Definition:Associative

Definition

Let $\circ$ be a binary operation.

Then $\circ$ is defined as being associative on $S$ iff:

$\forall x, y, z \in S: \left({x \circ y}\right) \circ z = x \circ \left({y \circ z}\right)$

Historical Note

The term associative was coined by William Hamilton in about 1844 while thinking about octonions, which aren't.

Sources

• Walter Ledermann: Introduction to the Theory of Finite Groups (1949): $\S 1$
• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$
• W.E. Deskins: Abstract Algebra (1964): $\S 1.4$: Definition $1.11 \ \text{(b)}$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.2$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 2$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.5$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.3$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.1$: Definitions $1.1 \ \text{(a)}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 26 \mu$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 28$
• John C. Baez: The Octonions (2002): 1 Introduction