I have these teo structures, $(N, <)$ and $(Q, <)$. And I want to know if they can come from the same language?
I'm confused about the definition I have for an La-structure. Specefically about domains. And I am hung up on how much what the set $A$ is (defined below) does or does not matter.
Here's the def i'm working with:
An La-structure is a pair $(M, I)$ as follows.
- $M \neq \{\}$
$I$ is a function with domain $A$ (I'll write what $A$ is next) such that for each $I \in N$ the following conditions hold:
If $c_i $ is in $A$ the $I( c_i )$ is in $M$
If $F_i $ is in $ A $ , then $I(F_i )$ is a function $I(F_i ): M^n \to M$, where $n$ equals the arity of $F_i $
If $(P_i )$ is in $ A $, then $I(P_i )$ is a subset of $ M^n$ where $n$ is the arity of $P_i $.
Okay. Now to define $ A $, as well as La-formulas:
- A finite sequence $\beta$ is a La-formulas iff $\beta$ is a formula, and the constant, predicate, and function symbols of our language are all occuring in $ A $.
The notations $(\mathbb N,{<})$ and $(\mathbb Q,{<})$ are slightly abbreviated compared to your formal definition.
You're working in a language containing a single binary predicate $p_1^2$, so $A=\{p_1^2\}$.
The structures the question is about would, more rigidly, be written $(\mathbb N,\{p_1^2\mapsto {<_\mathbb N}\})$ and $(\mathbb Q,\{p_1^2\mapsto {<_\mathbb Q}\})$, where e.g. $\{p_1^2\mapsto {<_\mathbb N}\}$ is the function that maps the single element of $A$ to the relation $\{(x,y)\in\mathbb N\times\mathbb N\mid x<y\}$.
Thus, in your two examples, the $M$ are $\mathbb N$ and $\mathbb Q$ respectively, $A$ is $\{p_1^2\}$ in both cases, and $I$ maps $p_1^2$ to the appropriate less-than relation for $M$.
This abbreviated format is very common: When there's a particular conventional (or implied) order to list the elements of $A$, we will usually simply write $(M,I(a_1),I(a_2),\ldots,I(a_n))$ instead of $(M,I)$. This is usually more readable than using a lot of boilerplate formalism to wrap the $I$ up as a function object.