Positive definite functions on semilattices

TL;DR

This paper defines positive definite functions on meet semilattices, explores their properties using matrix decompositions, and provides examples and counterexamples to illustrate these concepts.

Contribution

It introduces a new notion of positive definiteness for functions on meet semilattices and applies matrix decomposition techniques to analyze multivariate cases.

Findings

Characterization of positive definite functions on meet semilattices

Application of LDL^T decomposition to meet matrices

Examples and counterexamples of positive definite functions

Abstract

We introduce a notion of positive definiteness for functions $f\!:P\to\mathbb{R}$ defined on meet semilattices $(P,\preceq,\wedge)$ and prove several properties for these functions. In addition, we utilize the $LDL^{\rm T}$ decomposition of meet matrices in order to explore the properties of multivariate positive definite arithmetic functions $f\!:\mathbb{Z}_+^d\to\mathbb{R}$. Finally, we give a series of examples and counterexamples of positive definite functions.