: it’s all about stating that certain expressions are equal to other expressions
structures like groups, rings, and fields are defined entirely through equations, like associativity, commutativity, distributivity, etc
GAT extends algebraic theories but maintain the same fundamental restriction: they can only express equational axioms.
Non-example: what GAT cannot express
- existential statements: there exists x such that …
- inequalities: x < y, x != y
- logical implications that aren’t reducible to equations
- higher-order logic