However, commutativity does not imply associativity. Extending this relation by associativity and distributivity allows explicitly constructing the Weyl algebra.. Multiplication follows the usual laws of associativity and distributivity. This allows an easier use of associativity and commutativity. An operation that is mathematically associative, by definition requires no notational associativity. These axioms are closure, associativity, and the inclusion of an identity element and inverses. If one drops the requirement for the associativity, then one obtains a non-associative algebra. Equation (2) is an abstract expression of the associativity of matrix multiplication. Haswell's L1 data cache has an associativity of eight. Some authors define a ring without the requirement of associativity for multiplication. The theorem is true even more generally&colon; alternativity suffices in place of associativity. Its associativity follows from the associativity of the tensor product. Jaynes cites Abel for the first known use of the associativity functional equation. The victim cache exploits this property by providing high associativity to only these accesses. This is most problematic for caches that have low associativity.