  A polyhedral convex set optimization problem is given by a set-valued
objective mapping from the $n$-dimensional to the $q$-dimensional Euclidean
space whose graph is a convex polyhedron. This problem can be seen as the most
elementary subclass of set optimization problems, comparable to linear
programming in the framework of optimization with scalar-valued objective
function. Polyhedral convex set optimization generalizes both scalar and
multi-objective (or vector) linear programming. In contrast to scalar linear
programming but likewise to multi-objective linear programming, unbounded
problems can indeed have minimizers and provide a rich class of problem
instances. In this paper we extend the concept of finite infimizers from
multi-objective linear programming to not necessarily bounded polyhedral convex
set optimization problems. We show that finite infimizers can be obtained from
finite infimizers of a reformulation of the polyhedral convex set optimization
problem into a vector linear program. We also discuss two natural extensions of
solution concepts based on the complete lattice approach. Surprisingly, the
attempt to generalize the solution procedure for bounded polyhedral convex set
optimization problems introduced in [A. L\"ohne and C. Schrage. An algorithm to
solve polyhedral convex set optimization problems. {\em Optimization}
62(1):131--141, 2013.] to the case of not necessarily bounded problems uncovers
some problems, which will be discussed.
