Lifts for semigroups of endomorphisms of an independence algebra

João Araújo. "Lifts for semigroups of endomorphisms of an independence algebra." Colloquium Mathematicae 106.1 (2006): 39-56. <http://eudml.org/doc/286409>.

@article{JoãoAraújo2006,
abstract = {For a universal algebra , let End() and Aut() denote, respectively, the endomorphism monoid and the automorphism group of . 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 be an independence algebra of infinite rank and let S be a monoid of monomorphisms such that G = Aut() ≤ S ≤ End(). It is obvious that G is characteristic in S. Fitzpatrick and Symons proved that if is a set (that is, an algebra without operations), then |L(ϕ)| = 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 = ⟨Aut() ∪ E ∪ R⟩ ≤ End(), where E is any set of idempotents and R is the empty set or a set containing a special monomorphism α and a special epimorphism α*.},
author = {João Araújo},
journal = {Colloquium Mathematicae},
keywords = {universal algebra; independence algebra; endomorphism monoid; automorphism group},
language = {eng},
number = {1},
pages = {39-56},
title = {Lifts for semigroups of endomorphisms of an independence algebra},
url = {http://eudml.org/doc/286409},
volume = {106},
year = {2006},
}

TY - JOUR
AU - João Araújo
TI - Lifts for semigroups of endomorphisms of an independence algebra
JO - Colloquium Mathematicae
PY - 2006
VL - 106
IS - 1
SP - 39
EP - 56
AB - For a universal algebra , let End() and Aut() denote, respectively, the endomorphism monoid and the automorphism group of . 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 be an independence algebra of infinite rank and let S be a monoid of monomorphisms such that G = Aut() ≤ S ≤ End(). It is obvious that G is characteristic in S. Fitzpatrick and Symons proved that if is a set (that is, an algebra without operations), then |L(ϕ)| = 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 = ⟨Aut() ∪ E ∪ R⟩ ≤ End(), where E is any set of idempotents and R is the empty set or a set containing a special monomorphism α and a special epimorphism α*.
LA - eng
KW - universal algebra; independence algebra; endomorphism monoid; automorphism group
UR - http://eudml.org/doc/286409
ER -