Almost disjoint sets and Martin's axiom

1979 ◽  
Vol 44 (3) ◽  
pp. 313-318 ◽  
Author(s):  
Michael L. Wage
Keyword(s):  
Partial Order ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Almost Disjoint ◽  
Disjoint Sets

AbstractWe present a number of results involving almost disjoint sets and Martin's axiom. Included is an example, due to K. Kunen, of a c.c.c. partial order without property K whose product with every c.c.c. partial order is c.c.c.

Strongs-Sequences and Variations on Martin's Axiom

1985 ◽  
Vol 37 (4) ◽  
pp. 730-746 ◽  
Author(s):  
Juris Steprāns
Keyword(s):  
Pairwise Disjoint ◽  
Disjoint Family ◽  
Uncountable Family ◽  
Martin’S Axiom ◽  
Finite Set ◽  
Almost Disjoint ◽  
Disjoint Sets ◽  
Almost Disjoint Family ◽  
Require Equality ◽  
Single Set

As part of their study of βω — ω and βω1 — ω1, A. Szymanski and H. X. Zhou [3] were able to exploit the following difference between ω, and ω: ω1, contains uncountably many disjoint sets whereas any uncountable family of subsets of ω is, at best, almost disjoint. To translate this distinction between ω1, and ω to a possible distinction between βω1 — ω1, and βω — ω they used the fact that if a pairwise disjoint family of sets and a subset of each member of is chosen then it is trivial to find a single set whose intersection with each member is the chosen set. However, they noticed, it is not clear that the same is true if is only a pairwise almost disjoint family even if we only require equality except on a finite set. But any homeomorphism from βω1 — ω1 to βω — ω would have to carry a disjoint family of subsets of ω1, to an almost disjoint family of subsets of ω with this property. This observation should motivate the following definition.

scholarly journals Construction of dual modules using Martin's Axiom

2008 ◽  
Vol 320 (6) ◽  
pp. 2388-2404
Author(s):  
Rüdiger Göbel ◽  
Sebastian Pokutta
Keyword(s):  
Martin's Axiom ◽  
Martin’S Axiom

scholarly journals Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

2016 ◽  
Vol 68 (1) ◽  
pp. 44-66 ◽  
Author(s):  
David J. Fernández Bretón
Keyword(s):  
Abelian Group ◽  
Partial Orders ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Given

AbstractWe 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., , a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.

The equivalence of a generalized Martin's axiom to a combinatorial principle

1981 ◽  
Vol 46 (4) ◽  
pp. 817-821 ◽  
Author(s):  
William Weiss
Keyword(s):  
Combinatorial Principle ◽  
Martin's Axiom ◽  
Martin’S Axiom

AbstractA generalized version of Martin's axiom, called BACH, is shown to be equivalent to one of its combinatorial consequences, a generalization of P(c).

Countable dense homogeneous spaces under Martin’s axiom

1989 ◽  
Vol 65 (2) ◽  
pp. 153-164 ◽  
Author(s):  
Stewart Baldwin ◽  
Robert E. Beaudoin
Keyword(s):  
Homogeneous Spaces ◽  
Martin's Axiom ◽  
Martin’S Axiom
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