Lifts for semigroups of endomorphisms of an independence algebra
Tom 106 / 2006
Streszczenie
For a universal algebra ${\cal A}$, let $\mathop{\rm End}\nolimits({\cal A} )$ and $\mathop{\rm Aut}\nolimits({\cal A} )$ denote, respectively, the endomorphism monoid and the automorphism group of ${\cal A}$. Let $S$ be a semigroup and let $T$ be a characteristic subsemigroup of $S$. We say that $\phi \in \mathop{\rm Aut}\nolimits(S)$ is a lift for $\psi\in \mathop{\rm Aut}\nolimits(T)$ if $\phi|T=\psi$. For $\psi \in \mathop{\rm Aut}\nolimits(T)$ we denote by $L(\psi)$ the set of lifts of $\psi$, that is, $$ L(\psi )= \{\phi \in \mathop{\rm Aut}\nolimits(S) \mid \phi|_T=\psi\}. $$ Let ${\cal A}$ be an independence algebra of infinite rank and let $S$ be a monoid of monomorphisms such that $G=\mathop{\rm Aut}\nolimits({\cal A} )\leq S \leq \mathop{\rm End}\nolimits({\cal A} )$. It is obvious that $G$ is characteristic in $S$. Fitzpatrick and Symons proved that if ${\cal A}$ is a set (that is, an algebra without operations), then $|L(\phi)|= 1$. The author proved in a previous paper that the analogue of this result does not hold for all monoids of monomorphisms of an independence algebra. The aim of this paper is to prove that the analogue of the result above holds for semigroups $S=\langle \mathop{\rm Aut}\nolimits ({\cal A} ) \cup E \cup R\rangle\leq \mathop{\rm End}\nolimits({\cal A} )$, where $E$ is any set of idempotents and $R$ is the empty set or a set containing a special monomorphism $\alpha$ and a special epimorphism $\alpha^*$.
