Positive ∑ operations on ordinals and normal filters on greatly mahlo cardinals

Abstract

If ℱ is a normal filter on a regular uncountable cardinal κ, let ║f║ be the ℱ-norm of an ordinal function f. We introduce the class of positive ordinal operations and prove that if F is a positive operation then ║F(f)║ ≥ F(║f║). For each η < κ+ let fη be the canonical ηth function. We show that if F is a ∑ operation then F(fη) = fF(η).As an application we show that if κ is greatly Mahlo then there are normal filters on κ of order greater than κ+.