Geodesic
Metrized semigroups
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 6, Tome 279 (2001), pp. 24-60

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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. 
@article{ZNSL_2001_279_a2,
     author = {V. N. Berestovskii and V. M. Gichev},
     title = {Metrized semigroups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {24--60},
     publisher = {mathdoc},
     volume = {279},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_279_a2/}
}
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V. N. Berestovskii; V. M. Gichev. Metrized semigroups. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 6, Tome 279 (2001), pp. 24-60. http://geodesic.mathdoc.fr/item/ZNSL_2001_279_a2/