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 $\mathcal{L}=(1,0,+,.)$ 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 $A$, $B$ 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 $\mathcal{K}$, 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 $I$, the ultraproduct of $A_i$ 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 $t(x_1,\ldots,x_n)=t'(x_1,\ldots,x_n)$, 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 $((f_1=g_1) \wedge \ldots \wedge (f_n=g_n) \implies (f=g))$, $f's,g's$ terms of $\mathcal{L}$.

Theorem (Bikhoff): 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 $\mathcal{K}$ has?

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 there exists free objects?