Difference operators on lattices

Abstract

A differential operator of weight [Formula: see text] is the algebraic abstraction of the difference quotient [Formula: see text], including both the derivation as [Formula: see text] approaches to [Formula: see text] and the difference operator when [Formula: see text]. Correspondingly, differential algebra of weight [Formula: see text] extends the well-established theories of differential algebra and difference algebra. In this paper, we initiate the study of differential operators with weights, in particular difference operators, on lattices. We show that differential operators of weight [Formula: see text] on a lattice coincide with differential operators, while differential operators are special cases of difference operators. Distributivity of a lattice is characterized by the existence of certain difference operators. Furthermore, we characterize and enumerate difference operators on finite chains and finite quasi-antichains.