equacional

Equacional is the adjective used to describe things related to equations or the use of equations to define or reason about structures. In mathematics and logic, equational reasoning treats equality as a fundamental relation from which more complex statements are derived. An equational theory is a set of equations that hold in a given class of algebraic structures; it underpins universal algebra and the study of algebraic theories (varieties) as in Birkhoff's HSP framework.

Equational logic is a logical system focused on equations, often used to specify algebraic theories. In this setting, formulas are equations between terms over a fixed signature, and theorems are those derivable using substitution, symmetry, transitivity, and congruence.

In computer science, equational reasoning and term rewriting are central to functional programming and automated theorem proving. Equational unification seeks substitutions that make two terms equal under a given equational theory; many theories include operators with properties like associativity or commutativity (AC theories).

Examples include group, ring, and lattice theories, which are defined by a set of equations that the operations must satisfy. Equational approaches favor constructive, rewrite-based proofs and algebraic specifications that can be executed or analyzed by software tools.

Overall, equacional relates to methods and theories built from equations, emphasizing equality as a guiding principle for defining structures, proving properties, and implementing specifications.