Quasi-identity

Remarks

A quasi-identity with $n = 0$ is an identity $s \approx t$.

Example: the right cancellation law $x \cdot z \approx y \cdot z \rightarrow x \approx y$ is satisfied by every group. Indeed, if $a z = b z$ in a group, then multiplying on the right by $z^{-1}$ gives $a = b$.

Non-example: the monoid $(\Z, \cdot, 1)$ does not satisfy this quasi-identity. The assignment $x \mapsto 1$, $y \mapsto 2$, $z \mapsto 0$ satisfies the hypothesis $1 \cdot 0 = 2 \cdot 0$ but not the conclusion $1 = 2$.

In first-order terms, a quasi-identity stands for the universal sentence $\forall \bar x \, (s_1 = t_1 \wedge \cdots \wedge s_n = t_n \rightarrow s = t)$. Another example is $x^2 \approx 1 \rightarrow x \approx 1$, which a group satisfies exactly when it has no element of order $2$. This property cannot be expressed by identities, since the class of such groups is not closed under quotients: the additive group $\Z$ has it, and its quotient $\Z / 2\Z$ does not.