Associative triple trisystems and standard embeddings

In this paper, we introduce the concept of associative triple trisystems (ATTs), extending the classical theory of triple systems to the dialgebra framework. To provide concrete models for these structures, we utilize averaging operators to construct explicit examples of ATTs from classical associative triple systems. We develop the notion of di-endomorphisms for arbitrary modules, which serves as the foundation for constructing the standard embedding of ATTs into associative dialgebras. Through these embeddings, we establish the analogues of the classical relationships connecting associative, Jordan, and Lie triple systems within the context of trisystems.