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Convex geometries on root systems. (English) Zbl 1103.52001

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 \(F_4\).

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)
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References:

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