Algebras of binary isolating formulas for theories of modular products of graphs

This article discusses the problems of constructing and classifying algebras generated by modular products of cycles. We demonstrate that algebras of binary isolating, used to analyze relationships between binary formulas of a theory, can be naturally interpreted in terms of metric properties of graphs. A characteristic feature of the modular product is that with sufficiently large cycle parameters (m,n > 4), the diameter of such a graph does not exceed three. This makes it possible to define an algebra of binary formulas using only four labels. For small cycle parameters, the presence of simplices is identified and justified. Based on the analysis, we propose a generalized scheme combining modular products of cycles and their extended versions. It is proved that for m,n > 4, the algebra of binary isolating formulas for the theory of Cm∇Cn is isomorphic to the algebra of simplices of corresponding diameter. Explicit Cayley tables are constructed for products involving small cycles (C3–C6), leading to general descriptions of algebras Mo (odd) and Me (even). The proposed approach provides new opportunities for classifying theories and establishing correspondences between algebras and graphs, underlining its relevance for modern model theory and structural combinatorics.