Section: New Results

Tiles and inverse semigroups

In [10] it has already been observed that the theory of inverse semigroups (See  [43] for general presentation of inverse semigroup theory, and  [45] , [44] for graph-based representation of inverse semigroup elements.) is the adequate mathematical framework to define and study tiles and their languages. In this direction, we have shown that strings, trees and even many types of graphs can be unified into a notion of higher-dimensional strings [24] , [35] . Using techniques of partial algebra [4] , this notion recovers advanced results on formal languages of graphs of bounded tree-width (See  [38] for an up-to-date presentation of the formal language theory of graphs.), which shows the robustness of the approach.