What algebraic rules govern hidden order in one-dimensional quantum chains?

In one-dimensional quantum lattice systems, certain observable properties (order parameters) only become measurable in specific phases. The authors show these observables must obey strict algebraic multiplication rules—organizing into Lagrangian algebras—and that their values depend directly on how symmetries break in the ground state. This connection between abstract algebra and physical measurement works for both standard and exotic non-invertible symmetries.