Lifts for semigroups of monomorphisms of an independence algebra

João Araújo. "Lifts for semigroups of monomorphisms of an independence algebra." Colloquium Mathematicae 97.2 (2003): 277-284. <http://eudml.org/doc/286092>.

@article{JoãoAraújo2003,
abstract = {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.},
author = {João Araújo},
journal = {Colloquium Mathematicae},
keywords = {universal algebras; independence algebras; groups of automorphisms; semigroups of monomorphisms; lifts; maximal independent sets; endomorphism monoids},
language = {eng},
number = {2},
pages = {277-284},
title = {Lifts for semigroups of monomorphisms of an independence algebra},
url = {http://eudml.org/doc/286092},
volume = {97},
year = {2003},
}

TY - JOUR
AU - João Araújo
TI - Lifts for semigroups of monomorphisms of an independence algebra
JO - Colloquium Mathematicae
PY - 2003
VL - 97
IS - 2
SP - 277
EP - 284
AB - 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.
LA - eng
KW - universal algebras; independence algebras; groups of automorphisms; semigroups of monomorphisms; lifts; maximal independent sets; endomorphism monoids
UR - http://eudml.org/doc/286092
ER -