Finite Playground

To verify is human; to prove, divine.


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Three group-unrelated representation theorems

A representation theorem in general means a canonical way of expressing a class of objects using another class of objects, usually more fundamental or easier to grasp.  Beside the well-known group representation theory, which concerns about viewing (finite or Lie) groups as matrices, here are three lesser known (to me!) representation theorems.

Birkhoff’s Representation Theorem.  Every distributive lattice is isomorphic to a sublattice of the power set lattice of some set.

Riesz–Markov–Kakutani Representation Theorem.  Any positive linear functional on the space of compactly supported continuous functions on a locally compact Hausdorff space can be viewed as integration against a measure.

Kapovich-Millson Universality Theorem.  Any compact smooth manifold is diffeomorphic to a component of the configuration space of some planar linkage.