We begin with the classical viewpoint in which a Structure consists of a nonempty carrier together with single–valued basic operations. A Hyperstructure arises by promoting operations to act on (and return) subsets of a base set, i.e., on its powerset. Iterating the powerset operator P n times yields an n-Superhyperstructure: informally, the n-th powerset Pn(S) is obtained by n successive applications of P (cf. 1). We review the fundamental definitions and give compact, instructive examples. A Multi-Structure replaces classical operations with maps from tuples to finite multisets, thereby allowing multiple outputs per input in a controlled, simultaneous manner. A MetaStructure treats whole structures as elements and equips them with uniform, isomorphism–invariant operations that functorially construct new structures from existing ones. In this paper we define HyperMatrix, SuperHyperMatrix, MultiMatrix, Iterative MultiMatrix, MetaMatrix, and Iterated MetaMatrix—all as extensions of the classical notion of a matrix—and we offer a concise examination of their properties.
Toshio Fujita (Fri,) studied this question.