  In this paper, we analyze the toggle group on the set of antichains of a
poset. Toggle groups, generated by simple involutions, were first introduced by
Cameron and Fon-Der-Flaass for order ideals of posets. Recently Striker has
motivated the study of toggle groups on general families of subsets, including
antichains. This paper expands on this work by examining the relationship
between the toggle groups of antichains and order ideals, constructing an
explicit isomorphism between the two groups. We also focus on the rowmotion
action that has been well-studied in dynamical algebraic combinatorics,
describing it as the composition of antichain toggles. We also describe a
piecewise-linear analogue of toggling to Stanley's chain polytope. We examine
the connections with the piecewise-linear toggling Einstein and Propp
introduced for order polytopes and prove that almost all of our results for
antichain toggles extend to the piecewise-linear setting.
