signature

A signature $\Sigma$ is a set

$\bullet$

$\mathcal{R}_n$ is a (usually countable) set of $n$ -ary relation symbols.

$\bullet$

$\mathcal{F}_n$ is a (usually countable) set of $n$ -ary function symbols.

$\bullet$

$\mathcal{C}$ is a (usually countable) set of constant symbols.

We require that all these sets be pairwise disjoint.

Given a signature $\Sigma$ , a $\Sigma$ -structure is then a structure $\mathcal{A}$ , whose underlying set is some set $A$ , with elements $\mathcal{A}_c \in A$ for each constant symbol $c\in \Sigma$ , $n$ -ary operations $\mathcal{A}_f$ on $A$ for each $n$ -ary function symbol $f\in \Sigma$ , for each $n$ , and $m$ -ary relations $\mathcal{A}_R$ on $A$ for each $m$ -ary relation symbol $R\in \Sigma$ .

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 $\Sigma$ 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 $\lbrace e, ^{-1}, \cdot \rbrace$ , where
  1. $e$ (group identity symbol) is a constant symbol,
  2. $^{-1}$ (group inverse symbol) is a unary function symbol, and
  3. $\cdot$ (group multiplication symbol) is a binary function symbol.
• A signature of fields is a set $\lbrace 0,1, -, ^{-1}, +, \cdot \rbrace$ , where
  1. $0$ (additive identity symbol) and $1$ (multiplicative identity symbol) are constant symbols,
  2. $-$ (the additive inverse symbol) and $^{-1}$ (the multiplicative inverse symbol) are the unary function symbols, and
  3. $+$ (addition symbol) $\cdot$ (multiplication symbol) are binary function symbols.
• A signature of posets is a singleton $\lbrace \le \rbrace$ , where $\le$ (partial ordering symbol) is a binary relation symbol.
• A signature of vector spaces over a fixed field $k$ consists of the following
  1. $0$ (additive identity symbol) is the constant symbol,
  2. $+$ (vector addition symbol) is the binary function symbol, and
  3. $\cdot_r$ (multiplication by scalar $r$ symbol) is the unary function symbol, for each $r\in k$ .

Remark. Given a signature $\Sigma$ , 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 $\Sigma\cup L\cup V$ ). The language so-created is uniquely determined by $\Sigma$ . In the literature, it is a common practice to identify $\Sigma$ both as a signature and the unique language it generates.

Bibliography

1

W. Hodges, A Shorter Model Theory, Cambridge University Press, (1997).

2

D. Marker, Model Theory, An Introduction, Springer, (2002).