Congruences associated with DOL-schemes

Abstract:

For a DOL-scheme $(X,\varphi )$, where $X$ is a finite alphabet and $\varphi$ is an endomorphism of ${X^*}$, we study the properties of the congruence $\bar \varphi$ induced by $\varphi$ in terms of the properties of ${X^*}\varphi$. We prove that every submonoid of ${X^*}$ has a disjunctive subset (for any $X$) and deduce that $\bar \varphi$ is a syntactic congruence. As special cases, we consider the conditions on $\varphi$ which are equivalent to $\bar \varphi$ being perfect or uniquely perfect or linear. The latter is introduced in the paper together with a ramification.