Equational basis

Remarks

Equivalently, the class of all algebras satisfying $\Sigma$ is the variety generated by $\mathcal{K}$. By Birkhoff's completeness theorem, clause 2 can be read syntactically: every identity of $\mathcal{K}$ is derivable from $\Sigma$ in equational logic. The whole theory $\operatorname{Id}(\mathcal{K})$ is always an equational basis of $\mathcal{K}$, so the interest lies in small bases.

Example: in the signature $(\cdot, {}^{-1}, e)$, the set $\set{x(yz) \approx (xy)z, \ x e \approx x, \ x x^{-1} \approx e}$ is an equational basis of the class of groups. Clause 1 holds because every group satisfies these laws, and clause 2 holds because every algebra satisfying them is a group.

Non-example: in the same signature, $\set{x(yz) \approx (xy)z}$ is not an equational basis of the class of groups, since clause 2 fails. Let $A = \set{0, 1}$ with $a \cdot b = 0$ for all $a, b$, with $a^{-1} = a$, and with $e = 1$. Then $A$ satisfies associativity, since all products are $0$, but $1 \cdot e = 0 \ne 1$, so $A$ does not satisfy $x e \approx x$.