Lifts for semigroups of endomorphisms of an independence algebra

João Araújo¹

¹ Centro de Álgebra Universidade de Lisboa Av. Gama Pinto, 2 1649-003 Lisboa, Portugal and Universidade Aberta Rua da Escola Politécnica, no. 141–147 1269-001 Lisboa, Portugal

Colloquium Mathematicum, Tome 106 (2006) no. 1, pp. 39-56.

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 $G=\mathop{\rm Aut}\nolimits({\cal A} )\leq S \leq \mathop{\rm End}\nolimits({\cal A} )$. 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 $\alpha^*$.

Détail

DOI : 10.4064/cm106-1-4

Keywords: universal algebra cal mathop end nolimits cal mathop aut nolimits cal denote respectively endomorphism monoid automorphism group cal semigroup characteristic subsemigroup say phi mathop aut nolimits lift psi mathop aut nolimits phi psi psi mathop aut nolimits denote psi set lifts psi psi phi mathop aut nolimits mid phi psi cal independence algebra infinite rank monoid monomorphisms mathop aut nolimits cal leq leq mathop end nolimits cal obvious characteristic fitzpatrick symons proved cal set algebra without operations phi author proved previous paper analogue result does monoids monomorphisms independence algebra paper prove analogue result above holds semigroups langle mathop aut nolimits cal cup cup rangle leq mathop end nolimits cal where set idempotents empty set set containing special monomorphism nbsp alpha special epimorphism nbsp alpha *