A first-order theory in a fixed algebraic signature whose axioms are exclusively universally quantified equations between terms, and which is closed under the consequence relation of equational logic.

Principle

Describe algebraic structure purely by identities; closure under substitution, reflexivity, symmetry, transitivity of equality, and congruence rules yields all equational consequences.

Demonstration

The class of monoids is given by the equational theory with a binary symbol · and constant e, axioms (x·y)·z = x·(y·z), e·x = x, x·e = x; every model satisfying those identities is a monoid.

Misapplication

Treating a theory that requires existence quantifiers (for example, 'there exists an inverse for every element' without making inverse a function symbol) as equational; or adding non-equational axioms such as order relations or inequalities and still calling the theory equational.

Consequence

Its model class is a variety: closed under homomorphic images, subalgebras, and arbitrary direct products; equational theories admit free algebras and syntactic completeness with respect to equational consequence.

Reversal

A general first-order theory that uses relations, existential quantifiers, or disjunctions—such a theory is not an equational theory because its axioms cannot be rewritten purely as universally quantified equations.

Boundary

Restricted to signatures with function symbols and axioms that are universally quantified equalities; excludes axioms asserting existence, inequalities, ordering, or arbitrary relational constraints not equivalent to identities.

Semantic Tension

Between equational theories and universal Horn theories: every equational theory is a universal Horn theory, but universal Horn theories can express implications between equations that are not reducible to pure identities, creating a gap in expressive power.

Synthesis

An equational theory is the syntactic package of identities that define a variety: it is the purely equational specification whose models form a class closed under homomorphisms, subalgebras and products and that supports free objects built from syntactic terms.