Geometry of poset antimatroids

An antimatroid is an accessible set system closed under union. A poset antimatroid is a particular case of antimatroid, which is formed by the lower sets of a poset. Feasible sets in a poset antimatroid ordered by inclusion form a distributive lattice, and every distributive lattice can be formed in this way. We introduce the polydimension of an antimatroid as the minimum dimension d such that the antimatroid may be isometrically embedded into d-dimensional integer lattice Zd. We prove that every antimatroid of poly-dimension 2 is a poset antimatroid, and demonstrate both graph and geometric characterizations of such antimatroids. Finally, a conjecture concerning poset antimatroids of arbitrary poly-dimension d is presented.