Ultrafilter semigroups generated by direct sums

作者：Shuungula Onesmus; Zelenyuk Yevhen; Zelenyuk Yuliya

来源：Semigroup Forum, 2011, 82(2): 252-260.

DOI：10.1007/s00233-010-9262-x

• Stone-Cech compactification
• Ultrafilter semigroup
• Direct sum
• Local isomorphism

Let lambda be an infinite cardinal and for every ordinal alpha <lambda, let A (alpha) be a set with a distinguished element 0 (alpha) aA (alpha) . The direct sum of sets A (alpha) , alpha <lambda, is the subset X = circle plus (alpha <gimel) A(alpha) of the Cartesian product Pi(alpha <gimel) A(alpha) consisting of all x with finite supp (x)={alpha <lambda:x(alpha)not equal 0 (alpha) }. Endow X with a topology by taking as a neighborhood base at xaX the subsets of the form {yaX:y(alpha)=x(alpha) for all alpha <gamma} where gamma <lambda. Let Ult (X) denote the set of all nonprincipal ultrafilters on X converging to 0aX. There is a natural partial semigroup operation on X which induces a semigroup operation on Ult (X). We show that if direct sums X and Y are homeomorphic, then the semigroups Ult (X) and Ult (Y) are isomorphic.

• 出版日期2011-4