The cone of quasi-semimetrics and exponent matrices of tiled orders |
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Institution: | 1. Mathematics Department, Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP 05508-970, Brazil;2. Computer Science Department, Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP 05508-970, Brazil;3. Information Technology Department, Faculdade de Tecnologia de São Paulo, São Paulo, SP 01124-060, Brazil |
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Abstract: | Finite quasi semimetrics on n can be thought of as nonnegative valuations on the edges of a complete directed graph on n vertices satisfying all possible triangle inequalities. They comprise a polyhedral cone whose symmetry groups were studied for small n by Deza, Dutour and Panteleeva. We show that the symmetry and combinatorial symmetry groups are as they conjectured.Integral quasi semimetrics have a special place in the theory of tiled orders, being known as exponent matrices, and can be viewed as monoids under componentwise maximum; we provide a novel derivation of the automorphism group of that monoid. Some of these results follow from more general consideration of polyhedral cones that are closed under componentwise maximum. |
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Keywords: | Quasi-semimetric Polyhedral cone Exponent matrix Face lattice Symmetry Max-plus algebra |
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