A Munn tree type representation for the bifree locally inverse semigroup

There are two traditional approaches to the free inverse monoid. Scheiblich's approach as pairs (A,u) where A is a "closed" set of group words and u belongs to A, and Munn's approach where the elements of the free inverse monoid are represented as birooted edge-labeled digraphs (Munn trees). Scheiblich's approach has been generalized for the bifree locally inverse semigroup by Auinger. In this talk we generalize Munn's approach. The straight bound between inverse semigroups and groups is now set in terms of locally inverse semigroups and completely simple semigroups. However, there are substantial differences on the graphs that we need to consider for the locally inverse case when comparing with the Munn trees. The graphs we shall consider are no longer edge-labeled nor digraphs. Instead, are the vertices that have labels. But the more striking fact about these graphs is that the vertices, which shall be called blocks, have a complex structure: they are graphs themselves. In this talk we shall describe a model for the bifree locally inverse semigroup where the elements are represented as "block-graphs".