Definition:Multiplicative Notation
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Definition
Multiplicative notation, also called product notation, is a convention for representing a binary operation of an algebraic structure.
Let $S$ be such an algebraic structure, and let $x, y \in S$.
- $x y$ is used to indicate the result of the operation on $x$ and $y$. There is no symbol used to define the operation itself.
- $e$ or $1$ is used for the identity element.
- $x^{-1}$ is used for the inverse element.
- $x^n$ is used to indicate the $n$th power of $x$.
This notation is usual in group theory when discussing the general group.
It is also the usual notation in:
- ring theory for the ring product
- field theory for the field product.
Also see
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.1$. Binary operations on a set
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.5$. Examples of groups: $(2)$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.4$
- 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 $1.1 \ \text{(b)}$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 31$ Identity element and inverses
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Notation