Abstract
Free partially commutative inverse monoids are investigated. As in the case of free partially commutative monoids or groups (trace monoids or graph groups), free partially commutative inverse monoids are defined as quotients of free inverse monoids modulo a partially defined commutation relation on the generators. A quasi linear time algorithm for the word problem is presented. More precisely, we give an \(\mathcal {O}(n\log(n))\) algorithm for a RAM. \(\mathsf {NP}\) -completeness of the submonoid membership problem (also known as the generalized word problem) and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. It turns out that the word problem is decidable if and only if the complement of the partial commutation relation is transitive.
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