A signaturePlanetmathPlanetmathPlanetmathPlanetmath Σ is a set


where for each natural numberMathworldPlanetmath n>0,

n is a (usually countableMathworldPlanetmath) set of n-ary relation symbols.

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-structureMathworldPlanetmath is then a structure 𝒜, whose underlying set is some set A, with elements 𝒜cA for each constant symbol cΣ, n-ary operationsMathworldPlanetmath 𝒜f on A for each n-ary function symbol fΣ, for each n, and m-ary relationsMathworldPlanetmathPlanetmath 𝒜R on A for each m-ary relation symbol RΣ.

On the other hand, every structure is associated with a signature. For every structure, it has an underlying set, together with a collectionMathworldPlanetmath 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 infiniteMathworldPlanetmath, even uncountable, as this sometimes makes things easier, and just about everything still works when the signature is uncountable.


  • A signature of sets is the empty setMathworldPlanetmath.

  • A signature of pointed sets is a singleton consisting of a constant symbol.

  • A signature of groups is a set {e,-1,}, where

    1. (a)

      e (group identityPlanetmathPlanetmathPlanetmath symbol) is a constant symbol,

    2. (b)

      -1 (group inverse symbol) is a unary function symbol, and

    3. (c)

      (group multiplicationPlanetmathPlanetmath symbol) is a binary function symbol.

  • A signature of fields is a set {0,1,-,-1,+,}, where

    1. (a)

      0 (additive identity symbol) and 1 (multiplicative identityPlanetmathPlanetmath symbol) are constant symbols,

    2. (b)

      - (the additive inverse symbol) and -1 (the multiplicative inverse symbol) are the unary function symbols, and

    3. (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 spacesMathworldPlanetmath over a fixed field k consists of the following

    1. (a)

      0 (additive identity symbol) is the constant symbol,

    2. (b)

      + (vector addition symbol) is the binary function symbol, and

    3. (c)

      r (multiplication by scalar r symbol) is the unary function symbol, for each rk.

Remark. Given a signature Σ, the set L of logical symbols from first order logic, and a countably infiniteMathworldPlanetmath set V of variables, we can form a first order language, consisting of all formulasMathworldPlanetmath built from these symbols (in ΣLV). The languagePlanetmathPlanetmath 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.


  • 1 W. Hodges, A Shorter Model TheoryMathworldPlanetmath, 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