Let $\mathcal{L}$ be a first-order language without relation symbols, and let $\mathcal{K}$ be a class of $\mathcal{L}$-algebras. $\mathcal{K}$ is axiomatizable if there is a set $T$ of first-order formulas in the language $\mathcal{L}$ such that $A$ belongs to $\mathcal{K}$ if and only if $A$ is a model for $T$. As a simple example, if $\mathcal{L}=(1,0,+,\cdot)$ is the language of rings, then the class of fields is axiomatizable.
Certain closure properties of an axiomatizable class $\mathcal{K}$ of algebras reflect properties of $T$.
Among closures properties of $\mathcal{K}$, some important examples are:
a) $\mathcal{K}$ is $S$-closed if, given any algebra $A$ in this class and $B$ a subalgebra of $A$, then $B$ is in $\mathcal{K}$.
b) $\mathcal{K}$ is $H$-closed if, given any algebra $A$ in this class and $B$ an homomorphic image of $A$, then $B$ is in $\mathcal{K}$.
c) $\mathcal{K}$ is $\prod$-closed if, given any family of algebras $A_i$, $i \in I$, belonging to $\mathcal{K}$, then the direct product $A=\prod_{i \in I} A_i$ belongs to $\mathcal{K}$.
d) $\mathcal{K}$ is $\prod_f$-closed if, given a family of algebras $A_i$, $i \in I$, and $\mathcal{F}$ a filter in $I$, then the filtered product of $A_i$ belongs to $\mathcal{K}$.
e) $\mathcal{K}$ is $\prod_u$-closed if, given a family of algebras $A_i$, $i \in I$, and $\mathcal{F}$ an ultrafilter in $I$, then the ultraproduct of the $A_i$ belongs to $\mathcal{K}$.
A formula in the language of $\mathcal{L}$ is called an identity if it is the universal closure of a formula of type $t(x_1,\dotsc,x_n)=t'(x_1,\dotsc,x_n)$, where $t$, $t'$ are terms in $\mathcal{L}$. A formula is a quasi-identity if it is the universal closure of a formula of the form $((f_1=g_1) \land \ldots \land (f_n=g_n) \implies (f=g))$, $f$'s, $g$'s terms of $\mathcal{L}$.
Theorem (Birkhoff): If a non-empty class of algebras $\mathcal{K}$ is $H$-closed, $S$-closed and $\prod$-closed, then it can be axiomatized by a set of identities $T$. In this case $\mathcal{K}$ is called a variety of algebras.
Theorem (Malcev): If a non-empty class of algebras $\mathcal{K}$ is $S$-closed, $\prod_f$-closed and contains the one element algebra, then it can be axiomatized by a set $T$ of quasi-identities. In this case $\mathcal{K}$ is called a quasi-variety of algebras.
Question 1: Given a class of algebras $\mathcal{K}$, what conditions on $\mathcal{K}$ imply that it can be axiomatized by a class of $\mathcal{L}$-formulas of a special type (such as an identity or quasi-identity)? Conversely, if $\mathcal{K}$ is axiomatized by a set of formulas of a certain special type, what closure properties does $\mathcal{K}$ have?
My second question regards free objects. Let $\mathcal{K}$ be a class of algebras and $X$ a non-empty set. An algebra $F_\mathcal{K}(X)$ in $\mathcal{K}$ is a free $\mathcal{K}$-algebra in $X$ if $F_\mathcal{K}(X)$ is generated by $X$ and given any map $h: X \rightarrow A$, where $A$ is an algebra in $\mathcal{K}$, there exists a unique homomorphism $g: F_\mathcal{K}(X) \rightarrow A$ extending $h$.
For instance, free objects exists in the class of all $\mathcal{L}$-algebras: they are the term algebras.
More examples: a prevariety is a class of algebras $\mathcal{K}$ that contains the one element algebras and is $S$-closed and $\prod$-closed. If a prevariety contains at least one algebra with at least two elements, then given any set $X$, $\mathcal{K}$ contains the free object generated by $X$. The same holds, as is well known, when $\mathcal{K}$ is a variety, etc.
Question 2: For what classes of $\mathcal{L}$-algebras do there exist free objects?
Best Answer
Question 1: Given a class of algebras $\mathcal{K}$, what conditions on $\mathcal{K}$ imply that it can be axiomatized by a class of $\mathcal{L}$-formulas of a special type (such as an identity or quasi-identity)? Conversely, if $\mathcal{K}$ is axiomatized by a set of formulas of a certain special type, what closure properties $\mathcal{K}$ has?
There are many known theorems of this type, and this subject is not exhausted. (In fact, this MO post from five days ago asks about whether classes closed under a certain construction are axiomatized by sentences of a certain form.) Some well-known preservation theorems are:
Łoś–Tarski preservation theorem. A first-order theory is preserved under the formation of submodels iff the theory has a set of universal axioms.
Lyndon's positivity theorem. A first-order theory is preserved under the formation of homomorphic images iff the theory has a set of positive axioms.
Chang–Łoś–Suszko theorem. A first-order theory is preserved under unions of chains iff the theory has a set of $\forall\exists$ axioms.
Question 2: For what classes of $\mathcal{L}$-algebras there exists free objects?
I am going to assume that 'class' means 'class closed under isomorphism'.
Call a class $\mathcal C$ of $\mathcal L$-algebras an 'intermediate class' if it contains all the free algebras of the variety it generates. That is, $\mathcal C$ is an 'intermediate class' if for ${\mathcal V} = \operatorname{HSP}(\mathcal C)$ and ${\mathcal F}$ the subclass of of $\mathcal V$ consisting of $\mathcal V$-free objects, then ${\mathcal F}\subseteq {\mathcal C}\subseteq {\mathcal V}$.
Claim. $\mathcal C$ will contain free algebras of all ranks iff it is an intermediate class.
Reasoning. Assume that $\mathcal C$ is an intermediate class. The algebras of $\mathcal F$ are free in $\mathcal V$ and contained in the subclass $\mathcal C$, so they are free in the subclass ${\mathcal C}$ of $\mathcal V$. This shows that intermediate classes have free algebras of all ranks.
Conversely, if $\mathcal F$ has free algebras of all ranks for $\mathcal C$, it is not difficult to show that an algebra $F$ in $\mathcal F$ that is free in $\mathcal C$ over the set $X$ remains free over $X$ in each of the classes $\operatorname{H}({\mathcal C})$, $\operatorname{S}({\mathcal C})$, and $\operatorname{P}({\mathcal C})$. (The freeness property of $F$ over $X$ is preserved under the operators $\operatorname{H}$, $\operatorname{S}$, and $\operatorname{P}$.) Thus, the members of $\mathcal F$ are the free algebras of the variety $\operatorname{HSP}({\mathcal C})$ that is generated by $\mathcal C$. This shows that if $\mathcal C$ has free algebras of all ranks, then it is an intermediate class.