Concept Lattice Orbifolds – First Steps

Factorisations of some partially ordered sets and small categories

“…We will consider a category C as a (not necessarily simple) directed graph with vertex set Ob C and edge set Mor C . 1 We refer to the edges as arrows. The set of arrows between two vertices x, y ∈ Ob C is denoted by Mor C ( x, y ).…”

“…In this section we will define a structure that can be considered as a factorisation of pogroups. This work is inspired by binary relation orbifolds as discussed in [7,8,1]. However, these ideas are based on factor relations while we use a description here, that is more focused on the internal structure of the factor structure.…”

“…However, when we consider a partial order as a directed graph, the direction of the arrows is preserved during the factorisation in many cases, while the factor graph of a simple graph is not necessarily simple. Even if the factor relation can be used to anchor unfolding information [1], this structure is usually not visible as a relation.…”

“…Conseqeuntly, the factored cyclic order relation contains the two triplets ( 0, 0 ), ( 0, 1 ), ( 3,1 ) and ( 0, 0 ), ( 3, 1 ), ( 0, 1 ) . This violates the antisymmetry condition of cyclic orders and cyclically ordered groups.…”

Lattice-based and topological representations of binary relations with an application to music

“…In order to develop decomposition techniques, which provide a link between the local and the global, it thus seems promising to take both viewpoints into account. As a concrete starting point we would like to study the connection between simplicial complexes and concept lattice orbifolds, a technique to "fold" concept lattices using symmetries of the context, which has been introduced by Daniel Borchmann and Bernhard Ganter [BG09].…”