We investigate algebras of sets, and pairs (A,I)
consisting of an algebra A and an ideal I, which is a subset of A, that
possess an inner MB-representation. We compare inner
MB-representability of (A,I) with several properties
of (A,I) considered by Baldwin. We show that A is inner
MB-representable if and only if A =S(A \ H (A)),
where S(^{.}) is a Marczewski operation defined below
and H consists of
sets that are hereditarily in A. We study uniqueness issue of the
ideal in that representation.

See also related paper:

- M. Balcerzak, A. Bartoszewicz, J. Rzepecka, S. Wronski,
Marczewski fields and ideals,
*Real Anal. Exchange 26(2)*(2000-2001), 703-715. - M. Balcerzak, A. Bartoszewicz, K. Ciesielski,
On Marczewski-Burstin representations of certain
algebras of sets,
*Real Anal. Exchange 26(2)*(2000-2001), 581-591; MR 2002e:03078. - A. Bartoszewicz, K. Ciesielski,
MB-representations and topological algebras,
*Real Anal. Exchange 27(2)*(2001-2002), 749-755. Errata,*Real Anal. Exchange 28(1)*(2002-2003), to appear.

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