In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie to investigate symmetries of differential equations, rather than out of abstract algebra (such as quasigroup, for example). The modern theory of pseudogroups was developed by Élie Cartan in the early 1900s. Definition A pseudogroup imposes several conditions on a sets of homeomorphisms (respectively, diffeomorphisms) defined on open sets U of a given Euclidean space or more generally of a fixed topological space (respectively, smooth manifold). Since two homeomorphisms and compose to a homeomorphism from U to W, one asks that the pseudogroup is closed under composition and inversion. However, unlike those for a group, the axioms defining a pseudogroup are not purely algebraic; the further requirements are related to the possibility of restricting and of patching homeomorphisms (similar to the gluing axiom for sections of a sheaf). More precisely, a pseudogroup on a topological space is a collection of homeomorphisms between open subsets of satisfying the following properties: The domains of the elements in cover ("cover"). The restriction of an element in to any open set contained in its domain is also in ("restriction"). The composition ○ of two elements of , when defined, is in ("composition"). The inverse of an element of is in ("inverse"). The property of lying in is local, i.e. if : → is a homeomorphism between open sets of and is covered by open sets with restricted to lying in for each , then also lies in ("local"). As a consequence the identity homeomorphism of any open subset of lies in . Similarly, a pseudogroup on a smooth manifold is defined as a collection of diffeomorphisms between open subsets of satisfying analogous properties (where we replace homeomorphisms with diffeomorphisms). Two points in are said to be in the same orbit if an element of sends one to the other. Orbits of a pseudogroup clearly form a partition of ; a pseudogroup is called transitive if it has only one orbit. Examples A widespread class of examples is given by pseudogroups preserving a given geometric structure. For instance, if (X, g) is a Riemannian manifold, one has the pseudogroup of its local isometries; if (X, ω) is a symplectic manifold, one has the pseudogroup of its local