The upper interval topology, property M, and compactness

The use of intrinsic topology such as interval topology and order topology that were typically symmetric was discussed. The theory of continuous lattices provided strong motivation for the consideration of such topologies such as the Scott topology or the hull-kernel topology which were not symmetric. Another approach to the study of topology on ordered structures was to begin with a set X equipped both with a partial order ≤ and a topology. The results show that these orders were called as closed order and the resulting ordered topological space was called as pospace, when the assumption was satisfied

Generalized persistence and graded structures

We investigate the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the category of additive functors from this category to the category of Abelian groups is isomorphic to the category of modules graded over the set with a monoid action, and to the category of unital modules over a certain smash product. Furthermore, when the indexing set is a poset, we provide a new characterization for a generalized persistence module being finitely presented.Comment: 25 pages, to appear in Homology, Homotopy and Application