##  [Equational Theory](/equational-theory-0) 

 Definition

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

Principle

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

 

 

 

 

 





## Demonstration

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

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

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

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

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

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

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.