  Let F be a family of subsets of {1,2,...,n}. The width-degree of an element x
in at least one member of F is the width of the family {U in F | x in U}. If F
has maximum width-degree at most k, then F is locally k-wide. Bounds on the
size of locally k-wide families of sets are established. If F is locally k-wide
and centered (every U in F has an element which does not belong to any member
of F incomparable to U), then |F| <= (k+1)(n-k/2); this bound is best possible.
Nearly exact bounds, linear in n and k, on the size of locally k-wide families
of arcs or segments are determined. If F is any locally k-wide family of sets,
then |F| is linearly bounded in n. The proof of this result involves an
analysis of the combinatorics of antichains. Let P be a poset and L a
semilattice (or an intersection-closed family of sets). The P-size of L is
|L^P|. For u in L, the P-density of u is the ratio |[u)^P|/|L^P|. The density
of u is given by the [1]-density of u. Let p be the number of filters of P. L
has the P-density property iff there is a join-irreducible a in L such that the
P-density of a is at most 1/p Which non-trivial semilattices have the P-density
property? For P=[1], it has been conjectured that the answer is: "all" (the
union-closed sets conjecture). Certain subdirect products of lower-semimodular
lattices and, for P=[n], of geometric lattices have the P-density property in a
strong sense. This generalizes some previously known results. A fixed lattice
has the [n]-density property if n is large enough. The density of a generator U
of a union-closed family of sets L containing the empty set is estimated. The
estimate depends only on the local properties of L at U. If L is generated by
sets of size at most two, then there is a generator U of L with estimated
density at most 1/2.
