Hyperorders and transclusion: understanding dimensional hypertext

ZigZag is a unique hyperstructural paradigm designed by the hypertext pioneer Ted Nelson. It has piqued a lot of interest in the hypertext community in recent years because of its aim of revolutionizing electronic access to information and knowledge bases. In ZigZag information is stored in cells that are arranged into lists organized along unlimited numbers of intersecting sets of associations called dimensions. To this infrastructure a mechanism of transclusion is added, allowing the data stored in cells to span, and hence be utilized, in different contexts. Proponents of ZigZag claim that it is a flexible and universal structure for information representation, and yet the system has not been widely adopted and has been implemented even more rarely. In this paper we address the question of whether there are intrinsic theoretical reasons as to why this is the case. While the basic features and specifications of ZigZag are well known, we delve in to the less understood area of its theoretical underpinnings to tackle this question. By modeling ZigZag within the framework of set theory we reveal a new class of hyperstructure that contains no referencable link objects whatsoever, instead grouping non-referencable binary associations into disjunct but parallel sets of common semantics (dimensions). We go on to further specialize these "dimensional models" into sets of finite partial functions, which are closed over a single domain, isolating the new class of hyperstructures we are calling hyperorders. This analysis not only sheds light on the benefits and limitations of the ZigZag hypermedia system, but also provides a framework to describe and understand a wider family of possible hyperstructure models of which it is an early example. Characteristics of Zigzag's transclusion mechanisms are also investigated, highlighting a previously unrecognized distinction, and potential irrevocable conflict, between two distinct uses of content reuse: instance and identity transclusion.