  In this paper, we prove the conjectures of Gharakhloo and Welker (2023) that
the positive matching decomposition number (pmd) of a $3$-uniform hypergraph is
bounded from above by a polynomial of degree $2$ in terms of the number of
vertices. Moreover, we derive a lower bound for pmd specifically for complete
$3$-uniform hypergraphs. Additionally, we obtain an upper bound for pmd of
$r$-uniform hypergraphs. For a $r$-uniform hypergraphs $H=(V,E)$ such that
$\lvert e_i\cap e_j\rvert \leq 1$ for all $e_i,e_j \in E$, we give a
characterization of positive matching in terms of strong alternate closed
walks. For specific classes of a hypergraph, we classify the radical and
complete intersection Lov\'{a}sz$-$Saks$-$Schrijver ideals.
