author = "Gutman A.E. and Emelyanov E.Yu. and Kusraev A.G. and Kutateladze S.S.",

title = "Nonstandard analysis and vector lattices",

address = "Novosibirsk",

publisher = "Institute of Mathematics",

year = "1999",

pages = "x+380",

isbn = "5-86134-068-4",

language = "russian",

annote = "The book is devoted to applications of nonstandard methods of analysis to the theory of vector lattices. Much attention is paid to the problem of combining infinitesimal and Boolean-valued concepts to studying classical problems of the theory of vector lattices related to constructing concrete representations of abstract functional-analytic objects: Banach--Kantorovich spaces, dominated operators, vector measures, integral operators, etc. This book is intended for those interested in modern applications of nonstandard analysis to problems of functional analysis."

author = "Gutman A.E. and Emelyanov E.Yu. and Kusraev A.G. and Kutateladze S.S.",

title = "Nonstandard analysis and vector lattices",

address = "Dordrecht",

publisher = "Kluwer Academic Publishers",

year = "2000",

pages = "xii+307",

isbn = "978-94-010-5863-6",

doi = "10.1007/978-94-011-4305-9",

annote = "The book is devoted to applications of nonstandard methods of analysis to the theory of vector lattices. Much attention is paid to the problem of combining infinitesimal and Boolean-valued concepts to studying classical problems of the theory of vector lattices related to constructing concrete representations of abstract functional-analytic objects: Banach--Kantorovich spaces, dominated operators, vector measures, integral operators, etc. This book is intended for those interested in modern applications of nonstandard analysis to problems of functional analysis."

author = "Gutman A.E. and Emelyanov E.Yu. and Koptev A.V. and Kusraev A.G. and Kutateladze S.S. and Malyugin S.A.",

title = "Nonstandard analysis and vector lattices. 2nd ed., corr. and enl.",

address = "Novosibirsk",

publisher = "Institute of Mathematics",

year = "2005",

pages = "x+400",

isbn = "5-86134-127-3",

language = "russian",

annote = "The book is devoted to applications of nonstandard methods of analysis to the theory of vector lattices. Much attention is paid to the problem of combining infinitesimal and Boolean-valued concepts to studying classical problems of the theory of vector lattices related to constructing concrete representations of abstract functional-analytic objects: Banach--Kantorovich spaces, dominated operators, vector measures, integral operators, etc. This book is intended for those interested in modern applications of nonstandard analysis to problems of functional analysis."

author = "Gutman A.E. and Lisovskaya S.A.",

title = "The boundedness principle for lattice-normed spaces",

journal = "Sib. Matem. Zh.",

year = "2009",

volume = "50",

number = "5",

pages = "1050--1059",

language = "russian",

annote = "Three classical facts of the theory of normed spaces are considered: the boundedness principle, the Banach--Steinhaus theorem, and the uniform boundedness principle for a compact convex set. By means of Boolean-valued analysis, exact analogs of the theorems are proven for the case of lattice-normed spaces.",

keywords = "Banach--Steinhaus theorem, Banach--Kantorovich space, cyclically compact set, Boolean-valued analysis"

author = "Gutman A.E. and Lisovskaya S.A.",

title = "The boundedness principle for lattice-normed spaces",

journal = "Sib. Math. J.",

year = "2009",

volume = "50",

number = "5",

pages = "830--837",

doi = "10.1007/s11202-009-0093-5",

annote = "Three classical facts of the theory of normed spaces are considered: the boundedness principle, the Banach--Steinhaus theorem, and the uniform boundedness principle for a compact convex set. By means of Boolean valued analysis, exact analogs of the theorems are proven for the case of lattice-normed spaces.",

keywords = "Banach--Steinhaus theorem, Banach--Kantorovich space, cyclically compact set, Boolean valued analysis"

author = "Gutman A.E. and Lisovskaya S.A.",

howpublished = "Electronic",

title = "The boundedness principle for lattice-normed spaces",

booktitle = "Contemporary Analysis and Geometry. International Conference (Novosibirsk, September 14--20, 2009): Proceedings",

address = "Novosibirsk",

publisher = "Institute of Mathematics",

year = "2009",

pages = "29",

language = "russian",

annote = "Three classical facts of the theory of normed spaces are considered: the boundedness principle, the Banach--Steinhaus theorem, and the uniform boundedness principle for a compact convex set. By means of Boolean valued analysis, exact analogs of the theorems are proven for the case of lattice-normed spaces over a uniformly complete Kantorovich space."

author = "Gutman A.E.",

title = "An example of using $\Delta_1$ terms in Boolean-valued analysis",

journal = "Vladikavk. Math. J.",

year = "2012",

volume = "14",

number = "1",

pages = "47--63",

doi = "10.23671/VNC.2012.14.10953",

language = "russian",

annote = "Syntactic tools related to $\Delta_1$ terms are demonstrated by application to Boolean valued analysis. As an example, the question is considered of what approaches to defining the field $R$ of reals and what complete Boolean algebras $B$ provide the explicit inclusion of $R^{\land}$ into $R$ inside the Boolean-valued universe $V^{(B)}$.",

keywords = "set theory, conservative extension, real number, Boolean valued analysis, canonical embedding, $\sigma$-distributive Boolean algebra, $\Sigma_1$ formula."

author = "Gutman A.E.",

howpublished = "Electronic",

title = "The technique of definable terms in Boolean valued analysis",

booktitle = "Mal'tsev Meeting. International Conference (Novosibirsk, November 11--15, 2013): Proceedings",

address = "Novosibirsk",

year = "2013",

pages = "164",

annote = "A syntax technique related to the notion of $\Delta_1$ term is demonstrated by means of its applications to Boolean-valued analysis. As an example, the following question is considered: Which of the classical approaches to the definition of the field $R$ of reals and which Boolean algebras provide the explicit inclusion of $R^{\land}$ in $R$ inside the Boolean-valued universe $V^{(B)}$?"

author = "Gutman A.E. and Kusraev A.G. and Kutateladze S.S.",

title = "The growth points of Boolean valued analysis",

booktitle = "Geometry Days in Novosibirsk -- 2014. International conference dedicated to 85th anniversary of academician Yu.G.Reshetnyak (Novosibirsk, September 24--27, 2014): Proceedings",

address = "Novosibirsk",

publisher = "Sobolev Institute of Mathematics SB RAS",

year = "2014",

pages = "102",

annote = "Boolean valued analysis is a powerful method of extending the scope of mathematical theories by means of the special nonstandard models of set theory. This communication pays attention to the Continuum Hypothesis, Kantorovich spaces, and the machinery of Boolean valued analysis."

author = "Gutman A.E.",

title = "On the structure of the Boolean-valued universe",

journal = "Vladikavk. Math. J.",

year = "2018",

volume = "20",

number = "2",

pages = "38--48",

doi = "10.23671/VNC.2018.2.14718",

language = "russian",

annote = "The logical machinery is clarified which justifies declaration of hypotheses. In particular, attention is paid to hypotheses and conclusions constituted by infinitely many formulas. Formal definitions are presented for Boolean-valued algebraic system and model of a theory, for the system of terms of Boolean-valued truth value of formulas, for ascent and mixing. Logical interrelations are described between the ascent, mixing, and maximum principles. It is shown that every ascent with arbitrary weights can be transformed into an ascent with constant weight. The notion of restriction of an element of a Boolean-valued algebraic system is introduced and studied. It is proven that every Boolean-valued model of Set theory which meets the ascent principle has multilevel structure analogous to von Neumann's cumulative hierarchy.",

keywords = "Set theory, Boolean-valued model, universe, cumulative hierarchy"

author = "Gutman A.E.",

title = "Boolean-valued universe as an algebraic system. I: Basic principles",

journal = "Sib. Matem. Zh.",

year = "2019",

volume = "60",

number = "5",

pages = "1041--1062",

doi = "10.33048/smzh.2019.60.505",

language = "russian",

annote = "The paper is devoted to the study of Boolean-valued algebraic systems of set-theoretic signature. The technique of partial elements of these systems is developed. Some formal apparatus is presented for using partial elements and Boolean-valued classes in the truth values of formulas. The predicative Boolean-valued classes are studied that admit quantification. Logical interrelations are described between the basic properties of Boolean-valued systems: the transfer, mixing, and maximum principles.",

keywords = "Boolean-valued algebraic system, set theory, Boolean-valued analysis"

author = "Gutman A.E.",

title = "Boolean-valued universe as an algebraic system. I: Basic principles",

journal = "Sib. Math. J.",

year = "2019",

volume = "60",

number = "5",

pages = "810--827",

doi = "10.1134/S0037446619050057",

annote = "The paper is devoted to the study of Boolean-valued algebraic systems of set-theoretic signature. The technique of partial elements of these systems is developed. Some formal apparatus is presented for using partial elements and Boolean-valued classes in the truth values of formulas. The predicative Boolean-valued classes are studied that admit quantification. Logical interrelations are described between the basic properties of Boolean-valued systems: the transfer, mixing, and maximum principles.",

keywords = "Boolean-valued algebraic system, set theory, Boolean-valued analysis"

author = "Gutman A.E.",

title = "Cumulative structure of a Boolean-valued model of set theory",

booktitle = "International conference on Geometric Analysis in honor of the 90th anniversary of academician Yu.G.Reshetnyak (Novosibirsk, September, 22--28, 2019): Proceedings",

address = "Novosibirsk",

publisher = "Sobolev Institute of Mathematics SB RAS",

year = "2019",

isbn = "978-5-4437-0949-9",

pages = "64--66",

annote = "We show that every Bolean-valued universe has a multilevel structure analogous to the von Neumann cumulative hierarchy, in which, at each level, the ascents are added of the Boolean-valued functions defined on subsets of the previous levels. Another cumulative structure is obtained if we consider the ascents of constant functions only and add mixings at the limit steps. Such cumulative hierarchies clarify the structure of Boolean-valued systems and, in particular, make it possible to easily prove the uniqueness of a Boolean-valued universe up to isomorphism. We also present a general tool for adding ascents to Boolean-valued systems which builds the cumulative hierarchy starting from an arbitrary extensional system. This makes it possible to construct examples of Boolean-valued systems with unusual properties. By means of the tool, we show that each of the five conditions listed in the definition of a Boolean-valued universe, is essential and does not follow from the other conditions."

author = "Gutman A.E.",

title = "Boolean-valued universe as an algebraic system. II: Intensional hierarchies",

journal = "Sib. Matem. Zh.",

year = "2020",

volume = "61",

number = "3",

pages = "539--571",

doi = "10.33048/smzh.2020.61.305",

language = "russian",

annote = "For Boolean-valued algebraic systems of set-theoretic signature, the notions of transitivity, regularity, and $\sigma$-regularity are studied. The notion of a universe over an arbitrary extensional Boolean-valued system is introduced. A description is proposed of the structure of the universe by means of various hierarchies. The results are used for proving the uniqueness of a Boolean-valued universe up to a unique isomorphism.",

keywords = "Boolean-valued algebraic system, set theory, Boolean-valued analysis, universe, cumulative hierarchy"

author = "Gutman A.E.",

title = "Boolean-valued universe as an algebraic system. II: Intensional hierarchies",

journal = "Sib. Math. J.",

year = "2020",

volume = "61",

number = "3",

pages = "426--452",

doi = "10.1134/S0037446620030052",

annote = "For Boolean-valued algebraic systems of set-theoretic signature, the notions of transitivity, regularity, and $\sigma$-regularity are studied. The notion of a universe over an arbitrary extensional Boolean-valued system is introduced. A description is proposed of the structure of the universe by means of various hierarchies. The results are used for proving the uniqueness of a Boolean-valued universe up to a unique isomorphism.",

keywords = "Boolean-valued algebraic system, set theory, Boolean-valued analysis, universe, cumulative hierarchy"

author = "Gutman A.E.",

title = "Boolean-valued set-theoretic systems: General formalism and basic technique",

journal = "Mathematics",

year = "2021",

volume = "9",

number = "9",

article = "1056",

pages = "78",

doi = "10.3390/math9091056",

annote = "The article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols, internal classes, outer terms, and external Boolean-valued classes. Next, we enrich the collection of Boolean-valued research tools with the technique of partial elements and the corresponding joins, mixings, and ascents. Passing on to the set-theoretic signature, we prove that bounded formulas are absolute for transitive Boolean-valued subsystems. We also introduce and study intensional, predicative, cyclic, and regular Boolean-valued systems, examine the maximum principle, and analyze its relationship with the ascent and mixing principles. The main applications relate to the universe over an arbitrary extensional Boolean-valued system. A close interrelation is established between such a universe and the intensional hierarchy. We~prove the existence and uniqueness of the Boolean-valued universe up to a unique isomorphism and show that the conditions in the corresponding axiomatic characterization are logically independent. We also describe the structure of the universe by means of several cumulative hierarchies. Another application, based on the quantifier hierarchy of formulas, improves the transfer principle for the canonical embedding in the Boolean-valued universe.",

keywords = "Boolean-valued universe, algebraic system, set theory, cumulative hierarchy"

author = "Gutman A.E.",

title = "Boolean-Valued Analysis: See the Simple in the Complex",

booktitle = "International Conference «Order Analysis and Related Problems of Mathematical Modeling, XVI. Operator Theory and Differential Equations» (Vladikavkaz, September 20--25, 2021): Proceedings",

address = "Vladikavkaz",

publisher = "SMI VSC RAS and RNO-A",

year = "2021",

pages = "1",

language = "russian",

annote = "The audience is offered a story about Boolean-valued models of set theory. These are very unusual models with very unusual logic. Assertions in such models do not have to be true or false, and their truth can take intermediate values. This was the idea that helped to successfully solve the most famous mathematical problem of the 20th century, the Continuum Problem."

author = "Gutman A.E.",

title = "A sentence preservation theorem for Boolean algebras",

journal = "J. Math. Sci.",

year = "2023",

pages = "8",

doi = "10.1007/s10958-023-06599-4",

annote = "At the initial stages of studying the theory of Boolean algebras, before trying to prove or disprove any simple sentence, students are usually asked to test their intuition using Venn diagrams or truth tables. A natural question arises: is it necessary to invent a proof after a positive check of this kind? Isn't such a check itself a rigorous proof of the verified sentence? And if this is not true in the general case, for which sentences is this true? We answer the question and prove an analog of the Jech Theorem for arbitrary (not necessarily complete) Boolean algebras.",

keywords = "Boolean algebra, Venn diagram, truth table, Horn formula"

author = "Gutman A.E. and Kusraev A.G.",

title = "Boolean valued analysis and the Wickstead problem",

booktitle = "Mathematical Forum. Vol. 14. Modern Mathematics. Introductory Lectures (Project OTDE-Workshop)",

address = "Vladikavkaz",

publisher = "SMI VSC RAS",

year = "2023",

isbn = "978-5-904695-46-0",

pages = "11--48",

language = "russian",

annote = "The purpose of this mini-course, consisting of four lectures, is to sketch Boolean valued analysis and its application to one problem from the theory of linear operators in vector lattices.",

keywords = "Kantorovich space, Wickstead problem, Cauchy functional equation, field extension, Boolean valued model, descent and ascent, Boolean valued reals"