Lifts for semigroups of monomorphisms of an independence algebra

João Araújo ¹

¹ Universidade Aberta R. Escola Politécnica, 147 1269-001 Lisboa, Portugal and Centro de Álgebra Universidade de Lisboa Av. Gama Pinto, 2 1649-003 Lisboa, Portugal

Colloquium Mathematicum, Tome 97 (2003) no. 2, pp. 277-284

Résumé

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.

Détail

DOI : 10.4064/cm97-2-11

Keywords: universal algebra cal mathop end nolimits cal mathop aut nolimits cal denote respectively endomorphism monoid automorphism group nbsp cal nbsp semigroup characteristic subsemigroup nbsp say phi mathop aut nolimits lift psi mathop aut nolimits phi psi psi mathop aut nolimits denote psi set lifts nbsp psi psi phi mathop aut nolimits mid phi psi cal independence algebra infinite rank nbsp monoid monomorphisms mathop aut nolimits cal leq leq mathop end nolimits cal nbsp proved cal set algebra without operations phi analogous result vector spaces does natural question characterize independence algebras which phi note answer question