Interval-Dismantling for Lattices

TL;DR

This paper extends the concept of dismantling in lattices from single elements to intervals, using Formal Concept Analysis to identify and compute dismantling intervals with a new algorithm.

Contribution

It introduces a novel approach to dismantling lattices by intervals, establishing theoretical foundations and providing an algorithm for their identification.

Findings

Dismantling by intervals corresponds to closed subrelations in formal contexts.

Existence of a unique kernel for dismantling by intervals.

Algorithm for computing all dismantling intervals in a lattice.

Abstract

Dismantling allows for the removal of elements of a set, or in our case lattice, without disturbing the remaining structure. In this paper we have extended the notion of dismantling by single elements to the dismantling by intervals in a lattice. We utilize theory from Formal Concept Analysis (FCA) to show that lattices dismantled by intervals correspond to closed subrelations in the respective formal context, and that there exists a unique kernel with respect to dismantling by intervals. Furthermore, we show that dismantling intervals can be identified directly in the formal context utilizing a characterization via arrow relations and provide an algorithm to compute all dismantling intervals.