Equational theory

Remarks

Every equational theory contains $t \approx t$ for every term $t$, is closed under symmetry and transitivity, under applying an operation symbol to identities, and under substitution of terms for variables. By Birkhoff's completeness theorem, these closure properties characterise equational theories. For a set $\Sigma$ of identities, the equational theory of the class of all algebras satisfying $\Sigma$ is the set of consequences of $\Sigma$.

Example: for $\mathcal{K}$ the class of all commutative semigroups, $\operatorname{Id}(\mathcal{K})$ contains $x(yz) \approx (zy)x$, since both sides equal $xyz$ in every commutative semigroup.

Non-example: the set $\set{x y \approx y x}$ is not an equational theory. Every set of the form $\operatorname{Id}(\mathcal{K})$ contains the identity $x \approx x$, which is not in this set.