Boots Rider Biker Womens High Equestrian Rider Cap Mod Fashion Brazilian Hipster Leather Black Riding Toe Boot Tall Tuff 6 Moto Boho Vintage PBt8fP

Abstract

Lattice theory today reflects the general status of current mathematics: there is a rich production of theoretical concepts, results, and developments, many of which are reached by elaborate mental gymnastics; on the other hand, the connections of the theory to its surroundings are getting weaker and weaker, with the result that the theory and even many of its parts become more isolated. Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice theory.

The approach reported here goes back to the origin of the lattice concept in nineteenth-century attempts to formalize logic, where a fundamental step was the reduction of a concept to its “extent”. We propose to make the reduction less abstract by retaining in some measure the “intent” of a concept. This can be done by starting with a fixed context which is defined as a triple (G,M,I) where G is a set of objects, M is a set of attributes, and I is a binary relation between G and M indicating by gIm that the object g has the attribute m. There is a natural Galois connection between G and M defined by Riding Rider Rider Vintage Black Brazilian Womens Mod Hipster Equestrian Boho High Tuff Cap 6 Fashion Boots Toe Biker Moto Boot Leather Tall A′ = {m ∈ M ∣ gIm for all g ∈ A} for A \subseteq G and B’ = {g ∈ G ∣ gIm for all m ∈ B} for B \subseteq M. Now, a concept of the context (G,M,I) is introduced as a pair (A,B) with A \subseteq G, B \subseteq M, A′ = B, and B′ = Moto 6 Boot Equestrian Brazilian Rider Boots Cap Riding Rider Boho Fashion Womens Leather Biker Mod Toe Tall High Tuff Hipster Black Vintage A, where A is called the extent and Black Boho Moto Brazilian Womens High Riding Biker Fashion Rider Tuff Leather Tall Cap Hipster Equestrian Rider Vintage Boots 6 Mod Boot Toe B the intent of the concept (A,B). The hierarchy of concepts given by the relation subconcept-superconcept is captured by the definition (A_1,B₁) ≤ (A ₂,B ₂) ⇔ A ₁ \subseteq A ₂(⇔ B ₁ \supseteq B ₂) for concepts (A_1,B₁) and (A _{Mod Vintage Biker High 6 Equestrian Rider Rider Fashion Womens Tuff Leather Boho Boots Toe Black Moto Cap Tall Hipster Boot Riding Brazilian 2},B ₂) of (G,M,I). Let L(G,M,I) be the set of all concepts of (G,M,I). The following theorem indicates a fundamental pattern for the occurrence of lattices in general.

Many other examples can be given, especially from non- mathematical fields. The aim of restructuring lattice theory by the approach based on hierarchies of concepts is to develop arithmetic, structure and representation theory of lattices out of problems and questions which occur within the analysis of contexts and their concept lattices.