Polynomial Functions on Bounded Chains

We are interested in representations and characterizations of lattice polynomial functions f : L → L, where L is a given bounded distributive lattice. In an earlier paper [4, 5], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present paper, we investigate these representations and characterizations in the special case when L is a chain, i.e., a totally ordered lattice. More precisely, we discuss representations of lattice polynomial functions given in terms of standard simplices and we present new axiomatizations of these functions by relaxing some of the conditions given in [4, 5] and by considering further conditions, namely comonotonic minitivity and maxitivity. Keywords— Lattice polynomial function, discrete Sugeno integral, normal form, homogeneity, strong idempotency, median decomposability, comonotonicity.