On Structure, Relation, and Synthesis: the Algebra of Invariants as a Universal Language for Describing Systems

This paper proposes a unified formal language for describing structure, relations, and synthesis, applicable in topology, algebra, graph theory, and the theory of complex systems. The approach is centered on the representation of a relation as a quadruple , where is a universe of objects, is a monoid of admissible transformations, is an invariant, and is a valuation satisfying the axioms of symmetry, associativity, normalization, and distributivity. It is shown that all classical classes of relations—topological, geometric, and algebraic—emerge as particular instances of this construction under appropriate choices of and . As an illustration, we consider the spectral valuation associated with the pattern-intersection operator in the theory of hierarchical systems.