Posetoaction

Posetoaction is a term used in mathematics to denote the action of a symmetry group or monoid on a partially ordered set, in a way that respects the order.

Formally, let (P, ≤) be a poset and G a group acting on P via a map G × P → P, (g, x) ↦ g·x. The action is required to satisfy e·x = x for the identity e, (gh)·x = g·(h·x) for all g,h ∈ G and x ∈ P, and x ≤ y implies g·x ≤ g·y for all g ∈ G. Thus the action is compatible with the poset structure. If every g ∈ G acts as an automorphism of P (i.e., preserves all order relations), the action is called an action by automorphisms.

Key concepts associated with a posetoaction include orbits, stabilizers, and fixed points. For x ∈ P, the orbit Gx = {g·x : g ∈ G} describes the symmetry of x, while the stabilizer Gx = {g ∈ G : g·x = x} captures elements that fix x. The fixed-point set P^G = {x ∈ P : g·x = x for all g ∈ G} highlights points completely invariant under the action. In many settings, one may form a quotient or orbit poset under suitable conditions.

Common examples include the Boolean lattice of all subsets of a finite set, ordered by inclusion, with the symmetric group S_n acting by permuting the ground elements; another example is the lattice of subspaces of a vector space with a linear group acting via linear transformations. This framework, often called a G-poset, extends to monoid actions as well.

Applications of posetoactions appear in combinatorics, invariant theory, and equivariant topology, including counting up to symmetry (via Burnside-type arguments) and studying invariant subposets under group actions. See also group action, poset, order automorphism, and Burnside’s lemma.