A p-semilattice S is an algebra (S,\(\wedge,*,0)\) such that (S,\(\wedge)\) is a meet-semilattice with least element 0 and \(y\leq x^*\) iff \(y\wedge x=0\). S is algebraically closed (a.c.) iff any finite system of polynomial equations with constants from S has a solution in S itself provided it has one in some p-semilattice \(S_ 1\) extending S. In the same way one defines ''algebraically closed in \({\mathcal K}''\) for any class \({\mathcal K}\) of p-semilattices - considering only extensions \(S_ 1\) in \({\mathcal K}.\) The present paper determines explicitly the finite a.c. members of the natural Horn classes of p-semilattices which correspond to the well-known Lee classes of distributive p-algebras; they are certain direct products of subdirectly irreducible p-semilattices. The result is then used to characterize all a.c. members of the classes in question as direct limits of such direct products. The method used is mainly combinatorial analysis of the subdirect representations of a.c. p-semilattices.