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arXiv · 2010.11442

Lattice Diversities

David Bryant·Raúl Felipe·Mauricio Toledo-Acosta·Paul Tupper
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Abstract

Diversities are a generalization of metric spaces, where instead of the non-negative function being defined on pairs of points, it is defined on arbitrary finite sets of points. Diversities have a well-developed theory. This includes the concept of a diversity tight span that extends the metric tight span in a natural way. Here we explore the generalization of diversities to lattices. Instead of defining diversities on finite subsets of a set we consider diversities defined on members of an arbitrary lattice (with a 0). We show that many of the basic properties of diversities continue to hold. However, the natural map from a lattice diversity to its tight span is not a lattice homomorphism, preventing the development of a complete tight span theory as in the metric and diversity cases.

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David Bryant, Raúl Felipe, Mauricio Toledo-Acosta, Paul Tupper. 2020-10-22. Lattice Diversities. https://arxiv.org/abs/2010.11442

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