A selection of maximal elements under non-transitive indifferences

In this work we are concerned with maximality issues under intransitivity of the indifference. Our approach relies on the analysis of ‘‘undominated maximals’’ (cf., Peris & Subiza, 2002). Provided that an agent’s binary relation is acyclic, this is a selection of its maximal elements that can always be done when the set of alternatives is finite. In the case of semiorders, proceeding in this way is the same as using Luce’s selected maximals. We present a sufficient condition for the existence of undominated maximals for interval orders without any cardinality restriction. Its application to certain types of continuous semiorders is very intuitive and accommodates the well-known ‘‘sugar example’’ by Luce.