Idempotent uninorms on a complete chain

We characterize idempotent uninorms on a complete chain in terms of decreasing unary functions with a symmetry-related property. As a particular case, we retrieve and simplify a known characterization theorem for idempotent uninorms on the real unit interval. We also introduce the notion of left-continuity (resp. right-continuity) of an idempotent uninorm on a complete chain and characterize left-continuous (resp. right-continuous) idempotent uninorms in terms of decreasing unary functions with a second (resp. a third) symmetry-related property, unifying a characterization theorem for left-continuous (resp. right-continuous) idempotent uninorms on the real unit interval and a characterization theorem for idempotent uninorms on a finite chain known in the literature.