Resolutions of Convex Geometries

Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry. Contrary to what happens for similar constructions–compounds of hypergraphs, as in Chein, Habib and Maurer (1981), and compositions of set systems, as in Möhring and Radermacher)–, resolutions of convex geometries always yield a convex geometry. We investigate resolutions of special convex geometries: ordinal and affine. A resolution of ordinal convex geometries is again ordinal, but a resolution of affine convex geometries may fail to be affine. A notion of primitivity, which generalize the corresponding notion for posets, arises from resolutions: a convex geometry is primitive if it is not a resolution of smaller ones. We obtain a characterization of affine convex geometries that are primitive, and compute the number of primitive convex geometries on at most four elements. Several open problems are listed.

Reconstructing the Intrinsic Statistical Properties of Intermittent Locomotion Through Corrections for Boundary Effects

Locomotion characteristics are often recorded within bounded spaces, a constraint which introduces geometry-specific biases and potentially complicates the inference of behavioural features from empirical observations. We describe how statistical properties of an uncorrelated random walk, namely the steady-state stopping location probability density and the empirical step probability density, are affected by enclosure in a bounded space. The random walk here is considered as a null model for an organism moving intermittently in such a space, that is, the points represent stopping locations and the step is the displacement between them. Closed-form expressions are derived for motion in one dimension and simple two-dimensional geometries, in addition to an implicit expression for arbitrary (convex) geometries. For the particular choice of no-go boundary conditions, we demonstrate that the empirical step distribution is related to the intrinsic step distribution, i.e. the one we would observe in unbounded space, via a multiplicative transformation dependent solely on the boundary geometry. This conclusion allows in practice for the compensation of boundary effects and the reconstruction of the intrinsic step distribution from empirical observations.

Convexity in Ordered Matroids and the Generalized External Order

In 1980, Las Vergnas defined a notion of discrete convexity for oriented matroids, which Edelman subsequently related to the theory of anti-exchange closure functions and convex geometries. In this paper, we use generalized matroid activity to construct a convex geometry associated with an ordered, unoriented matroid. The construction in particular yields a new type of representability for an ordered matroid defined by the affine representability of its corresponding convex geometry. The lattice of convex sets of this convex geometry induces an ordering on the matroid independent sets which extends the external active order on matroid bases. We show that this generalized external order forms a supersolvable meet-distributive lattice refining the geometric lattice of flats, and we uniquely characterize the lattices isomorphic to the external order of a matroid. Finally, we introduce a new trivariate generating function generalizing the matroid Tutte polynomial.

Convex Geometries are Extremal for the Generalized Sauer-Shelah Bound

The Sauer-Shelah lemma provides an exact upper bound on the size of set families with bounded Vapnik-Chervonekis dimension. When applied to lattices represented as closure systems, this lemma outlines a class of extremal lattices obtaining this bound. Here we show that the Sauer-Shelah bound can be easily generalized to arbitrary antichains, and extremal objects for this generalized bound are exactly convex geometries. We also show that the problem of classification of antichains admitting such extremal objects is NP-complete.