Transformations of Antimatroid Closure Spaces (1998) [3 citations — 3 self]
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Abstract:
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.
Citations
| 232 | Matroid Theory – Welsh - 1976 |
| 33 | The theory of convex geometries, Geometriae Dedicata 19 – Edelman, Jamison - 1985 |
| 30 | Convexity in graphs and hypergraphs – Farber, Jamison - 1986 |
| 24 | On Devaney's definition of chaos – Banks, Brooks, et al. - 1992 |
| 23 | Meet-distributive lattices and the anti-exchange closure – Edelman - 1980 |
| 16 | Closure Lattices – Pfaltz - 1996 |
| 11 | A Use for Frequently Rediscovering a Concept – Monjardet - 1985 |
| 9 | Convexity in Directed Graphs – Pfaltz - 1971 |
| 8 | Matroids and Antimatroids --- A Survey – Dietrich - 1989 |
| 7 | A Tutte polynomial for partially ordered sets – Gordon - 1993 |
| 6 | Evaluating the binary partition function when – Pfaltz - 1995 |
| 5 | Some interpretations of abstract linear dependence in terms of projective geometry – MacLane - 1936 |
| 2 | Transformations of structures by convex homomorphisms – Pfaltz - 1983 |
| 1 | Semi-homomorphisms of semi-modular lattices – Pfaltz - 1969 |
