  We discuss several extension properties of matroids and polymatroids and
their application as necessary conditions for the existence of different
matroid representations, namely linear, folded linear, algebraic, and entropic
representations. Iterations of those extension properties are checked for
matroids on eight and nine elements by means of computer-aided explorations,
finding in that way several new examples of non-linearly representable
matroids. A special emphasis is made on sparse paving matroids on nine points
containing the tic-tac-toe configuration. We present a new, more clear
description of that family and we analyze extension properties on those
matroids and their duals.
