For a universal algebra 𝓐, let End(𝓐) and Aut(𝓐) denote, respectively, the endomorphism monoid and the automorphism group of 𝓐. Let S be a semigroup and let T be a characteristic subsemigroup of S. We say that ϕ ∈ Aut(S) is a lift for ψ ∈ Aut(T) if ϕ|T = ψ. For ψ ∈ Aut(T) we denote by L(ψ) the set of lifts of ψ, that is, L(ψ) = {ϕ ∈ Aut(S) | ϕ|T = ψ}. Let 𝓐 be an independence algebra of infinite rank and let S be a monoid of monomorphisms such that G = Aut(𝓐) ≤ S ≤ End(𝓐). In [2] it is proved that if 𝓐 is a set (that is, an algebra without operations), then |L(ϕ)| = 1. The analogous result for vector spaces does not hold. Thus the natural question is: Characterize the independence algebras in which |L(ϕ)| = 1. The aim of this note is to answer this question.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm97-2-11,
author = {Jo\~ao Ara\'ujo},
title = {Lifts for semigroups of monomorphisms of an independence algebra},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {277-284},
zbl = {1056.20048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm97-2-11}
}
João Araújo. Lifts for semigroups of monomorphisms of an independence algebra. Colloquium Mathematicae, Tome 96 (2003) pp. 277-284. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm97-2-11/