Geodesic
Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property
Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 44-66

Voir la notice de l'article provenant de la source Cambridge University Press

We answer two questions of Hindman, Steprāns, and Strauss; namely, we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover, we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases; we also construct (assuming Martin's Axiom for countable partial orders, i.e., $\operatorname{cov}\left( \text{M} \right)=\mathfrak{c})$ , a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.
DOI : 10.4153/CJM-2015-023-9
Mots-clés : 03E75, 54D35, 54D80, 05D10, 05A18, 20K99, ultrafilter, Stone–Čech compactification, sparse ultrafilter, strongly summable ultrafilter, union ultrafilter, finite sum, additive isomorphism, trivial sums property, Boolean group, abelian group 
Bretón, David J. Fernández. Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property. Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 44-66. doi: 10.4153/CJM-2015-023-9
@article{10_4153_CJM_2015_023_9,
     author = {Bret\'on, David J. Fern\'andez},
     title = {Strongly {Summable} {Ultrafilters,} {Union} {Ultrafilters,} and the {Trivial} {Sums} {Property}},
     journal = {Canadian journal of mathematics},
     pages = {44--66},
     year = {2016},
     volume = {68},
     number = {1},
     doi = {10.4153/CJM-2015-023-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-023-9/}
}
TY  - JOUR
AU  - Bretón, David J. Fernández
TI  - Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property
JO  - Canadian journal of mathematics
PY  - 2016
SP  - 44
EP  - 66
VL  - 68
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-023-9/
DO  - 10.4153/CJM-2015-023-9
ID  - 10_4153_CJM_2015_023_9
ER  - 
%0 Journal Article
%A Bretón, David J. Fernández
%T Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property
%J Canadian journal of mathematics
%D 2016
%P 44-66
%V 68
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-023-9/
%R 10.4153/CJM-2015-023-9
%F 10_4153_CJM_2015_023_9

[1] [1] Blass, A., Ultrafilters related to Hindman's finite-unions theorem and its extensions. In: Logic and Combinatorics, Contemp. Math., 65, American Mathematical Society, Providence, RI, 1987, pp. 89–124. http://dx.doi.Org/10.1090/conm/065/891244 Google Scholar

[2] [2] Blass, A.and Hindman, N., On strongly summable ultrafihers and union ultrafilters. Trans. Amer. Math. Soc. 304(1987), no. 1, 83–99. http://dx.doi.Org/10.1090/S0002-9947-1987-0906807-4 Google Scholar

[3] [3] Eisworth, T., Forcing and stable-ordered union ultrafilters. J. Symbolic Logic 67(2002), no. 1,449–464. http://dx.doi.Org/10.2178/jsl/1190150054 Google Scholar

[4] [4] D. Fernandez Bretón, J., Every strongly summable ultrafilter on ⊕ℤ is sparse. New York J. Math. 19(2013), 117–129. Google Scholar

[5] [5] Hindman, N., The existence of certain ultrafilters on and a conjecture of Graham and Rothschild. Proc. Amer. Math. Soc. 36(1972), no. 2, 341–346. Google Scholar

[6] [6] Hindman, N., Summable ultrafilters and finite sums. In: Logic and Combinatorics, Contemp. Math., 65, American Mathematical Society, Providence, RI, 1987, 263–274. http://dx.doi.Org/10.1090/conm/065/891252 Google Scholar

[7] [7] Hindman, N. and Legette Jones, L., Idempotents in βS that are only products trivially. New York J. Math. 20(2014), 57–80. Google Scholar

[8] [8] Hindman, N., Protasov, I., and Strauss, D., Strongly summable ultrafilters on Abelian groups. Mat. Stud. 10(1998), no. 2, 121–132. Google Scholar

[9] [9] Hindman, N., Steprāns, J., and Strauss, D., Semigroups in which all strongly summable ultrafilters are sparse. New York J. Math. 18(2012), 835–848. Google Scholar

[10] [10] Hindman, N. and Strauss, D., Algebra in the Stone-Čech compactification. Second ed., de Gruyter Textbook, Walter de Gruyter, Berlin, 2012. Google Scholar

[11] [11] Krautzberger, P., On strongly summable ultrafilters. New York J. Math. 16(2010), 629–649. Google Scholar

[12] [12] Krautzberger, P., On union ultrafilters. Order 29(2012), 317–343. http://dx.doi.Org/10.1007/si1083-011-9223-3 Google Scholar

[13] [13] Protasov, I. V., Finite groups in βG. Mat. Stud. 10(1998), no. 1,17–22. Google Scholar

Cité par Sources :