Rowmotion and generalized toggle groups

We generalize the notion of the toggle group, as defined in [P. Cameron-D. Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], 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 [P. Cameron-D. Fon-der-Flaass '95] and [J. Striker-N. Williams '12], 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.Comment: 26 pages, 5 figures, final journal versio

Truncations of inductively minimal geometries

AbstractInductively minimal geometries form an infinite family of incidence geometries on which finite symmetric groups act flag-transitively. They were introduced in Buekenhout et al. (in: N.L. Johnson (Ed.), Mostly Finite Geometries, Marcel Dekker, New York, 1997, pp. 185–190) and satisfy, among other, the (IP)2 and RWPRI conditions (see Bull. Belg. Math. Soc. Simon Stevin 5 (1998) 213–219). In the present paper we characterize the truncations of inductively minimal geometries which satisfy both of these conditions. We also determine all rank 2 residues in these truncations. This enables one to find the diagram of these truncations

Imbrex geometries

We introduce an axiom on strong parapolar spaces of diameter 2, which arises naturally in the framework of Hjelmslev geometries. This way, we characterize the Hjelmslev-Moufang plane and its relatives (line Grassmannians, certain half-spin geometries and Segre geometries). At the same time we provide a more general framework for a Lemma of Cohen, which is widely used to study parapolar spaces. As an application, if the geometries are embedded in projective space, we provide a common characterization of (projections of) Segre varieties, line Grassmann varieties, half-spin varieties of low rank, and the exceptional variety $\mathcal{E}_{6,1}$ by means of a local condition on tangent spaces

Sublattices of complete lattices with continuity conditions

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice IdS of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then IdS can be embedded into a direct product of finite lower bounded lattices. We also find a system of infinitary identities that characterize sublattices of complete, lower continuous, and join-semidistributive lattices. These conditions are satisfied by any (not necessarily finitely generated) lower bounded lattice and by any locally finite, join-semidistributive lattice. Furthermore, they imply M. Ern\'e's dual staircase distributivity. On the other hand, we prove that the subspace lattice of any infinite-dimensional vector space cannot be embedded into any countably complete, countably upper continuous, and countably lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.Comment: To appear in Algebra Universali

On organizing principles of Discrete Differential Geometry. Geometry of spheres

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse