equational

Equational is an adjective relating to equations or to the study and use of equations. In mathematics and logic, equational reasoning treats equations as the primary objects of manipulation, deriving new equalities from known ones through formal rules and transformations.

An equational theory is a set of equations that hold in a given class of algebraic structures. A variety is a class of algebras defined by a collection of identities, which are universally quantified equations. Equational logic provides a formal framework in which formulas are equations between terms, and inference rules enable the deduction of consequences from a theory. Key rules include reflexivity (every term equals itself), symmetry (if s = t then t = s), transitivity (if s = t and t = u then s = u), and substitution (if s = t, then f(..., s, ...) = f(..., t, ...)).

In computer science, equational specifications describe data types and operations by equations rather than procedures. Term rewriting systems orient equations into rewrite rules to simplify expressions and reason about equality algorithmically. Unification seeks substitutions that make two terms identical under a given set of equations. Equational reasoning under rewrite rules underpins automated theorem proving, symbolic computation, and formal methods. The Knuth–Bendix completion procedure is a classic method that attempts to transform a finite set of equations into a confluent, terminating rewrite system when possible.

Examples of equational principles include commutativity (a + b = b + a), associativity (a + (b + c) = (a + b) + c), and distributivity (a·(b + c) = a·b + a·c). The term equational derives from aequatio in Latin, meaning equalization.