Abstract
The lexicographic complexity of an asymmetric binary relation on a finite set is defined to be the minimal number of ternary criteria sufficient to construct a 1–1 lexicographic representation of that relation. Lexicographic complexity provides a measure of the degree of non-representability by utility function of an asymmetric binary relation. Tight upper and lower bounds are established for the lexicographic complexity of an asymmetric binary relation on a set of cardinality n. Four examples launch an exploration of the relationships between lexicographic complexity and intransitivity and between lexicographic complexity and incompleteness. Two examples of 1–1 lexicographic representations exhibit a strong lexicographic flavor in that, for each of the two examples, any two distinct reorderings of its coordinate functions result in representations of distinct binary relations.
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