  In this paper we show that the (co)chain complex associated with a
decomposition of the computational domain, commonly called a mesh in
computational science and engineering, can be represented by a block-bidiagonal
matrix that we call the Hasse matrix. Moreover, we show that
topology-preserving mesh refinements, produced by the action of (the simplest)
Euler operators, can be reduced to multilinear transformations of the Hasse
matrix representing the complex. Our main result is a new representation of the
(co)chain complex underlying field computations, a representation that provides
new insights into the transformations induced by local mesh refinements. Our
approach is based on first principles and is general in that it applies to most
representational domains that can be characterized as cell complexes, without
any restrictions on their type, dimension, codimension, orientability,
manifoldness, connectedness.
