The word problem for some classes of Adian inverse semigroups-II

We introduce the notion of a subgraph generated by an $R$-word $r$ of the Schützenberger graph of a positive word $w$, $S\mathrm{\Gamma }(w)$, where $w$ contains $r$ as its subword. We show that the word problem for a finitely presented Adian inverse semigroup $Inv⟨X\mathrm{\mid }R⟩$ is decidable if the subgraphs of $S\mathrm{\Gamma }(t)$, for all $t\in X^{+}$, generated by all the $R$-words over the presentation $⟨X\mathrm{\mid }R⟩$, are finite. As a consequence of this result, we show that the word problem is decidable for some classes of one relation Adian inverse semigroups.