Let $M$ be a positive homogeneously presented monoid ${ \langle L \mid R\,\rangle}_{ mo}$. If $M$ satisfies the cancellation condition and carries certain particular elements similar to the \emph{ fundamental elements} in Artin monoids, then the solvability of the conjugacy problem for $M$ implies that in the corresponding group ${ \langle L \mid R\,\rangle}$. In addition to these conditions, if $M$ satisfies the LCM condition (i.e. any two elements $\alpha$ and $\beta$ in $M$ admit the left (resp.~right) least common multiple), then the solution to the conjugacy problem for $M$ is known. We will give two kinds of examples that do not satisfy only the LCM condition. For these examples, we will give a solution to the conjugacy problem by improving the method given by E. Brieskorn and K. Saito.
Categories: math.GR