“commutative diagram”的概念、定义、翻译、参考文献-科学参考

commutative diagram

Mathematics

A commutative diagram is an arrangement of sets and functions such that any two paths from one given set to another give equal functions. A simple example is

$\begin{matrix} A & \overset{α}{\to} & B \\ \begin{matrix} f & ↓ \end{matrix} & \begin{matrix} ↓ & g \end{matrix} \\ X & \overset{β}{\to} & Y \end{matrix}$

where the commutativity of the diagram means $β \circ f = g \circ α$ , as both these compositions arise from paths from A to Y

Computer

A method for displaying equations between functions. For example, suppose that there is a function ϕ‎ of the form
$ϕ : X \to Y$
and what is needed is to represent or code the data in X and Y, and the function, by means of the data sets A and B, respectively. Functions α‎ and β‎ are chosen where
$α : A \to X and β : B \to Y$
and a function f : A → B is defined to be a representation or function for ϕ‎ on the code sets A and B if, for all a ∈ A, the following equation holds:
$ϕ α (a) = β f (a)$
This equation is depicted by the commutative diagram shown in the figure.
Equations and commutative diagrams of this form play an important role in relating different levels of abstraction, and are used to formulate the correctness of data-type implementations, compilers, and machine architectures. As equations grow in complexity, commutative diagrams become essential. See also computable algebra.
Commutative diagram.

单词	commutative diagram
释义	commutative diagram Mathematics A commutative diagram is an arrangement of sets and functions such that any two paths from one given set to another give equal functions. A simple example is $\begin{matrix} A & \overset{α}{\to} & B \\ \begin{matrix} f & ↓ \end{matrix} & \begin{matrix} ↓ & g \end{matrix} \\ X & \overset{β}{\to} & Y \end{matrix}$ where the commutativity of the diagram means $β \circ f = g \circ α$ , as both these compositions arise from paths from A to Y Computer A method for displaying equations between functions. For example, suppose that there is a function ϕ‎ of the form $ϕ : X \to Y$ and what is needed is to represent or code the data in X and Y, and the function, by means of the data sets A and B, respectively. Functions α‎ and β‎ are chosen where $α : A \to X and β : B \to Y$ and a function f : A → B is defined to be a representation or function for ϕ‎ on the code sets A and B if, for all a ∈ A, the following equation holds: $ϕ α (a) = β f (a)$ This equation is depicted by the commutative diagram shown in the figure. Equations and commutative diagrams of this form play an important role in relating different levels of abstraction, and are used to formulate the correctness of data-type implementations, compilers, and machine architectures. As equations grow in complexity, commutative diagrams become essential. See also computable algebra. Commutative diagram.
随便看	High Sierra standard high-speed photometry high-speed steel high-spin and low-spin complexes highspread highstand highstand systems tract high symmetry high-technology high-temperature superconductivity high-temperature superconductor high tension high-tension high-velocity cloud high-velocity scanning high-velocity star high-voltage test highway highwaymen H II region H-II Transfer Vehicle Hilbert, David Hilbert, David (1862–1943) Hilbert space Hilbert-style calculus