  In this work we study the framework of mathematical morphology on simplicial
complex spaces. Simplicial complexes are widely used to represent
multidimensional data, such as meshes, that are two dimensional complexes, or
graphs, that can be interpreted as one dimensional complexes. Mathematical
morphology is one of the most powerful frameworks for image processing,
including the processing of digital structures, and is heavily used for many
applications. However, mathematical morphology operators on simplicial complex
spaces is not a concept fully developed in the literature. Specifically, we
explore properties of the dimensional operators, small, versatile operators
that can be used to define new operators on simplicial complexes, while
maintaining properties from mathematical morphology. These operators can also
be used to recover many morphological operators from the literature. Matlab
code and additional material, including the proofs of the original properties,
are freely available at
\url{https://code.google.com/p/math-morpho-simplicial-complexes.}
