Lifts for semigroups of monomorphisms of an independence algebra
Volume 97 / 2003
Abstract
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 $G=\mathop{\rm Aut}\nolimits ({\cal A} )\leq S \leq \mathop{\rm End}\nolimits ({\cal A} )$. In [2] it is proved that if ${\cal A}$ is a set (that is, an algebra without operations), then $|L(\phi)|= 1$. The analogous result for vector spaces does not hold. Thus the natural question is: Characterize the independence algebras in which $|L(\phi)|=1$. The aim of this note is to answer this question.
