Abstract
An antimatroid is a family of sets which is accessible, closed under union, and includes an empty set. A number of examples of antimatroids arise from various kinds of shellings and searches on combinatorial objects, such as, edge/node shelling of trees, poset shelling, node-search on graphs, etc. (Discrete Math. 78 (1989) 223; Geom. Dedicata 19 (1985) 247; Greedoids, Springer, Berlin, 1980) [1–3]. We introduce a one-element extension of antimatroids, called a lifting, and the converse operation, called a reduction. It is shown that a family of sets is an antimatroid if and only if it is constructed by applying lifting repeatedly to a trivial lattice. Furthermore, we introduce two specific types of liftings, 1-lifting and 2-lifting, and show that a family of sets is an antimatroid of poset shelling if and only if it is constructed from a trivial lattice by repeating 1-lifting. Similarly, an antimatroid of edge-shelling of a tree is shown to be constructed by repeating 2-lifting, and vice versa.
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