# Domain of functions interpretated in model

I read (V. Manca, Logica matematica, 2011) that

the set $T_\Sigma(V)$ of the terms on a signature $\Sigma$ and variables $V$ is inductively defined letting, for any generic variable $x$, constant $c$ and ($k$-ary) functor $f$:$$x\in T_\Sigma(V)$$$$c\in T_\Sigma(V)$$$$t_1,\ldots,t_k\in T_\Sigma(V)\Rightarrow f(t_1,\ldots,t_k)\in T_\Sigma(V).$$

Does that imply that, when an interpretation is given in a model $\mathscr{M}$ to the constants, and variables varying in the universe $D$ of interpretation, the domain of $f^\mathscr{M}$ is $D^k$ for any $k$-ary functional symbol $f\in\Sigma$, or can the domain of $f^\mathscr{M}$ be a proper subset of $D^k$?

I have found in several resources the notation $f^\mathscr{M}:D^k\to D$, but, since some authors do use that notation even if $\text{dom }f^\mathscr{M}\subsetneq D^k$, that is not of help to me... I thank you very much for any clarification!

An interpretation $\mathcal I$ for a language $\mathcal L$ must assume :

• a domain $D$

• a "distinguished" element $c^{\mathcal I} \in D$ for any constant symbol $c$ of $\mathcal L$

• a subset $R^{\mathcal I} \subseteq D^k$ for each $k$-ary relation symbol $R_k$

• a function $f^{\mathcal I} : D^n \to D$ for any $n$-ary function symbol $f_n$.

In the "standard" semantics for FOL, the function symbols must be "total", i.e. defined for any $n$-uple of elements in $D$.

• Thank you so much for the detailed answer! – Self-teaching worker Feb 3 '15 at 16:42
• If you're being so thorough, you should also mention relation symbols. – tomasz Feb 4 '15 at 3:08

The domain of the interpretation of the function symbols is always assumed to be total.