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Automorphism Groups of Algebras of Finite Type

  • Matthew Gould (a1)

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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.

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References

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