The first difficulty occurs with associative operations like addition and multiplication.
2.
Each reducer has an identity element and an associative operation that combines two values.
3.
Overriding is an associative operation; it has the empty function as an identity element.
4.
Then " is an associative operation with identity element ( 0, 1 ).
5.
However, mathematicians agree on a particular order of evaluation for several common non-associative operations.
6.
This expression could also have been simply evaluated right to left, because addition is an associative operation.
7.
Associative operations are abundant in mathematics; in fact, many algebraic structures ( such as categories ) explicitly require their binary operations to be associative.
8.
In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears more than once in an expression.
9.
Since this holds true when performing addition and multiplication on any real numbers, it can be said that " addition and multiplication of real numbers are associative operations ".
10.
Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics : examples include "'composition as the associative operation on arrows.
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