A transitive closure operator will be an important new operator in future deductive database systems. We discuss the compilation of recursive rule clusters into formulas containing transitive closure operations and study three promising algorithms for the processing of transitive closure queries: the wavefront algorithm, the δ-wavefront algorithm and the level-relaxed δ-wavefront algorithm. The relative processing efficiency of these algorithms are analyzed and compared based on different database structures and accessing methods. Our study shows that the δ-wavefront algorithm performs consistently better than the wavefront algorithm, and the level-relaxed δ-wavefront algorithm has high potential of further reducing I/O accessing cost on the databases with clustered derivation paths. The study also provides some interesting heuristics on the database structures and implementation techniques in the processing of recursive database queries.