New Applications of Hyperstructures and Superhyperstructures: Queue, Markov Chains, Intervals, Logic, and Systems

Abstract

Mathematical structures can be systematically extended to hyperstructures and superhyperstructures
through the use of power sets and iterated power-set constructions. Such extensions provide a flexible
framework for modeling hierarchical, multilevel, and set-valued phenomena across diverse mathematical
and applied domains. Representative examples include superhypergraphs[1], superhyperalgebras, and
superhyperfuzzy sets[2], which have recently attracted growing attention. Despite this progress, research
on superhyperstructures is still at an early stage, and many potential applications, structural properties,
and related concepts remain unexplored. Motivated by this gap, this paper investigates hyperstructural
and superhyperstructural extensions of queues, Markov chains, intervals, logical systems, and discrete-time
systems. For each case, we formally define the corresponding hyper and superhyper models, analyze their
fundamental mathematical properties, and present illustrative examples that clarify their behavior and
hierarchical nature. These results aim to broaden the theoretical foundation of superhyperstructures and
stimulate further research into their applications and connections with existing mathematical frameworks.