  The relation between discrete topological field theories on triangulations of
two-dimensional manifolds and associative algebras was worked out recently. The
starting point for this development was the graphical interpretation of the
associativity as flip of triangles. We show that there is a more general
relation between flip-moves with two $n$-gons and $Z_{n-2}$-graded associative
algebras. A detailed examination shows that flip-invariant models on a lattice
of $n$-gons can be constructed {}from $Z_{2}$- or $Z_{1}$-graded algebras,
reducing in the second case to triangulations of the two-dimensional manifolds.
Related problems occure naturally in three-dimensional topological lattice
theories.
