Directed hypergraphs are vital for modeling complex polyadic relationships in domains such as discrete mathematics, computer science, network security, and systems modeling. However, their inherent complexity often impedes effective visualization and analysis, particularly for large graphs. This paper introduces a novel Transitivity Preserving Projection (TPP) to address the limitations of the computationally intensive Basu and Blanning projection (BBP), which can paradoxically increase complexity by flattening transitive relationships. TPP offers a minimal and complete representation of relationships within a chosen subset of elements, capturing only irreducible dominant metapaths to ensure the smallest set of edges while preserving all essential transitive and direct connections. This approach significantly enhances visualization by reducing edge proliferation and maintains the integrity of the original hypergraph's structure. We develop an efficient algorithm leveraging the set-trie data structure, reducing the computational complexity from an exponential number of metapath searches in BBP to a linear number of metapath searches with polynomial-time filtering, enabling scalability for real-world applications. Experimental results demonstrate TPP's superior performance, completing projections in seconds on graphs where BBP fails to terminate within 24 hours. By providing a minimal yet complete view of relationships, TPP supports applications in network security and supply