Definition f binary operation
Let be a set and be a binary operation. We say that is associative if it satisfies the following for all:. We see that the condition feels a lot less intuitive in function notation than with the infix notation, which is why infix notation is generally preferred for describing associativity in the context of binary operations.
A set equipped with an associative binary operation is termed a semigroup. If, further, there is a neutral element identity element for the associative binary operation, the set is termed a monoid. The associative law is typically viewed as a universally quantified identity. In this context, we discuss some invariants that can be associated with the identity.
For full proof, refer: Associative implies generalized associative. When a binary operation is associative, it turns out that we can drop parenthesization from products of many elements. That is, given an expression of the form:. The result is proved by induction, with the base case following from the definition of associativity. For this reason, we always use infix operator symbols for associative binary operations, and often even drop the operator symbol, so that the expression is just written as: Also, the re-parenthesization identities i.
The associativity pentagon is a pentagon whose vertices are the five different ways of associating a product of length four, with an edge between two vertices if moving from one to the other requires a single application of the associative law. This is a cyclic pentagon. The associativity pentagon is significant because, loosely, it generates all relations between the different ways of applying the associativity law to re-parenthesize expressions.
It also helps to prove results about the set of left-associative, middle-associative, and right-associative elements.
It is also related to the associator identity. In the presence of associativity, it is possible to unambiguously define positive powers of any element. Explicitly, is the -fold product. The powers satisfy the usual laws of powers: Note that this also implies that all powers of commute with each other. Note that to define powers, we do not actually need global associativity, but only power-associativity: Suppose is a commutative unital ring and is a possibly associative, possibly non-associative algebra over.
In other words, is a -module and is a possibly associative, possibly non-associative binary operation. Note first that in order to verify the associativity of multiplication globally, it suffices to verify associativity on a generating set for the additive group of as a -module.
This is because the associator function is -linear in each input:. In the special case that is freely generated as a -module, the following test works for associativity: If form a freely generating set of as a -module, and denote the structure constants , then associativity is the following identity for all:.
For a non-associative ring with multiplication , we can define the associator as:. An element is said to be left-associative or left nuclear with respect to a binary operation if any ordered triple starting with that element associates.
The set of left-associative elements in any magma is a subsemigroup called the left nucleus. If the magma contains a neutral element, it is a submonoid. Left-associative elements of magma form submagma. An element is said to be middle-associative or middle nuclear with respect to a binary operation if any ordered triple with that element in the middle, associates. What is 2 4?
Unfortunately, you now need to check all of the other possibilities. There is, however, a shorter way If the table is symmetric with respect to this line, the table is commutative. What is the identity element for the operation? Find the single element that will always return the original value.
The identity element is 4. You will have found the identity element when all of the values in its row and its column are the same as the row and column headings. Is associative for these values? Unfortunately, if you were asked the general question, "Is associative?
Unlike the commutative property, there is NO shortcut for checking associativity when working with a table. But remember, it only takes one arrangement which does not work to show that associativity fails. The re-posting of materials in part or whole from this site to the Internet is copyright violation and is not considered "fair use" for educators.