Metrized Semigroups

The notion of 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 Shannon entropy, a discontinuous antinorm on a Lie algebra determining a continuous antimetric on the Lie group, and an example of the converse situation. Problems are formulated. Bibliography: 47 titles.