PlanetMath: signature

(more info)

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signature

(Definition)

A signature $\Sigma$ is a set

$\displaystyle \Sigma:=\left(\bigcup_{n\in\omega}\mathcal{R}_n\right)\cup\left(\bigcup_{n\in\omega}\mathcal{F}_n \right)\cup\mathcal{C}$

where for each natural number

$\bullet$: $\mathcal{R}_n$ is a (usually countable) set of -ary relation symbols.
$\bullet$: $\mathcal{F}_n$ is a (usually countable) set of -ary function symbols.
$\bullet$: $\mathcal{C}$ is a (usually countable) set of constant symbols.

We require that all these sets be pairwise disjoint. Every structure is associated with a signature. 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 symbol $\varnothing$ .
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. (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 (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 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 symbol) is the unary function symbol, for each $r\in k$ .

Remark. Given a signature $\Sigma$ , the set of logical symbols from first order logic, and a countably infinite set 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.