Interval-Valued Rank in Finite Ordered Sets

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 with respect to 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.