Complete Quotient Boolean Algebras
نویسنده
چکیده
For I a proper, countably complete ideal on P(X) for some set X , can the quotient Boolean algebra P(X)/I be complete? This question was raised by Sikorski [Si] in 1949. By a simple projection argument as for measurable cardinals, it can be assumed that X is an uncountable cardinal κ, and that I is a κ-complete ideal on P(κ) containing all singletons. In this paper we provide consequences from and consistency results about completeness. Throughout, κ will denote an uncountable cardinal, and by an ideal over κ we shall mean a proper, κ-complete ideal on P(κ) containing all singletons. If κ is a measurable cardinal and I a prime ideal over κ, then of course P(κ)/I is complete, being the two-element Boolean algebra. The following theorem shows that completeness in itself has strong consistency strength:
منابع مشابه
ON THE REPRESENTATION OF <r-COMPLETE BOOLEAN ALGEBRAS
A <r-complete Boolean algebra is a Boolean algebra in which for every sequence of elements a$-, i = l, • • • , there is an element U?an, the countable union of the a», such that aiQU?an for every i, and such that if diQx for every i then U?anQx. The dual operation, countable intersection, can be introduced through complementation, and the distributive law afMJi*'a» = Uf {aC\an) and its dual can...
متن کاملCdmtcs Research Report Series Eective Presentability of Boolean Algebras of Cantor{bendixson Rank 1 Eective Presentability of Boolean Algebras of Cantor{bendixson Rank 1
We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B=I is of Cantor{Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of in nite Cantor{Bendixson rank.
متن کاملOn Some Varieties of MTL-algebras
The study of perfect, local and bipartite IMTL-algebras presented in [29] is generalized in this paper to the general non-involutive case, i. e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras, B...
متن کاملSymmetric Chain Decomposition for Cyclic Quotients of Boolean Algebras and Relation to Cyclic Crystals
The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the sl2 lowering ...
متن کاملThe Completeness of the Isomorphism Relation for Countable Boolean Algebras
We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an esse...
متن کامل