Mal'cev's characterization of quasivarieties

Remarks

The forward direction is a check: a quasi-identity is preserved by subalgebras and by products, coordinate by coordinate, and by ultraproducts by Łoś's theorem. For the converse, let $A$ satisfy every quasi-identity that holds in $K$, and let $a \ne b$ in $A$. For each finite set $\Phi$ of equations true in $A$ among names of its elements, the quasi-identity "$\bigwedge \Phi \rightarrow a \approx b$" fails in $A$, so it fails in some $B_\Phi \in K$. An ultraproduct of the algebras $B_\Phi$ over a suitable ultrafilter gives a member of $K$ with a homomorphism from $A$ that separates $a$ and $b$. The product of these maps over all pairs $a \ne b$ embeds $A$ into a product of members of $K$, so $A \in K$.

The same argument shows that the quasivariety generated by any class $K$ is $\mathbf{ISPP_U}(K)$, the class of algebras isomorphic to subalgebras of products of ultraproducts of members of $K$. Compare Birkhoff's HSP theorem, where closure under homomorphic images is added and identities replace quasi-identities.

Closure under ultraproducts cannot be dropped. Residually finite groups are closed under the first three constructions, but not under ultraproducts, and they do not form a quasivariety.

Going deeper

For a student

The theorem lets one show that a class is a quasivariety without finding its axioms, and that a class is not a quasivariety by exhibiting a failure of closure. In the theory of finite algebras, the quasivariety generated by one finite algebra $M$ is $\mathbf{ISP}(M)$, because ultraproducts of copies of a finite algebra are isomorphic to it. This class is the starting point of the theory of natural dualities. In abstract algebraic logic, the algebraic counterparts of many deductive systems are quasivarieties, and the theorem is the tool to recognise them.