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 Sch\"{u}tzenberger graph of a positive word $w$, $S\Gamma(w)$, where $w$ contains $r$ as its subword. We show that the word problem for a finitely presented Adian inverse semigroup $Inv\langle X|R \rangle$ is decidable if the subgraphs of $S\Gamma(t)$, for all $t\in X^+$, generated by all the $R$-words over the presentation $\langle X|R\rangle$, 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.