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Automorphism Groups of Algebras of Finite Type

Published online by Cambridge University Press:  20 November 2018

Matthew Gould*
Affiliation:
Vanderbilt University, Nashville, Tennessee
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By “algebra” we shall mean a finitary universal algebra, that is, a pair 〈A; F〉 where A and F are nonvoid sets and every element of F is a function, defined on A, of some finite number of variables. Armbrust and Schmidt showed in [1] that for any finite nonvoid set A, every group G of permutations of A is the automorphism group of an algebra defined on A and having only one operation, whose rank is the cardinality of A. In [6], Jónsson gave a necessary and sufficient condition for a given permutation group to be the automorphism group of an algebra, whereupon Plonka [8] modified Jonsson's condition to characterize the automorphism groups of algebras whose operations have ranks not exceeding a prescribed bound.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Armbrust, M. and Schmidt, J., Zum Cayleyschen Darstellungssatz, Math. Ann. 154 (1964), 7073.Google Scholar
2. Birkhoff, G., Sobre los grupos de automorfismos, Rev. Un. Mat. Argentina 11 (1946), 155157.Google Scholar
3. Gould, M., Multiplicity type and subalgebra structure in universal algebras, Pacific J. Math. 26 (1968), 469485.Google Scholar
4. Gould, M., A note on automorphisms of groupoids, Algebra Universalis 2 (1972), 3638.Google Scholar
5. Grätzer, G., Universal algebra (D. Van Nostrand Co., Princeton, N.J., 1968).Google Scholar
6. Jónsson, B., Algebraic structures with prescribed automorphism groups, Colloq. Math. XIX (1968), 14.Google Scholar
7. Jónsson, B., Topics in universal algebra (lecture notes, Vanderbilt University, 1969-70).Google Scholar
8. Plonka, E., On a problem of B. Jónsson concerning automorphisms of a general algebra, Colloq. Math. XIX (1968), 58.Google Scholar
9. Vopěnka, P., Pultr, A., and Hedrlin, Z., A rigid relation exists on any set, CM.U.C. 6 (1965), 149155.Google Scholar