Metrized semigroups

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.