  Lattice non-invertible symmetries have rich fusion structures and play
important roles in understanding various exotic topological phases. In this
paper, we explore methods to generate new lattice non-invertible
transformations/symmetries from a given non-invertible seed
transformation/symmetry. The new lattice non-invertible symmetry is constructed
by composing the seed transformations on different sites or sandwiching a
unitary transformation between the transformations on the same sites. In
addition to known non-invertible symmetries with fusion algebras of
Tambara-Yamagami $\mathbb Z_N\times\mathbb Z_N$ type, we obtain a new
non-invertible symmetry in models with $\mathbb Z_N$ dipole symmetries. We name
the latter the dipole Kramers-Wannier symmetry because it arises from gauging
the dipole symmetry. We further study the dipole Kramers-Wannier symmetry in
depth, including its topological defect, its anomaly and its associated
generalized Kennedy-Tasaki transformation.
