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Nilpotent local automorphisms of an independence algebra

Published online by Cambridge University Press:  14 November 2011

Lucinda M. Lima
Lucinda M. Lima
Affiliation:
Departamento de Matemática Pura, Universidade do Porto, Praça Gomes Teixeira, 4000 Porto, Portugal; Department of Mathematics, University of York, Heslington, York Y01 5DD, U.K.
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Abstract

Let A be a strong independence algebra of infinite rank m. Let ℒ(A) be the inverse monoid of all local automorphisms of A. The Baer–Levi semigroup B over the algebra A is defined to be the subsemigroup of ℒ(A) consisting of all the elements α with dom α = A and corank im α = m. Let Km be the subsemigroup of ℒ(A) generated by B−1B. Then Km is inverse and it is generated (as a semigroup) by N2, the subset of ℒ(A) consisting of all nilpotent elements of index 2. The 2-nilpotent depth, Δ2(Km) of Km is defined to be the smallest positive integer t such that Km = N2∪…∪(N2)t. In fact, Δ2(Km) is either 2 or 3 and a criterion is found which distinguishes between the two cases.

If N denotes the set of all nilpotents in ℒ(A), then the subsemigroup generated by N is also Km. In fact, Km is proved to be exactly N2.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 3 , 1994 , pp. 423 - 436
Copyright
Copyright © Royal Society of Edinburgh 1994

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