Rowmotion and generalized toggle groups.

*(English)*Zbl 1401.05315Summary: We generalize the notion of the toggle group, as defined in P. J. Cameron and D. G. Fon-Der-Flaass [Eur. J. Comb. 16, No. 6, 545–554 (1995; Zbl 0831.06001)] and further explored in J. Striker and N. Williams [Eur. J. Comb. 33, No. 8, 1919–1942 (2012; Zbl 1260.06004)], from the set of order ideals of a poset to any family of subsets of a finite set. We prove structure theorems for certain finite generalized toggle groups, similar to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We apply these theorems and find other results on generalized toggle groups in the following settings: chains, antichains, and interval-closed sets of a poset; independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a graph; matroids and convex geometries. We generalize rowmotion, an action studied on order ideals in Cameron and Fon-der-Flaass [loc. cit.] and J. Striker and N. Williams [loc. cit.], to a map we call cover-closure on closed sets of a closure operator. We show that cover-closure is bijective if and only if the set of closed sets is isomorphic to the set of order ideals of a poset, which implies rowmotion is the only bijective cover-closure map.