Abstract. The main algebraic result of this paper contains a characterization of the weak subalgebra lattice of a unary partial algebra of a given finite unary type. This lattice must satisfy the conditions from [Bar] and moreover, one combinatorial condition concerning its atoms and join--irreducible elements. In a subsequent part [Pió2] we solve this problem for infinite unary types. Recall that in [Pió1] we reduced our algebraic problem for finite unary types to the following question: let $\bf G$ be a graph (which may have infinite sets of vertices and edges) and let $n$ be a natural number; when can $\bf G$ be directed ({\rm i.e.} when can its edges be directed) in such a way that at most $n$ directed edges start from each vertex? In the present paper we first solve this graph problem and hence we easily obtain the solution of our algebraic problem for finite unary types.

AMS Subject Classification (1991): 08A55, 08A60, 05C20, 05C90, 08A30

Received July 15, 1998, and in final form January 21, 1999. (Registered under 2694/2009.)