Generalizations of lattices via non-deterministic operators

It is well-known that in the lattice theory we can use indistinctly pairs of elements or finite subsets to characterize them. However, this is not true when we work with multilattices. For this reason in this paper we introduce two new structures from the ordered point of view, called universal multisemilattice and universal multilattice, and we propose an equivalent algebraic characterization for them. These new structures are generalizations, on one hand, of semilattice and lattice and, on the other hand, of multisemilattice and multilattice, respectively. The algebraic characterizations have the same advantages as the two introduced by us in Martinez et al. The most important purpose of this paper is to deepen the theoretical study of universal multisemilattices and universal multilattices.