Definition:Associative Operation
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Definition
Let $S$ be a set.
Let $\circ : S \times S \to S$ be a binary operation.
Then $\circ$ is associative if and only if:
- $\forall x, y, z \in S: \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$
Examples
Arbitrary Non-Associative Order 3 Structure
Consider the algebraic structure of order $3$ defined by the Cayley table:
- $\begin{array}{c|cccc} \circ & a & b & c \\ \hline a & b & c & b \\ b & b & a & c \\ c & a & c & c \\ \end{array}$
\(\ds \paren {a \circ a} \circ b\) | \(=\) | \(\ds b \circ b\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a\) | ||||||||||||
\(\ds a \circ \paren {a \circ b}\) | \(=\) | \(\ds a \circ c\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds b\) |
demonstrating non-associativity.
Also note that $a \circ b \ne b \circ a$, so $\circ$ is non-commutative as well.
Also see
Historical Note
The term associative was coined by William Hamilton in about $1844$ while thinking about octonions, which aren't.
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Chapter $\text{I}$: Semi-Groups and Groups: $1$: Definition and examples of semigroups
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.4$: Definition $1.11 \ \text{(b)}$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.2$. Commutative and associative operations
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 2$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.5$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 3$. Definition of an Integral Domain
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(a)}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 27$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Operations
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 28$. Associativity and commutativity: Definition $1$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: associative
- 1999: J.C. Rosales and P.A. García-Sánchez: Finitely Generated Commutative Monoids ... (previous) ... (next): Chapter $1$: Basic Definitions and Results
- John C. Baez: The Octonions (2002): 1 Introduction