signature

signature

A

signature

Σ

is a set

Σ

:=

(

⋃

n

∈

ω

ℛ

n

)

∪

(

⋃

n

∈

ω

ℱ

n

)

∪

𝒞

where for each

natural number

n

>

0

,

∙

ℛ

n

is a (usually

countable

) set of

n

-ary relation symbols.

∙

ℱ

n

is a (usually countable) set of

n

-ary function symbols.

∙

𝒞

is a (usually countable) set of constant symbols.

We require that all these sets be

pairwise disjoint

.

Given a signature

Σ

, a

Σ

-

structure

is then a structure

𝒜

, whose underlying set is some set

A

, with

elements

𝒜

c

∈

A

for each constant symbol

c

∈

Σ

,

n

-ary

operations

𝒜

f

on

A

for each

n

-ary function symbol

f

∈

Σ

, for each

n

, and

m

-ary

relations

𝒜

R

on

A

for each

m

-ary relation symbol

R

∈

Σ

.

On the other hand, every structure is associated with a signature. For every structure, it has an underlying set, together with a

collection

of “designated” objects that “define” the structure. These objects may be elements of the underlying set, operations on the set, or relations on the set. For each such “designated” object, pick a symbol for it. Make sure all symbols used are distinct from one another. Then the collection of all such symbols is a signature for the structure.

For most structures that we encounter, the set

Σ

is finite, but we allow it to be

infinite

, even

uncountable

, as this sometimes makes things easier, and just about everything still works when the signature is uncountable.

Examples

:

•

A signature of sets is the

empty set

.

•

A signature of

pointed sets

is a singleton consisting of a constant symbol.

•

A signature of groups is a set

{

e

,

-

1

,

⋅

}

, where

(a)

e

(group

identity

symbol) is a constant symbol,

(b)

-

1

(

group inverse

symbol) is a unary function symbol, and

(c)

⋅

(group

multiplication

symbol) is a binary function symbol.

•

A signature of fields is a set

{

0

,

1

,

-

,

-

1

,

+

,

⋅

}

, where

(a)

0

(

additive identity

symbol) and

1

(

multiplicative identity

symbol) are constant symbols,

(b)

-

(the

additive inverse

symbol) and

-

1

(the multiplicative inverse symbol) are the unary function symbols, and

(c)

+

(

addition

symbol)

⋅

(multiplication symbol) are binary function symbols.

•

A signature of posets is a singleton

{

≤

}

, where

≤

(partial ordering symbol) is a

binary relation

symbol.

•

A signature of

vector spaces

over a fixed field

k

consists of the following

(a)

0

(additive identity symbol) is the constant symbol,

(b)

+

(

vector addition

symbol) is the binary function symbol, and

(c)

⋅

r

(multiplication by scalar

r

symbol) is the unary function symbol, for

each

r

∈

k

.

Remark

. Given a signature

Σ

, the set

L

of

logical symbols

from

first order logic

, and a

countably infinite

set

V

of variables, we can form a

first order language

, consisting of all

formulas

built from these symbols (in

Σ

∪

L

∪

V

). The

language

so-created is uniquely determined by

Σ

. In the literature, it is a common practice to identify

Σ

both as a signature and the unique language it

generates

.

References

1

W. Hodges,

A Shorter

Model Theory

, Cambridge University Press, (1997).

2

D. Marker,

Model Theory, An

Introduction

, Springer, (2002).

Title

signature

Canonical name

Signature

Date of creation

2013-03-22 13:51:48

Last modified on

2013-03-22 13:51:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03C07

Synonym

language

Synonym

non-logical symbols

Defines

constant symbol

Defines

function symbol

Defines

relation symbol

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