  We consider the concept of rank as a measure of the vertical levels and
positions of elements of partially ordered sets (posets). We are motivated by
the need for algorithmic measures on large, real-world
hierarchically-structured data objects like the semantic hierarchies of
ontological databases. These rarely satisfy the strong property of gradedness,
which is required for traditional rank functions to exist. Representing such
semantic hierarchies as finite, bounded posets, we recognize the duality of
ordered structures to motivate rank functions which respect verticality both
from the bottom and from the top. Our rank functions are thus interval-valued,
and always exist, even for non-graded posets, providing order homomorphisms to
an interval order on the interval-valued ranks. The concept of rank width
arises naturally, allowing us to identify the poset region with point-valued
width as its longest graded portion (which we call the "spindle"). A standard
interval rank function is naturally motivated both in terms of its extremality
and on pragmatic grounds. Its properties are examined, including the
relationship to traditional grading and rank functions, and methods to assess
comparisons of standard interval-valued ranks.
