In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.
Formally, a (single-sorted) signature can be defined as a 4-tuple
\sigma=\left(S\operatorname{func
S\operatorname{func
S\operatorname{rel
+, x
\leq,\in
0,1
\operatorname{ar}:S\operatorname{func
n
n.
0
A signature with no function symbols is called a , and a signature with no relation symbols is called an .[1] A is a signature such that
S\operatorname{func
S\operatorname{rel
\sigma=\left(S\operatorname{func
|\sigma|=\left|S\operatorname{func
The is the set of all well formed sentences built from the symbols in that signature together with the symbols in the logical system.
In universal algebra the word or is often used as a synonym for "signature". In model theory, a signature
\sigma
L
L
\sigma
|L|
\aleph0
As the formal definition is inconvenient for everyday use, the definition of a specific signature is often abbreviated in an informal way, as in:
"The standard signature for abelian groups is
\sigma=(+,-,0),
-
Sometimes an algebraic signature is regarded as just a list of arities, as in:
"The similarity type for abelian groups is
\sigma=(2,1,0).
Formally this would define the function symbols of the signature as something like
f0
f1
f2
In mathematical logic, very often symbols are not allowed to be nullary, so that constant symbols must be treated separately rather than as nullary function symbols. They form a set
S\operatorname{const
S\operatorname{func
\operatorname{ar}
An example for an infinite signature uses
S\operatorname{func
S\operatorname{rel
F,
fa
a.
In the context of first-order logic, the symbols in a signature are also known as the non-logical symbols, because together with the logical symbols they form the underlying alphabet over which two formal languages are inductively defined: The set of terms over the signature and the set of (well-formed) formulas over the signature.
In a structure, an interpretation ties the function and relation symbols to mathematical objects that justify their names: The interpretation of an
n
f
A
A
fA:An\toA,
n
RA\subseteqAn.
An=A x A x … x A
n
A
f
n
R
n
For many-sorted logic and for many-sorted structures, signatures must encode information about the sorts. The most straightforward way of doing this is via that play the role of generalized arities.[3]
Let
S
x
\to.
The symbol types over
S
S\cup\{ x ,\to\}
s1 x … x sn,
s1 x … x sn\tos\prime,
n
s1,s2,\ldots,sn,s\prime\inS.
n=0,
s1 x … x sn
A (many-sorted) signature is a triple
(S,P,\operatorname{type})
S
P
\operatorname{type}
P
S.
. James C. Abbot . Trends in Lattice Theory . Princeton/NJ . Van Nostrand . 1967 . George Grätzer . George Grätzer . IV. Universal Algebra . 173 - 210. Here: p.173.