Free inverse semigroups

1. There is no innnite sequence g 1 =) =) g i =) 2. If h 1 (= g =) h 2 then there is g 2 G X such that h 1 =) g (= h 2. Proof. For (1) just note that each =)-step diminishes by one the number of vertices of a nite graph. For (2), if the two subgraphs to be rewritten in g are edge-disjoints, then deene g as the graph obtained from g by doing both rewritings (note that the order in which they are done does not matter). If the two subgraphs to be rewritten have common edges, g must have a subgraph like a ? a ?! a ?. Observe that again g deened as before works. From the two statements of Lemma 3, a purely combinatorial argument shows that a \global" version of 3(2) also holds: If h 1 (= g =) h 2 then there is g such that h 1 =) g (= h 2. In fact, something seemingly stronger, but actually equivalent to it, can be proved (for a discussion of these rewriting concepts, and the missing proofs, see 3]): Lemma 4. If g 1 () g 2 then there is g such that h 1 =) g (= h 2. Theorem 1. Up to isomorphism, the free inverse semigroup on X consists of all isomorphism classes of birooted word-trees on X. Proof. Lemmas 3 and 4 show that each ()-class of graphs in G X has a canonical representative: Consider any g in the class, and apply repeatedly =) until it is no more applicable. By Lemma 3(1) this process stops, and from Lemma 4, the element obtained, denoted by nf(g), can be proved to be unique. Clearly nf is an isomorphism. Also is an isomorphism with inverse : Given t 2 T X , observe that (w t) = w t , and using Lemma 2 it follows that I ` (t) = (w t) = w t = t. Similarly it can be shown that is the identity in G X. Hence