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Generating countable sets of surjective functions

100%

Mitchell J., Péresse Y.

Fundamenta Mathematicae

2011

tom 213

nr 1

67-93

We prove that any countable set of surjective functions on an infinite set of cardinality ℵₙ with n ∈ ℕ can be generated by at most n²/2 + 9n/2 + 7 surjective functions of the same set; and there exist n²/2 + 9n/2 + 7 surjective functions that cannot be generated by any smaller number of surjections. We also present several analogous results for other classical infinite transformation semigroups such as the injective functions, the Baer-Levi semigroups, and the Schützenberger monoids.

Highly transitive subgroups of the symmetric group on the natural numbers

100%

Darji U., Mitchell J.

Colloquium Mathematicum

2008

tom 112

nr 1

163-173

Highly transitive subgroups of the symmetric group on the natural numbers are studied using combinatorics and the Baire category method. In particular, elementary combinatorial arguments are used to prove that given any nonidentity permutation α on ℕ there is another permutation β on ℕ such that the subgroup generated by α and β is highly transitive. The Baire category method is used to prove that for certain types of permutation α there are many such possibilities for β. As a simple corollary, if $2 ≤ κ ≤ 2^{ℵ₀}$, then the free group of rank κ has a highly transitive faithful representation as permutations on the natural numbers.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Fundamenta Mathematicae

2 Mitchell J.

1 Darji U.

1 Péresse Y.

1 2011

1 2008

Wyniki wyszukiwania

Generating countable sets of surjective functions

Highly transitive subgroups of the symmetric group on the natural numbers