Structural Saturation on a Discrete Carrier

This work proposes a unified scheme of structural saturation for subsets of a discrete carrier. The present research builds upon the general formal apparatus of distinguishability horizons, closure defects and refinement developed in [1]. If the previous work focused on the axiomatic description of stable forms and conditions for their consistency, the aim of the present work is to pass to a constructive mechanism: the explicit specification of rules of forced completion R, the construction of the local saturation operator TR, and the investigation of the algebraic structure arising on the lattice of R-closed forms. The first fully developed specialization of the general scheme is the finite linear model. On the carrier XN = {p0, p1, . . . , pN} a discrete ternary relation "to lie between" is defined, which does not assume metric, coordinates, density or continuity. The interval closure BCl generated by it is proved to be an extensive, monotone and idempotent operator. Its fixed points on the finite chain are completely characterized: they are the empty set and discrete intervals. Thus the linear form is understood not as a pre-given geometric primitive, but as a stable form on one carrier with respect to a concrete saturation rule. In addition to the classification of forms, the work investigates the algebraic structure of the family of stable forms. The endomorphisms and automorphisms of the closure operator End(ClR) and Aut(ClR) are introduced. It is shown that the action of endomorphisms on the lattice of closed forms generalizes classical distributivity, acting as equivariance with respect to the lattice join. In the linear model this structure is realized by natural shifts and reflections of intervals. After the linear model, the general language of R-saturation is introduced, allowing one to compare different geometric specializations without changing the carrier. Then the Pasch specialization RPasch is formulated, intended for the description of forced completions in discrete triangular configurations. It fixes the direction of further construction of finite models of intersections, traversal and consistency of sides, but is not identified in the present work with an already constructed complete Pasch geometry.