On the existence of strongly normal ideals overP _κ λ

Abstract

For every uncountable regular cardinalκ and any cardinalλ≧κ,P _κ λ denotes the set$\left\{ {x \subseteqq \lambda :\left| x \right|< \kappa } \right\}$. Furthermore, < denotes the binary operation defined inP _κ λ byx<y iffx⊂y∧¦x∣<¦y∩κ∣.

By anideal over P _κ λ we mean a proper, non-principal,κ-complete ideal overP _κ λ extending the ideal dual to the filter generated by$\left\{ {\left\{ {x \in P_\kappa \lambda :y \subseteqq x} \right\}:y \in P_\kappa \lambda } \right\}$. For any idealI overP _κ λ,I ⁺ denotes the setP _κ λ−I, andI ^* the filter dual toI.

An idealI overP _κ λ is said to benormal iff every functionf:P _κ λ→λ with the property that {x∈P _κ λ:f(x)∈x}∈I ⁺ is constant on a set inI ⁺.I is said to bestrongly normal iff every functionf:P _κ λ→P _κ λ with the property that {x∈P _κ λ:x∩xκ≠Ø∧f(x)<x}∈I ⁺ is constant on a set inI ⁺. It is easy to see that every strongly normal ideal is normal. However, the converse of this is false.

In this paper, we completely characterize those pairs (κ, λ) for whichP _κ λ bears a strongly normal ideal, and describe the smallest such ideal for these pairs. As well, we show that for each of these pairs, the operator∇ _< defined on the set of all ideals overP _κ λ by

where

$$\nabla _< \left\{ {X_a :a \in P_\kappa \lambda } \right\} = \left\{ {x \in P_\kappa \lambda :x \cap \kappa = \phi \vee \left( {\exists a< x} \right)\left( {x \in X_a } \right)} \right\}$$

is idempotent. Our results include the following theorems.