tulksets

Tulksets are an abstract mathematical concept used to model entities that belong to multiple, nested contexts. A tulkset is defined by a base set X together with a distinguished collection T of subsets of X, called tulk layers. Each tulk layer represents a level of inclusion, with higher layers containing or refining lower ones. This structure captures hierarchical, overlapping membership that arises in complex systems.

In typical constructions, X is any set of items, and T is closed under finite unions and under taking common refinements. A basic operation is tulk combination, which forms a new tulkset by aggregating the base sets and intersecting corresponding layers to preserve the hierarchy. Tulksets accommodate both disjoint and nested groupings, offering a flexible language for multi-context data.

History and status: The term tulkset appears in speculative mathematical literature and has not achieved a standard, widely accepted formalization. It is mainly used as a conceptual aid for discussing nested partitions and overlapping affiliations in networks and data structures.

Applications: Researchers have suggested tulksets for organizing metadata, representing community structures in graphs, and modeling access controls with multi-level permissions. They serve as a bridge between set theory and lattice-based approaches, enabling reasoning about how different contexts intersect and combine.