Tyelementy

Tyelementy is a theoretical framework used to describe systems in which three basic components interact under a fixed set of rules. The concept is employed in abstract mathematics, theoretical computer science, and modeling studies of triadic relations. The name suggests a triadic structure built from fundamental elements and their prescribed interactions.

A tyelementy structure consists of a set of three distinguished elements, typically labeled A, B, and C, together with a defined set of interactions that specify the results of combining these elements in pairs or triples. A minimal formalization emphasizes closure (each interaction yields an element from {A, B, C}) and a triadic compatibility condition that constrains how pairwise interactions extend to triple interactions. Variants may include additional structure such as order, symmetry, or multiple interaction channels, but all keep the core triadic premise intact.

A common simple model defines pairwise interactions by a fixed rule or permutation of the three elements. For example, AB may yield C, BC yields A, and CA yields B, with the reverse orders following corresponding rules. Triplet interactions are defined separately and, like the pairwise ones, return a member of {A, B, C} according to a specified triadic rule. Such rules create predictable patterns while allowing departures from binary intuition.

Applications of tyelementy concepts include analysis of triadic network motifs, simplified chemical reaction schemes with three interacting species, and coordination or consensus protocols in three-party distributed systems. The framework also serves as a pedagogical tool to contrast triadic behavior with traditional binary algebra and to explore how multi-element interactions shape system dynamics.