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Set Ideals with Complete Symmetry Group and Partition Ideals
Author(s) -
Mishkin Valery
Publication year - 1999
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609399006359
Subject(s) - mathematics , partition (number theory) , pure mathematics , combinatorics , discrete mathematics
For a wide class of set ideals (including, for example, all uniform ideals), a criterion of completeness of their symmetry groups is provided in terms of ideal quotients (polars). We apply it to partition ideals, and derive the extended Sierpiński–Erdös duality principle. We demonstrate that if the measure and category ideals I 0 and I 1 on the real line R are partition (or, equivalently, if just I 0 is partition), then not only are they isomorphic via an involution, but they also have complete (and distinct) symmetry groups coinciding, respectively, with the symmetry groups of the polarsI 0 ⊥andI 1 ⊥ . The measure and category ideals on R (and in more general spaces) are partition (Oxtoby) ideals assuming Martin's Axiom. In this case their polars are, respectively, the ideals generated by c ‐Sierpiński and c ‐Lusin sets. It is well known that the isomorphism of the measure and category ideals is not provable in ZFC. We show that the isomorphism of their symmetry groups is likewise unprovable. 1991 Mathematics Subject Classification 20B35, 03E50, 04A20.