algebraic fundamentally means equational

• : 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