Positive definite functions on semilattices

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.