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Inverse monoids presented by a single sparse relator

Steven P Lindblad, University of Nebraska - Lincoln


We study a class of inverse monoids of the form M = Inv 〈X : w = 1〉, where the single relator w has a combinatorial property that we call sparse. For a sparse word w, we prove that the Schützenberger complex of the identity of M has a particularly nice topology. We analyze the manner in which the Schützenberger complex is constructed using an iterative procedure, due to Stephen, analogous to the Todd-Coxeter procedure. We define an appropriate notion of dual graph of the Schützenberger complex, and prove that this dual graph is a tree if w is sparse. We use this tree to construct a pushdown automaton that encodes the information contained in the Schützenberger complex. This pushdown automaton provides us with an algorithm that, given a word u ∈ (XX−1)*, determines whether or not u = 1 in M. This, together with results of Stephen and the fact due to Ivanov, Margolis, and Meakin that the inverse monoid is E-unitary, shows that the word problem is solvable for M. Finally, we provide an implementation, in the C++ programming language, of the algorithm to determine whether or not u = 1 in M. ^

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Recommended Citation

Lindblad, Steven P, "Inverse monoids presented by a single sparse relator" (2003). ETD collection for University of Nebraska - Lincoln. AAI3116590.