@article { Gutman19971101,
  author = "Gutman A.E. and Losenkov G.A.",
  title = "Function representation of the Boolean-valued universe",
  journal = "Matem. tr.",
  year = "1998",
  volume = "1",
  number = "1",
  pages = "54--77",
  language = "russian",
  annote = "For an abstract Boolean-valued system, a function analog is proposed that is a model whose elements are functions and the basic logical operations are calculated ``pointwise.'' The new notion of continuous polyverse is introduced and studied which is a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse.",
  keywords = "Boolean-valued analysis, function representation, Stone space, continuous bundle, continuous section"
}
@article { Gutman19971102,
  author = "Gutman A.E. and Losenkov G.A.",
  title = "Function representation of the Boolean-valued universe",
  journal = "Siberian Adv. Math.",
  year = "1998",
  volume = "8",
  number = "1",
  pages = "99--120",
  annote = "For an abstract Boolean-valued system, a function analog is proposed that is a model whose elements are functions and the basic logical operations are calculated ``pointwise.'' The new notion of continuous polyverse is introduced and studied which is a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse.",
  keywords = "Boolean-valued analysis, function representation, Stone space, continuous bundle, continuous section"
}
@inbook { Gutman19991215,
  author = "Gutman A.E. and Losenkov G.A.",
  chapter = "2",
  title = "Function representation of a Boolean valued universe",
  booktitle = "Nonstandard analysis and vector lattices",
  address = "Novosibirsk",
  publisher = "Institute of Mathematics",
  year = "1999",
  pages = "97--125",
  language = "russian",
  annote = "Contemporary methods of Boolean-valued analysis, due to their nature, involve rather bulky logical technique. We can say that, from a pragmatic viewpoint, this technique might distract the user-analyst from a concrete aim: to apply the results of Boolean-valued analysis for solving analytical problems. Various function spaces are common in functional analysis, and so the intention is natural of replacing an abstract Boolean-valued system by some function analog, a model whose elements are functions and in which the basic logical operations are calculated ``pointwise.'' In the present chapter, a solution is proposed to the above problem. To this end, we introduce and study the new notion of continuous polyverse, the latter being a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse."
}
@inbook { Gutman20000411,
  author = "Gutman A.E. and Losenkov G.A.",
  chapter = "2",
  title = "Function representation of a Boolean valued universe",
  booktitle = "Nonstandard analysis and vector lattices",
  address = "Dordrecht",
  publisher = "Kluwer Academic Publishers",
  year = "2000",
  pages = "81--104",
  doi = "10.1007/978-94-011-4305-9_2",
  annote = "Contemporary methods of Boolean-valued analysis, due to their nature, involve rather bulky logical technique. We can say that, from a pragmatic viewpoint, this technique might distract the user-analyst from a concrete aim: to apply the results of Boolean-valued analysis for solving analytical problems. Various function spaces are common in functional analysis, and so the intention is natural of replacing an abstract Boolean-valued system by some function analog, a model whose elements are functions and in which the basic logical operations are calculated ``pointwise.'' In the present chapter, a solution is proposed to the above problem. To this end, we introduce and study the new notion of continuous polyverse, the latter being a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse."
}
@article { Gutman20010219,
  author = "Gutman A.E. and Ryabko D.B.",
  title = "The nonstandard hull of a normed space in a Boolean-valued universe",
  journal = "Matem. tr.",
  year = "2001",
  volume = "4",
  number = "2",
  pages = "42--52",
  language = "russian",
  annote = "In this article, we extend some results of infinitesimal analysis on normed spaces and the field of reals to the functional representation of a Boolean-valued universe. In particular, we prove equivalence of the following three conditions for an arbitrary polyverse over $Q$: a point $q\in Q$ is not $\sigma$-isolated; the stalk of the polyverse at $q$ is countably saturated; and the nonstandard hull of every normed space in the stalk of the polyverse at $q$ is complete.",
  keywords = "Boolean-valued analysis, infinitesimal analysis, nonstandard analysis, polyverse, nonstandard hull"
}
@article { Gutman20010220,
  author = "Gutman A.E. and Ryabko D.B.",
  title = "The nonstandard hull of a normed space in a Boolean-valued universe",
  journal = "Siberian Adv. Math.",
  year = "2002",
  volume = "12",
  number = "2",
  pages = "38--47",
  annote = "In this article, we extend some results of infinitesimal analysis on normed spaces and the field of reals to the functional representation of a Boolean-valued universe. In particular, we prove equivalence of the following three conditions for an arbitrary polyverse over $Q$: a point $q\in Q$ is not $\sigma$-isolated; the stalk of the polyverse at $q$ is countably saturated; and the nonstandard hull of every normed space in the stalk of the polyverse at $q$ is complete.",
  keywords = "Boolean-valued analysis, infinitesimal analysis, nonstandard analysis, polyverse, nonstandard hull"
}
@article { Gutman20011224,
  author = "Gutman A.E. and Ryabko D.B.",
  title = "Completeness criterion for the nonstandard hull of a normed space in a Boolean-valued universe",
  journal = "Proc. Acad. Sci.",
  year = "2002",
  volume = "384",
  number = "2",
  pages = "153--155",
  language = "russian",
  annote = "In this article, we extend some results of infinitesimal analysis on normed spaces and the field of reals to the functional representation of a Boolean-valued universe. In particular, we prove equivalence of the following three conditions for an arbitrary polyverse over $Q$: a point $q\in Q$ is not $\sigma$-isolated; the stalk of the polyverse at $q$ is countably saturated; and the nonstandard hull of every normed space in the stalk of the polyverse at $q$ is complete."
}
@article { Gutman20011225,
  author = "Gutman A.E. and Ryabko D.B.",
  title = "Completeness criterion for the nonstandard hull of a normed space in a Boolean-valued universe",
  journal = "Doklady Math.",
  year = "2002",
  volume = "65",
  number = "3",
  pages = "337--338",
  annote = "In this article, we extend some results of infinitesimal analysis on normed spaces and the field of reals to the functional representation of a Boolean-valued universe. In particular, we prove equivalence of the following three conditions for an arbitrary polyverse over $Q$: a point $q\in Q$ is not $\sigma$-isolated; the stalk of the polyverse at $q$ is countably saturated; and the nonstandard hull of every normed space in the stalk of the polyverse at $q$ is complete."
}
@article { Gutman20020204,
  author = "Gutman A.E. and Ryabko D.B.",
  title = "Function representation of Kantorovich spaces by means of a Boolean-valued models",
  journal = "Vladikavk. Math. J.",
  year = "2002",
  volume = "4",
  number = "1",
  pages = "34--49",
  language = "russian",
  annote = "In the paper, the notion is introduced of outer section of a polyverse (a function representation of a Boolean-valued universe) and a new function representation is obtained of K-spaces and vector lattices as outer sections. In particular, an isomorphism is constructed between an arbitrary vector lattice and an outer subset of the field of reals in the corresponding Boolean-valued universe. Within the new function representation, analogs are discovered of the main motions and facts of the theory of vector lattices. It is also clarified which of the properties of K-spaces under consideration admit ``pointwise criteria.''"
}
@inproceedings { Gutman20020518,
  author = "Gutman A.E. and Ryabko D.B.",
  howpublished = "Electronic",
  title = "Functional representation of a Dedekind-complete Reisz space in a Boolean valued universe",
  booktitle = "The 4th Conference on Function Spaces at SIUE (USA, May 14--19, 2002): Proceedings",
  address = "USA, Illinois, Edwardsville",
  year = "2002",
  pages = "16",
  annote = "Using a functional model of a Boolean-valued universe, the basic notions are introduced of infinitesimal analysis within a Boolean-valued universe, thus enriching the synthesis of the two main branches of nonstandard analysis, infinitesimal and Boolean-valued. The results obtained are applied to the theory of Riesz spaces."
}
@inbook { Gutman20050906,
  author = "Gutman A.E. and Losenkov G.A.",
  chapter = "2",
  title = "Function representation of a Boolean valued universe",
  booktitle = "Nonstandard analysis and vector lattices. 2nd ed., corr. and enl.",
  address = "Novosibirsk",
  publisher = "Institute of Mathematics",
  year = "2005",
  pages = "95--123",
  language = "russian",
  annote = "Contemporary methods of Boolean-valued analysis, due to their nature, involve rather bulky logical technique. We can say that, from a pragmatic viewpoint, this technique might distract the user-analyst from a concrete aim: to apply the results of Boolean-valued analysis for solving analytical problems. Various function spaces are common in functional analysis, and so the intention is natural of replacing an abstract Boolean-valued system by some function analog, a model whose elements are functions and in which the basic logical operations are calculated ``pointwise.'' In the present chapter, a solution is proposed to the above problem. To this end, we introduce and study the new notion of continuous polyverse, the latter being a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse."
}