The concept of a median semilattice is generalized in the following way: a meet semilattice \(S\) is called \(n\)-median semilattice iff all principal ideals in \(S\) are distributive lattices and any \(n\)-element subset of \(S\) has an upper bound whenever each of its \((n-1)\)-element subsets has an upper bound. Examples of \((n+1)\)-median semilattices which are not \(n\)- median are given. For a meet-semilattice \(S\) in which every principal ideal is a lattice, conditions are given under which the quota number of \(S\) has a prescribed value; the quota number of a meet semilattice \(S\) is the infimum of all quotients \(k/m\) for which \(\bigvee_ I \bigwedge_{i\in I} x_ i\) with \(I\subseteq\{1,\dots,m\}\), \(| I|=k\), exists for all \(x_ 1,\dots,x_ m\in S\). Further, some recursive descriptions are given which enable to characterize an \(n\)- median semilattice \(S\) with the help of \(k\)-median subsemilattices, where \(k