论文标题
晶格多样性
Lattice Diversities
论文作者
论文摘要
多样性是对度量空间的概括，在这些点成对上定义的非阴性函数是在任意有限的点集中定义的。多样性具有完善的理论。这包括多样性紧密跨度的概念，该跨度以自然的方式扩展了度量的紧密跨度。在这里，我们探讨了多样性对晶格的概括。我们考虑在任意晶格的成员（以0）为定义的多样性，而不是定义一组有限子集的多样性。我们表明，多样性的许多基本属性继续存在。但是，从晶格多样性到其紧密跨度的自然图并不是晶格的同态，从而阻止了完全紧密的跨度理论的发展。
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.
