Orbitallattices

Orbitallattices is a concept in abstract algebra and dynamical systems that refers to the lattice structure formed by organizing points according to their orbits under a group action. Given a group G acting on a set X, the orbits partition X into disjoint blocks G·x. The orbit lattice consists of all G-invariant subsets of X, i.e., unions of orbits, ordered by inclusion. In this setting the join operation is union and the meet operation is intersection; the atoms of the lattice are the individual orbits. Consequently, if the action has k distinct orbits, the orbit lattice is naturally isomorphic to the Boolean lattice on k atoms, i.e., the power set of the orbit set.

In topological dynamics, one often considers a refinement by orbit closures. The orbit closure lattice is then formed by the closures Cl(G·x) of orbits; this lattice can reveal how symmetries interact with the topology of X and need not be Boolean. The distinction between the plain orbit lattice and the orbit closure lattice reflects the difference between purely set-theoretic orbits and their topological closures.

Examples illustrate the concept. For a finite group acting on a finite set, the orbit lattice is finite and Boolean, with atoms corresponding to the distinct orbits. If the action is transitive, the lattice has just two elements: the empty set and the whole space. A typical algebraic example is a group acting on itself by conjugation, where the orbits are conjugacy classes and invariant subsets are unions of these classes, forming a Boolean lattice.

Orbitallattices connects to invariant theory, partition lattices, and the study of symmetry in geometry and dynamics. See also group action, orbit, stabilizer, lattice of partitions.