LAUSR

In this paper we identify and study several lattice structures in the context of directed topology. The set of d-structures on a topological space is a Heyting algebra. The implication is constructed explicitely. There is a Galois connection between the lattice of subsets of the space and the lattice of d-structures which clarifies the idea of removing a subset of the space which has only constant dipaths. Hence, variation of d-structures and variation of the “forbidden area” may be considered in one structure. Moreover, the lattice of d-structures gives rise to a lattice of directed paths between a fixed pair of points. That lattice permits us to discuss a perspective on generalised persistence. Furthermore, we consider a lattice structure in the hierarchy of structures on the n-cube.