On Submodular and Supermodular Functions on Lattices and Related Structures Dan A. Simovici University of Massachusetts Boston Department of Computer Science Boston, USA dsim@cs.umb.edu Abstract We give single-operations characterizations for sub- modular and supermodular functions on lattices that have monotonicity properties. We associate to such functions metrics on lattices and we investigate corresponding met- rics on the sets of partitions. Keywords-lattice; semilattice; submodularity; entropy; I. Introduction Submodular functions are useful in combinatorial op- timization problems [3], [4], [6]. They are defined as functions of the form f : P (S) −→ R, where S is a finite set and P (S) is the set of subsets of S, which satisfy the submodular inequality, that is, f (X ∩ Y )+ f (X ∪ Y )  f (X )+ f (Y ) for X, Y ∈ P (S). This inequality is equivalent to the “diminishing return property” of these functions which means that for every X, Y ∈P (S) such that X ⊆ Y and x ∈ S − Y , we have f (X ∪{x}) − f (X )  f (Y ∪{x}) − f (Y ). In this note we study submodular functions and their duals (known as supermodular functions) defined on lattices and we link these functions with a generalization of conditional entropy in lattices and with certain classes of metrics defined on these structures. The characterization of submodular or supermodular functions that have a monotonicity-linked property (obtained in Section II) is formulated using only one of the lattice operations, which opens the possibility of extending the notion of modularity to semilattices. Section III is dedicated to submodular and supramod- ular functions defined on partition lattices. The extension of semimodularity to functions defined on semilattices is of interest for the generalization of the notion of entropy (and of metrics derived from this notion) from partition lattices to other algebraic structures that play a role in designing data mining algorithms. II. Submodular Functions on Lattices A semilattice is a semigroup (S, ⋄), where ⋄ is a com- mutative and idempotent operation. This is a pervasive algebraic structure, with numerous applications in mathe- matics and computer science. A lattice is an algebraic structure (L, ∨, ∧) such that both (L, ∨) and (L, ∧) are semilattices and the two operations ∨ and ∧ satisfy the absorption laws (x ∨ y) ∧ y = y and (x ∧ y) ∨ y = y, for x, y ∈ L. Every semilattice (S, ⋄) generates a partial order on S defined by x  y if and only if x ⋄ y = y. Thus, a (L, ∨, ∧) generates two partial orders on L,“ 1 ” and “ 2 ” defined by x  1 y if x ∨ y = y, and x  2 y if x ∧ y = y for x, y ∈ L. Note that, by the absorption laws, we have u  1 v if and only if v  2 u for u, v ∈ L. The partial orders  1 and  2 are said to be dual of each other. Unless stated explicitly otherwise, we shall use the partial order  1 on lattices and will denote it simply by “”. If (P, ) and (Q, ) are two partially ordered sets (posets), then f : P −→ Q is a monotonic function if u  v implies f (u)  f (v) for u, v ∈ P . If u<v implies f (u) <f (v), then f is said to be strictly mono- tonic. The set of monotonic (strictly monotonic) functions from P to Q is denoted by MON(P, Q) (sMON(P, Q), respectively). If u  v implies f (u)  f (v) for u, v ∈ S, then f is an anti-monotonic function; the function f is strictly anti-monotonic if u<v implies f (u) >f (v) for all u, v ∈ P . The set of anti-monotonic (strictly