Abstracts of papers published in 1981

A. Pelc, Solution of a problem of Ulam on countable sequences of sets, Fundamenta Mathematicae 114 (1981), pp. 113-118.


Let E be a set of cardinality continuum and {An: n natural} an arbitrary sequence of subsets of E. Let B denote the sigma-algebra of subsets of E generated by the family {An: n natural} and B* the sigma-algebra of subsets of E2 generated by the family {An x Am: n,m natural}. S.M. Ulam stated a problem whether there exists an injection from E to E2 transforming B into B* and conversely. We give a negative answer to this question and formulate a condition on {An: n natural} under which the answer is positive.

A. Pelc, Ideals on the real line and Ulam's problem, Fundamenta Mathematicae 112 (1981), pp. 165-170.


We prove that the union of a countable family of continuum-complete fields of subsets of the reals, each of which contains all singletons but not all sets of reals, cannot contain all sets of reals.

A. Pelc, Invariant measures on commutative semigroups, Proc. "Open Days in Model Theory and Set Theory", Jadwisin, Poland, September 1981, pp. 249-258.


Our main result is the following: Assume that (S,+) is a commutative, cancellative semigroup with unity, whose cardinality is greater or equal than a real-valued measurable cardinal. Then there exists a universal semiregular invariant measure on S.