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A representation theorem for weak automorphisms of a universal algebra

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Abstract

Given a group G and a descending chainG 0,G 1,...,G n, of normal subgroups ofG, we prove that there exists a universal algebra\(\mathfrak{A}\), such that the chain ...⊲Wn(\(\mathfrak{A}\))⊲...⊲W1(\(\mathfrak{A}\)})⊲ W0(\(\mathfrak{A}\))⊲W(\(\mathfrak{A}\)) is isomorphic to the chain ...⊲G n⊲ ...⊲G 1⊲G 0⊲G, where W(\(\mathfrak{A}\)) is the group of weak automorphisms of\(\mathfrak{A}\), and Wn(\(\mathfrak{A}\)) is the group of weak automorphisms of\(\mathfrak{A}\) that leaves alln-ary operations fixed.

We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.

These results are a generalization of those proved by J. Sichler, in the special case when G=G0, and G1=G2=...=Gn=....

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Authors and Affiliations

  1. University of Dayton, Dayton, Ohio, USA

    Aparna W. Higgins

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  1. Aparna W. Higgins
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Additional information

This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.

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Higgins, A.W. A representation theorem for weak automorphisms of a universal algebra. Algebra Universalis 20, 179–193 (1985). https://doi.org/10.1007/BF01278595

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  • DOI: https://doi.org/10.1007/BF01278595

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