The infinite algebras with 3-transitive groups of weak automorphisms are investigated. Among others it is shown that if an infinite algebra with 3-transitive group of weak automorphisms has a nontrivial idempotent polynomial operation then either it is locally functionally complete or it is polynomially equivalent to a vector space over the two element field or it is a simple algebra that is semi-affine with respect to an elementary 2-group. In the second and third cases the group of weak automorphisms cannot be 4-transitive.
@article{107802,
author = {L\'aszl\'o Szab\'o},
title = {Infinite algebras with 3-transitive groups of weak automorphisms},
journal = {Archivum Mathematicum},
volume = {037},
year = {2001},
pages = {245-256},
zbl = {1069.08001},
mrnumber = {1879447},
language = {en},
url = {http://dml.mathdoc.fr/item/107802}
}
Szabó, László. Infinite algebras with 3-transitive groups of weak automorphisms. Archivum Mathematicum, Tome 037 (2001) pp. 245-256. http://gdmltest.u-ga.fr/item/107802/
Homogeneous algebras are functionally complete, Algebra Universalis 11 (1980), 149–158. | MR 0588208
Algebras with $p$-uniform principal congruences, Studia Sci. Math. Hungar. 16 (1981), 229–235. | MR 0703660
On weak isomorphisms and weak homomorphisms of abstract algebras, Colloq. Math. 14 (1966), 163–167. | MR 0184889 | Zbl 0192.09504
Remarks on a paper of L. Szabó and Szendrei, Á., Acta Sci. Math. (Szeged) 42 (1980), 95–98. | MR 0576941
Independence in abstract algebras, Colloq. Math. 14 (1966), 169–188. | MR 0184890
Algebras with doubly transitive automorphism groups, in: Finite Algebra and Multiple-Valued Logic (Proc. Conf. Szeged, 1979), Colloq. Math. Soc. J. Bolyai, vol. 28, North-Holland, Amsterdam, 1981; 521–535. | MR 0648631
Automorphism groups and functional completeness, Algebra Universalis 15 (1982), 385–400. | MR 0689771
Local completeness I, Algebra Universalis 18 (1984), 308–326. | MR 0745495
A theorem concerning the composition of functions of several variables ranging over a finite set, J. Symbolic Logic 25 (1960), 203–208. | MR 0144815
Almost all algebras with triply transitive automorphism groups are functionally complete, Acta Sci. Math. (Szeged) 41 (1979), 391–402. | MR 0555434
Interpolation in algebras with doubly primitive automorphism groups, Elektron. Informationsverarbeit. Kybernetik 19 (1983), 603–610. | MR 0746848
Basic permutation groups on infinite sets, Acta Sci. Math. (Szeged) 47 (1984), 61–70. | MR 0755563
Triply transitive algebras, Acta Sci. Math. (Szeged) 51 (1987), 221–227. | MR 0911573
Algebras with transitive automorphism groups, Algebra Universalis 31 (1994), 589–598. | MR 1274075
On simple surjective algebras, Acta Sci. Math. (Szeged) 59 (1994), 17–23. | MR 1285425
Simple, surjective, transitive algebras, in: General Algebra and Ordered Sets (Proc. of the Summer School, 1994), Dept. of Algebra and Geometry, Palacký University Olomouc, Olomouc, Czech Republic; 138-143. | MR 1342550
Characteristically simple algebras, Acta Sci. Math. (Szeged) 63 (1997), 51–70. | MR 1459779
Algebras that are simple with weak automorphisms, Algebra Universalis 42 (1999), 205–233. | MR 1736715
Finite permutation group, Academic Press, New York and London, 1964. | MR 0183775