  A lattice is a partially ordered set supporting a meet (or join) operation
that returns the largest lower bound (smallest upper bound) of two elements.
Just like graphs, lattices are a fundamental structure that occurs across
domains including social data analysis, natural language processing,
computational chemistry and biology, and database theory. In this paper we
introduce discrete-lattice signal processing (DLSP), an SP framework for data,
or signals, indexed by such lattices. We use the meet (or join) to define a
shift operation and derive associated notions of filtering, Fourier basis and
transform, and frequency response. We show that the spectrum of a lattice
signal inherits the lattice structure of the signal domain and derive a
sampling theorem. Finally, we show two prototypical applications: spectral
analysis of formal concept lattices in social science and sampling and Wiener
filtering of multiset lattices in combinatorial auctions. Formal concept
lattices are a compressed representation of relations between objects and
attributes. Since relations are equivalent to bipartite graphs and hypergraphs,
DLSP offers a form of Fourier analysis for these structures.
