Quasivariety

Remarks

Example: the class of cancellative semigroups is a quasivariety. It is defined by the associative law, which is a quasi-identity with no hypotheses, and by the two quasi-identities $z x \approx z y \rightarrow x \approx y$ and $x z \approx y z \rightarrow x \approx y$.

Non-example: the class of fields, in the signature $(+, -, \cdot, 0, 1)$, is not a quasivariety. Every quasivariety contains the one-element algebra, since all terms take the same value there, so every conclusion $s \approx t$ holds. But a field needs $0 \ne 1$. Also, every quasivariety is closed under direct products, and $\Q \times \Q$ is not a field.

Each variety is a quasivariety, defined by identities, which are quasi-identities with no hypotheses. The smallest quasivariety containing a class $K$ is denoted $Q(K)$ or $\mathbf{ISPP_U}(K)$, by Mal'cev's characterization.