  We are interested in representations and characterizations of lattice
polynomial functions f:L^n -> L, where L is a given bounded distributive
lattice. In companion papers [arXiv 0901.4888, arXiv 0808.2619], 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 [arXiv 0901.4888, arXiv 0808.2619] and by
considering further conditions, namely comonotonic minitivity and maxitivity.
