A Perspective on the Algebra, Topology, and Logic of Electrical Networks

Abstract

This paper presents a unified algebraic, topological, and logical framework for electrical one-port networks based on \v{S}are's $m$-theory. Within this formalism, networks are represented by $m$-words (jorbs) over an ordered alphabet, where series and parallel composition induce an $m$-topology on $m$-graphs with a theta mapping $\vartheta$ that preserves one-port equivalence. The study formalizes quasi-orders, shells, and cores, showing their structural correspondence to network boundary conditions and impedance behavior. The $\lambda--\Delta$ metric, together with the valuation morphism $\Phi$, provides a concise descriptor of the impedance-degree structure. In the computational domain, the framework is extended with algorithmic procedures for generating and classifying non-isomorphic series-parallel topologies, accompanied by programmatic Cauer/Foster synthesis workflows and validation against canonical examples from Ladenheim's catalogue. The resulting approach enables symbolic-to-topological translation of impedance functions, offering a constructive bridge between algebraic representation and electrical realization. Overall, the paper outlines a self-consistent theoretical and computational foundation for automated network synthesis, classification, and formal verification within the emerging field of Jorbology.