  The rowmotion action on order ideals or on antichains of a finite partially
ordered set has been studied (under a variety of names) by many authors.
Depending on the poset, one finds unexpectedly interesting orbit structures,
instances of (small order) periodicity, cyclic sieving, and homomesy. Many of
these nice features still hold when the action is extended to $[0,1]$-labelings
of the poset or (via detropicalization) to labelings by rational functions (the
birational setting).
  In this work, we parallel the birational lifting already done for order-ideal
rowmotion to antichain rowmotion. We give explicit equivariant bijections
between the birational toggle groups and between their respective liftings. We
further extend all of these notions to labellings by noncommutative rational
functions, setting an unpublished periodicity conjecture of Grinberg in a
broader context.
