The definition of a signature I'm working with is a quadruple $\sigma = (C,F,R,\sigma')$ with $C$ serving as a set of constant symbols, $F$ serving as a set of function symbols, and $R$ serving as a set of relation symbols. Finally we have the arity function $\sigma':F \cup R \rightarrow \mathbb{N}$.
Question: What is an appropriate signature to describe a vector space?
Since a vector space can be viewed as an abelian group $V$ together with a field $\mathcal{K}$, is the following appropriate?
- $C = \{0_V, 0_K, 1_K\}$
- $F = \{+_V, +_K, \cdot_K\}$
- $R = \emptyset$
- Define $\sigma'$ to be as follows: $\sigma'(+_V) = \sigma'(+_K) = \sigma'(\cdot_K) = 2$.
No, your signature (I think the more common term is "language") is not appropriate for vector spaces, because you want to be able to multiply scalars with vectors, but scalars need not be elements of the vector space. The signature frequently used for vector spaces is:
1. $C=\{0\}$
2. $F=\{+\}\cup K$ (i.e. one unitary function symbol for each element of the scalar field)
3. $R=\emptyset$
4. $\sigma^{\prime}(+)=2, \sigma^{\prime}(x)=1$ for $x\in K$
For $x\in K, \bar v\in V$, $x(\bar v)$ will be interpreted as $x\cdot\bar v$.