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Automorphism-primal algebras generate verbose varieties. (English) Zbl 1323.08005

A finite algebra is automorphism-primal if its clone of term functions coincides with the clone of all operations that preserve its automorphisms. It was proved by H. Werner that a finite algebra is automorphism-primal if and only if it is quasiprimal, every subuniverse is the set of fixed points of a group of automorphisms, and every isomorphism between nontrivial subalgebras can be extended to an automorphism of the whole algebra. As an example, every finite field is automorphism-primal. It is proved that the variety generated by an automorphism-primal algebra is verbose, i.e. that every fully invariant congruence on each of its members is verbal. The proof use natural dualities as developed by B. Davey.

MSC:

08B10 Congruence modularity, congruence distributivity
08A30 Subalgebras, congruence relations
08A35 Automorphisms and endomorphisms of algebraic structures
08C20 Natural dualities for classes of algebras
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References:

[1] Bergman, C.: Universal Algebra. Fundamentals and Selected Topics, Pure and Applied Mathematics (Boca Raton), vol. 301. CRC Press, Boca Raton, FL (2012) · Zbl 1229.08001
[2] Bergman C., Berman J.: Fully invariant and verbal congruence relations. Algebra Universalis 70, 71-94 (2013) · Zbl 1280.08001 · doi:10.1007/s00012-013-0238-z
[3] Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist, Cambridge Studies in Advanced Mathematics, vol. 57. Cambridge University Press, Cambridge (1998) · Zbl 0910.08001
[4] Werner, H.: Discriminator Algebras. Studium zur Algebra und ihre Andwendungen 6. Akademie Verlag, Berlin (1978) · Zbl 1280.08001
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