Graded Galois Lattices and Closed Itemsets

TL;DR

This paper introduces graded Galois lattices and graded closed itemsets, extending knowledge representation tools and providing constructive algorithms based on domain theory and Banach lattices.

Contribution

It proposes a new class of Galois lattices called graded Galois lattices and develops methods to compute graded formal concepts and closed itemsets.

Findings

Defined graded Galois lattices and graded closed itemsets.

Developed constructive algorithms for computing these structures.

Connected the methods to domain theory and Banach lattices.

Abstract

The Galois lattice is a graphic method of representing knowledge structures. The first basic purpose in this paper is to introduce a new class of Galois lattices, called graded Galois lattices. As a direct result, one can obtain the notion of graded closed itemsets (sets of items), to extend the definition of closed itemsets. Our second important goal in this paper, is related to set a constructive method, computing the graded formal concepts and graded closed itemsets. We mean by a constructive method, a method that builds up a complete solution from scratch by sequentially adding components to a partial solution until the solution is complete. Besides of computational aspects, our methods in this paper are based on the strong results obtained by special mappings in the realm of domain theory. To reach the fertilized consequences and constructive algorithms, we need to push the study to the structures of Banach lattices.