  We study duality transformations of the star-square relation and the
generalized star-triangle relation for Ising-like integrable lattice spin
models. The integrable models are obtained via gauge/YBE correspondence which
connects the supersymmetric gauge theories and lattice spin models of
statistical mechanics. By the use of integral identities coming from the
duality of three-dimensional supersymmetric gauge theories, we construct
hyperbolic, lens hyperbolic, trigonometric, and rational solutions to the
duality transformations. These duality transformations allow us to construct
spin lattice models with four-spin (the star-square relation) and three-spin
(the generalized star-triangle relation) interactions.
