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The aim of this paper is to classify order-preserving functions according totheir arity gap. Noteworthy examples of order-preserving functions areso-called aggregation functions. We first explicitly classify the Lov\'aszextensions of pseudo-Boolean functions according to their arity gap. Then weconsider the class of order-preserving functions between partially orderedsets, and establish a similar explicit classification for this function class.

The arity gap of order-preserving functions and extensions of pseudo-Boolean functions