Undecidability of the word problem for Yamamura’s HNN-extension under nice conditions

Abstract

The solvability of the word problem for Yamamura’s HNN-extensions $$[S;A_{1},A_{2};\varphi ]$$ has been proved in some particular cases. However, we show that, contrary to the group case, the word problem for $$[S;A_{1}A_{2};\varphi ]$$ is undecidable even if we consider S to have finite $$\mathscr {R}$$ -classes, $$A_{1}$$ and $$A_{2}$$ to be free inverse subsemigroups of finite rank and with zero, and $$\varphi ,\varphi ^{-1}$$ to be computable.