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:
“With the breakdown of the traditional institutions which convey values, more of the burdens and responsibility for transmitting values fall upon parental shoulders, and it is getting harder all the time both to embody the virtues we hope to teach our children and to find for ourselves the ideals and values that will give our own lives purpose and direction.”
—Neil Kurshan (20th century)