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,+,.)$$\mathcal{L}=(1,0,+,\cdot)$ is the language of rings, then the class of fields is axiomatizable.
a) $\mathcal{K}$ is $S$-closed if, given any algebra $A$ in this class and $B$ a subalgebra of $A$, then $B$ is in $\mathcal{K}$.
c) $\mathcal{K}$ is $\prod$-closed if, given any family of algebras $A_i$, $i \in I$, belonging to $\mathcal{K}$, then the direct product $A=\prod_{i \in I} A_i$ belongs to $\mathcal{K}$.
e) $\mathcal{K}$ is $\prod_u$-closed if, given a family of algebras $A_i$, $i \in I$, and $\mathcal{F}$ an ultrafilter in $I$, then the ultraproduct of the $A_i$ belongs to $\mathcal{K}$.
A formula in the language of $\mathcal{L}$ is called an identity if it is the universal closure of a formula of type $t(x_1,\ldots,x_n)=t'(x_1,\ldots,x_n)$$t(x_1,\dotsc,x_n)=t'(x_1,\dotsc,x_n)$, where $t,t'$$t$, $t'$ are terms in $\mathcal{L}$. A formula is a quasi-identity if it is the universal closure of a formula of the form $((f_1=g_1) \wedge \ldots \wedge (f_n=g_n) \implies (f=g))$$((f_1=g_1) \land \ldots \land (f_n=g_n) \implies (f=g))$, $f's,g's$$f$'s, $g$'s terms of $\mathcal{L}$.
Theorem (BikhoffBirkhoff): 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.
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 $\mathcal{K}$ hashave?
Question 2: For what classes of $\mathcal{L}$-algebras do there existsexist free objects?