Order isomorphisms for preferences with intransitive indifference

An order isomorphism from one preference relation to another is a function between their underlying sets that preserves both preference and indifference. A utility function is an order isomorphism, as is a well-known two-function representation for preference relations with intransitive indifference. Necessary and sufficient conditions are given for the existence of an order isomorphism from a given preference relation to the set of sequences of zeros and ones ordered by Pareto dominance. This means of preference representation is shown via examples to be quite general.