Signature (algebra)

See also: Signature

In Mathematical, a signature for an algebraic structure has on a subjacent unit S is a list of operations (with their arites) which characterizes has . The signatures are an important concept in universal Algèbre, and are also employed in Théorie of the models, Théorie of the categories and Théorie of the types.

General algebra

A signature consists of two lists, framed by $\backslash \; langle$ and $\backslash \; rangle$, from which the elements are separated by commas.

• a list starts with S followed by the symbols of the operations which characterize has . An operation F of arite N , where N is a natural entirety, is a function F : S ^N→ S . Elements distinguished from S , a neutral element or an element unit, are regarded as arite operations 0.
• the second list, composed of arites of the operations, is called the standard has . Arites are listed in the same order as the corresponding operations.

Example

a additive Groupe on $G$ has the signature $\backslash \; langle\; G,\; +,\; -,\; 0\; \backslash \; rangle$ of the type $\backslash \; langle\; 2,1,0\; \backslash \; rangle$.

Linear algebra

The signature of a quadratic Forme indicates the signs in front of the elements of its decomposition in linear squares of forms indépendantes.
That is included/understood better on an example:

Example

That is to say $q\; (X,\; there,\; Z)\; =\; 2\; there\; Z\; +\; 3\; X\; Z\; +\; X\; y$.
There is $q\; (X,\; there,\; Z)\; =\; \{\backslash \; color\; \{Blue\}\; +\}\; \backslash \; frac\; \{1\}\; \{4\}\; \backslash \; left\; (x+y+5z\; \backslash \; right)\; ^2\; \{\backslash \; color\; \{Blue\}\; -\}\; \backslash \; frac\; \{1\}\; \{4\}\; \backslash \; left\; (x-y-z\; \backslash \; right)\; ^2\; \{\backslash \; color\; \{Blue\}\; -\}\; 6\; z^2$, with $\backslash \; left\; (x+y+5z\; \backslash \; right)\; ^2$, $\backslash \; left\; (x-y-z\; \backslash \; right)\; ^2$ and $z^2$ linearly independent.
Consequently, the signature of $q$ is $\backslash \; langle\; +,\; -,\; -\; \backslash \; rangle$, also noted $\backslash \; langle\; 1,2\; \backslash \; rangle$

See too

• universal algebra

Refer

Monograph on line:

• Burris, Stanley NR., and H.P. Sankappanavar, H.P., 1981. has Race in Universal Algebra. Springer-Verlag. ISBN 3540905782. See in particular pp. 22-24.

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