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## Colloquium Mathematicum

2006 | 106 | 1 | 39-56
Tytuł artykułu

### Lifts for semigroups of endomorphisms of an independence algebra

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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). It is obvious that G is characteristic in S. Fitzpatrick and Symons proved that if 𝓐 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(𝓐) ∪ E ∪ R⟩ ≤ End(𝓐), 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*.
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Tom
Numer
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39-56
Opis fizyczny
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wydano
2006
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autor
• Centro de Álgebra, Universidade de Lisboa, Av. Gama Pinto, 2, 1649-003 Lisboa, Portugal
• Universidade Aberta, Rua da Escola Politécnica, no. 141-147, 1269-001 Lisboa, Portugal
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