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We explore non-invertible symmetries in two-dimensional lattice models withsubsystem $\mathbb Z_2$ symmetry. We introduce a subsystem $\mathbbZ_2$-gauging procedure, called the subsystem Kramers-Wannier transformation,which generalizes the ordinary Kramers-Wannier transformation. Thecorresponding duality operators and defects are constructed by gaugings on thewhole or half of the Hilbert space. By gauging twice, we derive fusion rules ofduality operators and defects, which enriches ordinary Ising fusion rules withsubsystem features. Subsystem Kramers-Wannier duality defects are mobile inboth spatial directions, unlike the defects of invertible subsystem symmetries.We finally comment on the anomaly of the subsystem Kramers-Wannier dualitysymmetry, and discuss its subtleties.

Subsystem Non-Invertible Symmetry Operators and Defects