Transformations of Antimatroid Closure Spaces

Investigation of the transformations of vector spaces, whose most abstract formulations are called matroids, is basic in mathematics; but transformations of discrete spaces have received relatively little attention. This paper develops the concept of transformations of discrete spaces in the context of antimatroid closure spaces. The nature of these transformations are quite different from those encountered in linear algebra because the underlying spaces are strikingly different. The transformation properties of "closed", "continuous" and "order preserving" are defined and explored. The classic graph transformations, homomorphism and topological sort, are examined in the context of these properties. Then we define a deletion which we believe plays a central role in discrete transformations. Antimatroid closure spaces, when partially ordered, can be interpreted as lattices. We show that deletions induce lower semihomomorphisms between the corresponding lattices.