  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.
