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Zapiski Nauchnykh Seminarov POMI, 2001, Volume 279, Pages 24–60
(Mi znsl1452)
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This article is cited in 2 scientific papers (total in 2 papers)
Metrized semigroups
V. N. Berestovskii, V. M. Gichev Omsk State University
Abstract:
The notion metrized order (antimetric) on a topological group is characterized by three equivalent systems of axioms and connected with pointed locally generated semigroups. In the present paper, these notions are discussed and new results are announced; the main result is an analog of the following fact in metric geometry: every left-invariant inner metric on a Lie group is Finsler (maybe, nonholonomic). In the situation considered, a norm is replaced by an antinorm, and a metric by an antimetric. Examples are given, showing the complexity of these structures and their prevalence. Among them are: a nonholonomic antimetric on Heisenberg group, an antimetric on a nonnilpotent group admitting dilatations, a pointed locally generated semigroup in the Hilbert space with trivial tangent cone, antinorms connected with the Brunn–Minkowski inequality and the Shannon entorpy, a discontinuous antinorm on a Lie algebra defining a continuous antimetric on the Lie group, and an example of the converse situation. Several problems are formulated.
Received: 25.12.2000
Citation:
V. N. Berestovskii, V. M. Gichev, “Metrized semigroups”, Geometry and topology. Part 6, Zap. Nauchn. Sem. POMI, 279, POMI, St. Petersburg, 2001, 24–60; J. Math. Sci. (N. Y.), 119:1 (2004), 10–29
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https://www.mathnet.ru/eng/znsl1452 https://www.mathnet.ru/eng/znsl/v279/p24
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Abstract page: | 254 | Full-text PDF : | 103 |
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