Weaker forms of increasingness of binary operations and their role in the characterization of meet and join operations

Due to the lack of transitivity of the pseudo-order relation of a proper trellis, its meet and join operations are not increasing. In this paper, we identify weaker forms of increasingness of binary operations that are satisfied by the meet and/or join operation of a trellis or lattice. Some of these forms coincide with the classical notion of increasingness when the trellis is a lattice. Furthermore, we demonstrate the role of these weaker forms in the characterization of the meet and join operations of a trellis, lattice or chain as specific idempotent operations.