General and left-continuous operators on lattice-based sums

Abstract

Lattice-based sum provides a procedure to obtain posets and lattices from families of posets and lattices, respectively. Establishing sufficient conditions to ensure the lattice structure was the most significant challenge achieved in previous works. Next steps are to consider structures with general operators defined on the lattices of the family, introduce a sum of these operators on the obtained lattice-based sum and study the properties preserved by this new definition. We will prove that the natural definition preserve, in general, the monotonicity, associativity, commutativity, etc. This paper also introduces a new mechanism focused on preserving the left-continuity property of the operators defined on the lattices. This new approach also preserves the associativity and the infimum of non-empty subsets, and takes into account (infinite) complete lattices, unlike the previous works.