A representation theorem for stratified complete lattices

TL;DR

This paper introduces a representation theorem for stratified complete lattices, characterizing models and strong models as inverse limits of simpler lattices, and explores their fixed point properties and symmetry conditions.

Contribution

It provides a novel inverse limit representation for stratified complete lattices, offering new insights into their structure and fixed point behavior, with applications to logic programming semantics.

Findings

Models are isomorphic to inverse limits of locally completely additive projections.

Strong models are isomorphic to inverse limits of completely additive projections.

The set of fixed points of weakly monotone functions forms a complete lattice.

Abstract

We consider complete lattices equipped with preorderings indexed by the ordinals less than a given (limit) ordinal subject to certain axioms. These structures, called stratified complete lattices, and weakly monotone functions over them, provide a framework for solving fixed point equations involving non-monotone operations such as negation or complement, and have been used to give semantics to logic programs with negation. More precisely, we consider stratified complete lattices subject to two slightly different systems of axioms defining `models' and `strong models'. We prove that a stratified complete lattice is a model iff it is isomorphic to the stratified complete lattice determined by the limit of an inverse system of complete lattices with `locally completely additive' projections. Moreover, we prove that a stratified complete lattice is a strong model iff it is isomorphic to the stratified complete lattice determined by the limit of an inverse system of complete lattices with completely additive projections. We use the inverse limit representation to give alternative proofs of some recent results and to derive some new ones for models and strong models. In particular, we use the representation theorem to prove that every model gives rise to another complete lattice structure, which in limit models corresponds to the lexicographic order. Moreover, we prove that the set of all fixed points of a weakly monotone function over a model, equipped with the new ordering, is a complete lattice. We also consider symmetric models that satisfy, together with each axiom, the dual axiom, and use the inverse limit representation to prove that every strong model is symmetric.