TY - JOUR
T1 - Theorems on sets not belonging to algebras
AU - Grinblat, L.
PY - 2004/5/26
Y1 - 2004/5/26
N2 - Let A1,…, An, An+1 be a finite sequence of algebras of sets given on a set X, (Formula Presented), with more than (Formula Presented) pairwise disjoint sets not belonging to An+1. It has been shown in the author's previous articles that in this case (Formula Presented). Let us consider, instead of An+1, a finite sequence of algebras An+1,…, An+l. It turns out that if for each natural i ≤ l there exist no less than (Formula Presented) pairwise disjoint sets not belonging to An+i, then (Formula Presented). Besides this result, the article contains: an essentially important theorem on a countable sequence of almost σ-algebras (the concept of almost σ-algebra was introduced by the author in 1999), a theorem on a family of algebras of arbitrary cardinality (the proof of this theorem is based on the beautiful idea of Halmos and Vaughan from their proof of the theorem on systems of distinct representatives), a new upper estimate of the function v(n) that was introduced by the author in 2002, and other new results.
AB - Let A1,…, An, An+1 be a finite sequence of algebras of sets given on a set X, (Formula Presented), with more than (Formula Presented) pairwise disjoint sets not belonging to An+1. It has been shown in the author's previous articles that in this case (Formula Presented). Let us consider, instead of An+1, a finite sequence of algebras An+1,…, An+l. It turns out that if for each natural i ≤ l there exist no less than (Formula Presented) pairwise disjoint sets not belonging to An+i, then (Formula Presented). Besides this result, the article contains: an essentially important theorem on a countable sequence of almost σ-algebras (the concept of almost σ-algebra was introduced by the author in 1999), a theorem on a family of algebras of arbitrary cardinality (the proof of this theorem is based on the beautiful idea of Halmos and Vaughan from their proof of the theorem on systems of distinct representatives), a new upper estimate of the function v(n) that was introduced by the author in 2002, and other new results.
KW - Algebra on a set
KW - Almost σ-algebra
KW - Pairwise disjoint sets
KW - Ultrafilter
UR - http://www.scopus.com/inward/record.url?scp=16244416559&partnerID=8YFLogxK
U2 - 10.1090/S1079-6762-04-00129-5
DO - 10.1090/S1079-6762-04-00129-5
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AN - SCOPUS:16244416559
SN - 1079-6762
SN - 2688-1594
VL - 10
SP - 51
EP - 57
JO - Electronic Research Announcements of the American Mathematical Society
JF - Electronic Research Announcements of the American Mathematical Society
IS - 6
ER -