Relationships Among Ideals
If I and J are ideals on X and Y respectively, I and J are Rudin–Keisler isomorphic if they are the same ideal except for renaming of the elements of their underlying sets (ignoring negligible sets). More formally, the requirement is that there be sets A and B, elements of I and J respectively, and a bijection φ : X \ A → Y \ B, such that for any subset C of X, C is in I if and only if the image of C under φ is in J.
If I and J are Rudin–Keisler isomorphic, then P(X) / I and P(Y) / J are isomorphic as Boolean algebras. Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are called trivial isomorphisms.
Read more about this topic: Ideal (set Theory)
Famous quotes containing the word ideals:
“The nineteenth century was completely lacking in logic, it had cosmic terms and hopes, and aspirations, and discoveries, and ideals but it had no logic.”
—Gertrude Stein (1874–1946)