Emergent algebras as generalizations of differentiable algebras, with applications

We propose a generalization of differentiable algebras, where the underlying differential structure is replaced by a uniform idempotent right quasigroup (irq). Then a emergent algebra is a algebra A over the uniform irq X such that all operations and algebraic relations from A can be constructed or deduced from combinations of operations in the uniform irq, possibly by taking limits which are uniform with respect to a set of parameters. In this approach, the usual compatibility condition between algebraic information A and differential information D, expressed as the differentiability of operations from A with respect to D, is replaced by the ”emergence” of algebraic operations and relations from the minimal structure of a uniform irq. Two applications are considered: we prove a bijection between contractible groups and distributive uniform irqs and that symmetric spaces in the sense of Loos may be seen as uniform quasigroups with a distributivity property.