Which sentences survive the passage from $X$ to the set of all functions $I \rightarrow X$?

Question

Suppose $X$ is a mathematical structure with a single underlying set which we will also denote $X$, equipped with some functions and relations. Letting $I$ denote an arbitrary non-empty set, we see that the set of all functions $I \rightarrow X$ can be made into a mathematical structure $Y$ in an obvious way. For example, if $X$ equipped with a binary operation $+$, then for all $f,g \in Y$ we define $(f+g)(i)=f(i)+g(i).$ Similarly, if $X$ is equipped with a binary relation symbol $\leq$, then for all $f,g \in Y$ we define $$f \leq g \leftrightarrow \forall i \in I(f(i) \leq g(i)).$$

Now in the passage from $X$ to $Y$, certain sentences are preserved. For example, if $+$ is commutative on $X$, well this is the sentence $$\forall x,y \in X(x+y=y+x).$$ Its easy to see that this is preserved, in the sense that $$\forall x,y \in X(x+y=y+x) \rightarrow \forall f,g \in Y(f+g=g+f).$$

In general, it seems obvious that universally quantified equalities survive the passage; but, that's not all.

Is there a nice characterization of those sentences which, if satisfied by $X$, are also satisfied by $Y$?

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Discussion. The three sentences for a partial order are all preserved, namely

1. $\forall x(x \leq x)$
2. $\forall x,y,z(x \leq y \wedge y \leq z \rightarrow x \leq z)$
3. $\forall x,y(x \leq y \wedge y \leq x \rightarrow x = y)$

However, linearity is not; and, this is of the form

• $\forall x(x \leq y \vee y \leq x).$

So, this suggests that $\vee$ is maybe "suspect." However, sometimes its okay: for example, if we have three constant symbols $0,1$ and $2$ and we know that $X$ satisfies $(0 = 1) \vee (1 = 2),$ well then so to does $Y$.

Answer 1

Let me start with some definitions:

A positive literal is an atomic formula and a negative literal is the negation of an atomic formula.

A clause is a disjunction of literals.

A Horn clause is a clause in which no more than one literal is positive.

A sentence is universal Horn is if it is the universal closure of a Horn clause.

It is not hard to see that universal Horn sentences that hold in $X$ are also satisfied in the $Y$. For example, the axioms for rings or partial orders can be written as universal Horn sentences but being a linear order or a field cannot be done since this are not preserved under products as you noticed.