TWO CHARACTERIZATIONS of ANTIMATROIDS* 1. Introduction
ANALELE S¸TIINT¸IFICE ALE UNIVERSITAT¸II˘ \AL.I. CUZA" DIN IAS¸I (S.N.) MATEMATICA,˘ Tomul LIX, 2013, f.2 DOI: 10.2478/v10157-012-0048-1 TWO CHARACTERIZATIONS OF ANTIMATROIDS* BY HUA MAO Abstract. NextClosure algorithm is a fast and good algorithm in formal concept analysis. With the assistance of NextClosure algorithm, this article provides a charac- terization of antimatroids. Additionally, this article introduces a characterization of k- truncated antimatroids. The characterization can be realized by an algorithm which works with polynomial time delay. Mathematics Subject Classification 2010: 05B35, 52A01, 68R05. Key words: antimatroid, k-truncated antimatroid, convex geometry, NextClosure algorithm, formal concept analysis. 1. Introduction Since 1940 (cf. [3]), when the antimatroiods were introduced, people have continuously searched for other characterizations of antimatroids (cf. [1], [2], [6], [8], [9], [11], [12]). According to our knowledge, we add up the following points. (α) Kempner and Levit point out in [8] that there are many equivalent axiomatizations of antimatroids that may be separated into two categories: antimatroids defined as set systems and antimatroids defined as languages. They introduce in [8] an algorithmic characterization of antimatroids based on the idea of optimization using set functions defined as minimum values of linkages between a set and the elements from the set complement. *This research is supported by NSF of China (11101115, 61202178, 61073121) and NSF of Hebei Province (F2012402037, A2013201119). 454 HUA MAO 2 (β) There is an algorithmic characterization of antimatroids based on the language definition in algorithmic idea in [2]. (γ) There is one-to-one correspondence between antimatroids and con- vex geometries (cf.
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