The word problem for some classes of one relation Adian inverse semigroups

Abstract

We show that if the Schützenberger graph of every positive word, that contains an $R$-word only once as it's subword, is finite over an Adain presentation $⟨X|u=v⟩$, then the Schützenberger graph of every positive word is finite over the presentation $⟨X|u=v⟩$. This enable us to solve the word problem ffor some classes of one relation Adian inverse semigroups.