Definition:Binary Operation/Also defined as

Binary Operation: Also defined as

Some authors specify that a binary operation $\circ$ is defined such that the codomain of $\circ$ is the same underlying set as that which forms the domain.

That is:

$\circ: S \times S \to S$

and thus gloss over the fact that a binary operation defined in such a way is closed.

Such a treatment can obscure the detail of the development of the theory of algebraic substructures; closedness is a fundamental concept in this context.

Sources

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 2$: Product sets, mappings
• 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 $1.10$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.1$. Binary operations on a set
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.5$
• 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
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Operations
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous): $\S 11$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.8$: Cartesian Product
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): algebraic operation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): algebraic operation
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers