 the entire reduction relation may consist of associativity and commutativity rules.Consequently, some authors define the reduction relation → as the indexed union of some relations; for instance if formula_2, the notation used is (A, →1, →2).As a mathematical object, an ARS is exactly the same as an unlabeled state transition system, and if the relation is considered as an indexed union, then an ARS is the same as a labeled state transition system with the indices being the labels.