COLIN implies LIN for emergent algebras

TL;DR

This paper proves that in emergent algebras, right-distributivity implies left-distributivity, showing such structures must derive from commutative groups with dilations, which is a surprising restriction.

Contribution

It establishes that the (COLIN) condition implies (LIN) in emergent algebras, linking right-distributivity to commutative group structures.

Findings

Right-distributivity implies left-distributivity in emergent algebras.

Emergent algebras satisfying (COLIN) are derived from commutative groups with dilations.

Non-commutative emergent algebras can satisfy (LIN) but not (COLIN).

Abstract

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.