Complex Systems

An Invitation to Higher Arity Science Download PDF

Carlos Zapata-Carratalá
School of Mathematics, The University of Edinburgh, United Kingdom
(lead author,
Society for Multidisciplinary and Fundamental Research

Xerxes D. Arsiwalla
Pompeu Fabra University, Barcelona, Spain
Wolfram Research, Inc., United States of America


Analytical thinking is dominated by binary ideas. From pairwise interactions to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific paradigms. In this paper, we introduce arity as the generic conceptualization of the order of an interaction between a discrete collection of entities and argue that there is a rich universe of higher arity ideas beyond binarity waiting to be explored. To illustrate this, we discuss several higher-order phenomena appearing in a wide range of research areas, paying special attention to instances of ternary interactions. From the point of view of formal sciences and mathematics, higher arity thinking opens up new paradigms of algebra, symbolic calculus and logic. In particular, we delve into the special case of ternary structures, as that itself reveals ample surprises: new notions of associativity (or lack thereof) in ternary operations of cubic matrices, ternary isomorphisms and ternary relations, the integration problem of 3-Lie algebras, and generalizations of adjacency in 3-uniform hypergraphs. All these are open problems that strongly suggest the need to develop new ternary mathematics. Finally, we comment on potential future research directions and remark on the transdisciplinary nature of higher arity science.

Keywords: higher-order structures; complex systems; interdisciplinary research; generalized associativity; ternary algebra; hypergraphs; matrix algebra; category theory

Cite this publication as:
C. Zapata-Carratalá and X. D. Arsiwalla, “An Invitation to Higher Arity Science,” Complex Systems, 32(4), 2024 pp. 409–468.