Abstract
The aim of this note is to emphasize the connection between the idempotent word problem for inverse monoid presentations and certain left closed subsets of the free group. The wide use of automata and languages throughout the paper justifies our choice of considering the free group as a subset of the free monoid. Automata theory is used to provide a decidability result concerning rational languages. As a consequence, an alternative proof to the theorem of Meakin and Margolis on idempotent-pure presentations [7] is obtained, and some new cases are established. Moreover, an example of a finitely presented inverse monoid with undecidable idempotent word problem is given.
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