Is there a concept of Free Model for any given classical first order theory?

Question

Just as in the wikipedia for 'Free object' there are free groups, monoids and other algebraically flavored structures, and given a set of axioms in the language of a signature and standard predicate logic, is there a distinguised structure built with syntactic means satisfying them, and having a nice universal property akin to that of the mentioned free objects? Where would be the details?

Answer 1

The most general first-order context in which free models exist is the universal Horn theories. These are the theories which are axiomatized by universal Horn sentences, which have the form $$\forall \overline{x}\, \left(\bigwedge_{i=1}^n \varphi_i(\overline{x})\rightarrow \psi(\overline{x})\right),$$ where all of the formulas $\varphi_i(\overline{x})$ and $\psi(\overline{x})$ are atomic.

Note that in the case $n = 0$, the left-hand side of the implication is the empty conjunction $\top$, so the sentence is equivalent to $\forall\overline{x}\,\psi(\overline{x})$, for atomic $\psi(\overline{x})$. Thus every equational algebraic theory (groups, commutative rings, etc.) is a universal Horn theory, since they are axiomatized by universally quantified equations.

To build the free model of a universal Horn theory $T$ on generators $X$, we build up a set $\Delta$ of atomic formulas in the variables $X$, starting from the empty set, and closing under the usual rules for equality, together with the closure conditions given by the sentences axiomatizing $T$.

Explicitly,

1. Put $t = t$ in $\Delta$ for every term $t$.
2. If $s = t$ is in $\Delta$, put $t = s$ in $\Delta$.
3. If $s = t$ and $t = u$ are in $\Delta$, put $s = u$ in $\Delta$.
4. If $s_i = t_i$ is in $\Delta$ for all $1\leq i\leq k$, and $R(s_1,\dots,s_k)$ is in $\Delta$ for a $k$-ary relation $R$, put $R(t_1,\dots,t_k)$ in $\Delta$.
5. For every sentence $\forall \overline{x}\, \left(\bigwedge_{i=1}^n \varphi_i(\overline{x})\rightarrow \psi(\overline{x})\right)$ in the axiomatization of $T$, if $\varphi_i(t_1,\dots,t_k)$ is in $\Delta$ for all $1\leq i\leq n$, put $\psi(t_1,\dots,t_k)$ in $\Delta$.

Having obtained the minimal closed set $\Delta$, we construct the term algebra $\mathcal{T}_X$ on $X$, take the quotient $\mathcal{T}_X/\sim$ by the equivalence relation $s\sim t\iff s=t\in \Delta$, and for each $k$-ary relation symbol $R$ and tuple of equivalence classes $[s_1],\dots,[s_k]$, set $R([s_1],\dots,[s_k]) \iff R(s_1,\dots,s_k)\in \Delta$.

Extra comments:

• The classes of structures axiomatizable by universal Horn theories are exactly those elementary classes which are closed under substructure and product.
• Even for general universal theories, you immediately run into problems building free models. For example, consider the language with two predicates $P$ and $Q$, and the universal axiom $\forall x\, (P(x)\lor Q(x))$. In the "free model" on $1$ generator $a$, if $a$ satisfies $P$, it can't be mapped to an element satisfying $Q$ but not $P$. And if $a$ satisfies $Q$, it can't be mapped to an element satisfying $P$ but not $Q$. The problem is essentially with disjunction - in the free model construction for universal Horn theories, we could satisfy all the axioms without ever having to make any choices.
• In categorical logic, the essentially algebraic theories are a slight generalization of first-order universal Horn theories in languages without relation symbols.
• In the comments, Noah mentions some other kinds of models in first-order model theory which are canonical in certain senses. I love all of these kinds of models dearly, but I wanted to give a few caveats to these notions: prime and atomic models only exist for certain theories (in a countable language, it's those in which the isolated types are dense in the types spaces over $\emptyset$), saturated models are only canonical (in the sense that they're unique up to isomorphism) in a fixed cardinality, and in general you need set theoretic hypotheses (or stability) for them to exist, and universal models are not even unique up to isomorphism.
• Most important caveat: In the category of models of a general first-order theory, you shouldn't expect to find objects satisfying universal properties in the usual sense. Universal properties assert uniqueness of arrows, and that's extremely uncommon in the elementary context. For example, a prime model of $T$ is one which embeds elementarily into every model of $T$, but this elementary embedding is typically not unique (so a prime model is not an initial object in the category of models of $T$ and elementary embeddings). The prime model of the theory of algebraically closed fields of characteristic $0$ is $\overline{\mathbb{Q}}$, which admits lots of elementary embeddings into any model of ACF$_0$ (pick any such embedding, and then precompose with any automorphism in $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Similarly, saturated models of size $\kappa$ are unique up to isomorphism, but not unique up to unique isomorphism.