11. A full transitive closure is not needed; a commutative transitive closure and even weaker forms suffice. 12. More precisely, it is the transitive closure of the relation " is the mother of ". 13. Let \ Rightarrow ^ { * } be the reflexive transitive closure of the relation \ Rightarrow. 14. The Ancestral relation R ^ * is equal to the transitive closure R ^ + of R. 15. Therefore, the transitive reduction can be constructed in the same asymptotic time bounds as the transitive closure . 16. In computational complexity theory, the complexity class L is first-order logic with the commutative, transitive closure . 17. If is a DAG, its transitive closure is the graph with the most edges that represents the same reachability relation. 18. The "'trace "'is defined as the symmetric, reflexive and transitive closure of \ sim. 19. Finally, one takes the reflexive and transitive closure of " E ", which is then a monoid congruence. 20. First-order logic is too restrictive for some database applications, for instance because of its inability to express transitive closure .
