  Emergent algebras, first time introduced in arXiv:0907.1520 , are families of
quasigroup operations indexed by a commutative group, which satisfy some
algebraic relations and also topological (convergence and continuity)
relations. Besides sub-riemannian geometry arXiv:math/0608536, they appear as a
semantics of a family of graph-rewrite systems related to interaction
combinators arXiv:2007.10288, or lambda calculus arXiv:1305.5786 . In
arXiv:1807.02058 there is a lambda calculus version of emergent algebras.
  In this article we prove that for emergent algebras the condition (COLIN), or
right-distributivity for emergent algebras, implies (LIN), or
left-distributivity for emergent algebras. It means that any emergent algebra
which is right-distributive has to come from a commutative group endowed with a
family of dilations.
  This is surprising, because there are many examples of emergent algebras
which satisfy (LIN), but not (COLIN), namely those who are associated to
non-commutative conical groups, in particular to non-commutative Carnot groups.

Group Theory;Differential Geometry;Metric Geometry