On the representation of internal uninorms on a bounded lattice

Abstract

Internal uninorms on a bounded lattice always output one of the two input values and are nothing else but idempotent uninorms when the lattice is a chain. In this paper, we study the existence and representation of internal uninorms with a given neutral element e on a bounded lattice. We obtain a necessary condition for the existence of such an internal uninorm. In the case that those elements that are incomparable with the neutral element (collected in the set Ie) constitute a chain, we prove that this necessary condition also becomes a sufficient one and allows to represent any internal uninorm on a bounded lattice in terms of an internal uninorm on the bounded chain obtained by deleting Ie from the lattice and an internal quasi-uninorm on Ie connected via a family of lower sets and a family of upper sets indexed by Ie.