Free inverse monoids are co-context-free
Tara Macalister Brough, Marianne Johnson, Mark Kambites, Carl-Fredrik Nyberg-Brodda
TL;DR
This work proves that the co-word problem of the free inverse monoid on a finite generating set is context-free, extending prior rank-1 results to all finite ranks. The authors employ grammar-based descriptions and Munn-tree analysis to connect word equivalence in $FIM(X)$ with structural properties of Munn trees, and they decompose the co-word problem into a free-group component plus edge-difference components that are captured by a context-free grammar. The main contributions include explicit grammars for idempotents and for the $K_1$ component of the co-word problem, and a complete CFG construction for the co-word problem, showing it is context-free. This establishes the co$\, ext{CF}$ property for free inverse monoids of finite rank and provides a constructive, generator-independent method for recognizing co-word non-equivalence, with implications for inverse-monoid theory and related language-theoretic properties.
Abstract
We prove (using grammars) that the free inverse monoid of every finite rank has co-context-free word problem. Equivalently, the co-word problem of the free inverse monoid of every finite rank is context-free.