Pilkington, Annette Convex geometries on root systems. (English) Zbl 1103.52001 Commun. Algebra 34, No. 9, 3183-3202 (2006). Let \(X\) be a set and let \(M\) be a collection of subsets of \(X\) with the properties: (i) \(\emptyset\in M\) and \(X\in M\); (ii) \(A_i\in M\) for \(i\in I\) implies \(\bigcap_{i\in I}A_i\in M\). We call \(M\) an alignment of \(X\). If \(A\subset X\), the closure of \(A\), with respect to the alignment \(M\), is defined to be \(\overline A=\bigcap_{\{B\in M\mid B\supset A\}}B\). The subsets in \(M\) that are of the form \(\overline A\), are called convex sets. We say \(M\) is anti-exchange, or satisfies the anti-exchange condition, if given any convex set \(K\) and two unequal points \(p\) and \(q\) in \(X\), neither in \(K\), then \(q\in\overline{K\cup\{p\}}\) implies that \(p\notin\overline{K\cup\{q\}}\). Equivalently, we call the operator \(A\to\overline A\) a convex closure operator if \(M\) satisfies the anti-exchange condition. In this paper the author investigates several convex closure operators on a finite root system. It is shown that a natural closure operator on the positive root system of a finite Weyl group satisfies the anti-exchange condition for all root systems except type \(F_4\). Reviewer: Chen Chengdong (Shanghai) Cited in 11 Documents MSC: 52A01 Axiomatic and generalized convexity 17B20 Simple, semisimple, reductive (super)algebras 20F55 Reflection and Coxeter groups (group-theoretic aspects) 06A15 Galois correspondences, closure operators (in relation to ordered sets) Keywords:convex geometries; Coxeter groups; matroids; root systems; closure operators PDFBibTeX XMLCite \textit{A. Pilkington}, Commun. Algebra 34, No. 9, 3183--3202 (2006; Zbl 1103.52001) Full Text: DOI References: [1] Bourbaki N., Groupes ét Algebres de Lie, Ch. 4–6 (1968) · Zbl 0186.33001 [2] DOI: 10.1007/BF01445101 · Zbl 0793.20036 · doi:10.1007/BF01445101 [3] Dyer M. J., Comp. Math. 89 pp 91– (1993) [4] DOI: 10.1007/BF00149365 · Zbl 0577.52001 · doi:10.1007/BF00149365 [5] DOI: 10.1112/jlms/s2-21.1.62 · Zbl 0427.20040 · doi:10.1112/jlms/s2-21.1.62 [6] Humphreys J. E., Reflection Groups and Coxeter Groups (1990) · Zbl 0725.20028 [7] Moody R., Lie Algebras with Triangular Decompositions (1995) · Zbl 0874.17026 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.