer for injective Banach lattices.

Given a finite collection of disjointness preserving linear operators with values in an f-algebra, the mapping defined as their pointwise product is a disjointness preserving multilinear operator. The central result asserts that an arbitrary order bounded disjointness preserving multilinear operator from the Cartesian product of vector lattices into an arbitrary vector lattice has a similar structure. Using this fact, we establish a multilinear version of the Hart theorem on the associated lattice homomorphism, and also give some consequences on the structure of disjointness preserving homogeneous polynomials.

The paper is devoted to the characterization and weighted shift representation of regular homogeneous polynomials between vector lattices admitting a decomposition into a sum of monomials in lattice homomorphisms. The main tool is the factorization theorem for order bounded disjointness preserving multilinear operators obtained earlier by the authors.

Necessary and sufficient conditions are found under which the sum of N order bounded disjointness preserving operators is n-disjoint with n and N naturals. It is shown that the decomposition of an order bounded n-disjoint operator into a sum of disjointness preserving operators is unique up to “Boolean permutation,” the meaning of which is clarified in the course of the presentation.

Цель настоящей статьи - дать обзор некоторых новых идей и недавних результатов в теории интегрирования скалярных функций относительно векторной меры, а также общих теорем о функциональном представлении квазибанаховых решеток. Приводится набросок чисто порядкового интеграла типа Канторовича - Райта скалярных функций относительно векторной меры, заданной на δ-кольце и принимающей значения в порядково σ-полной векторной решетке. Также представлено интегрирование типа Бартла - Данфорда - Шварца по мере, определенной на δ-кольце со значениями в квазибанаховой решетке. В контексте банаховых решеток решающую роль играют пространства интегрируемых и слабо интегрируемых функций относительно векторной меры. При решении задачи о функциональном представлении квазибанаховых решеток, подход, основанный на двойственности, не работает, но существуют два естественных кандидата для пространства слабо интегрируемых функций: максимальное квазибанахово расширение и область определения наименьшего расширения интегрального оператора. Используя эту идею, можно построить новые пространства слабо интегрируемых функций, которые играют существенную роль в задаче о функциональном представлении квазибанаховых решеток. В частности, показано, что при изучении квазибанаховых решеток, когда метод двойственности не применим, интеграл Канторовича - Райта оказывается более гибким инструментом, чем интеграл Бартла - Данфорда - Шварца.

The purpose of this paper is two-fold: first, to outline a purely order-based integral of the type of the Kantorovich–Wright integral of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a K σ-space (that is, a Dedekind σ-complete vector lattice) and, secondly, prove new theorems on the representation of Dedekind complete vector lattices and quasi-Banach lattices in the form of lattices of functions integrable or “weakly” integrable with respect to an appropriate vector measure. In particular, it is shown that, in studying quasi-Banach lattices, when the duality method does not apply, the Kantorovich–Wright integral is more flexible than the Bartle–Dunford–Schwartz integral.